efficient treatment allocation in 2 2 multicenter trials when costs and variances are heterogeneous at the design stage of a study it is crucial to compute the sample size needed for treatment effect estimation with maximum precision and power the optimal design depends on the costs which may be known at the design stage and on the outcome variances which are unknown a balanced design optimal for homogeneous costs and variances is typically used an alternative to the balanced design is a design optimal for the known and possibly heterogeneous costs and homogeneous variances called costs considering design both designs suffer from loss of efficiency compared with optimal designs for heterogeneous costs and variances for 2 2 multicenter trials we compute the relative efficiency of the balanced and the costs considering designs relative to the optimal designs we consider 2 heterogeneous costs and variance scenarios in 1 scenario 2 treatment conditions have small and 2 have large costs and variances in the other scenario 1 treatment condition has small 2 have intermediate and 1 has large costs and variances within these scenarios we examine the relative efficiency of the balanced design and of the costs considering design as a function of the extents of heterogeneity of the costs and of the variances and of their congruence congruent when the cheapest treatment has the smallest variance incongruent when the cheapest treatment has the largest variance we find that the costs considering design is generally more efficient than the balanced design and we illustrate this theory on a 2 2 multicenter trial on lifestyle improvement of patients in general practices

group inverse for the block matrices with an invertible subblock let m abcd a is square be a square block matrix with an invertible subblock over a skew field k in this paper we give the necessary and sufficient conditions for the existence as well as the expressions of the group inverse for m under some conditions

displacement rank of the drazin inverse in this paper we study the displacement rank of the drazin inverse both sylvester displacement and the generalized displacement are discussed we present upper bounds for the ranks of the displacements of the drazin inverse the general results are applied to the group inverse of a structured matrix such as close to toeplitz generalized cauchy toeplitz plus hankel and bezoutians

convergence and quotient convergence of iterative methods for solving singular linear equations with index one singular systems with index one arise in many applications such as markov chain modelling in this paper we use the group inverse to characterize the convergence and quotient convergence properties of stationary iterative schemes for solving consistent singular linear systems when the index of the coefficient matrix equals one we give necessary and sufficient conditions for the convergence of stationary iterative methods for such problems next we show that for the stationary iterative method the convergence and the quotient convergence are equivalent

the cohomology rings of moduli spaces of bundles over riemann surfaces the cohomology of the moduli space ji n d of semistable bundles of coprime rank n and degree d over a fixed riemann surface m of genus g 2 has been much studied over the last two decades and a great deal is known about it it is known that the cohomology has no torsion 2 the betti numbers have been computed using various different methods 2 5 11 and a set of generators for the cohomology ring is known 2 the purpose of this article is to provide a method of deciding when an expression in these generators is a relation and in particular when the rank n is two to prove two conjectures about the relations between the generators due to mumford and to newstead and ramanan newstead and ramanan conjectured that certain relations hold among the generators whereas mumford conjectured that a certain set of relations is a complete set in fact the conjecture of newstead and ramanan has two parts the first of which was proved by gieseker in 1982 4 the second part of their conjecture which is the part examined in this article was given an independent proof recently by thaddeus 19 who has a very different method of obtaining information about the relations in the cohomology ring the computation of the betti numbers of the moduli spaces i n d in 2 stimulated a general method of computing betti numbers of quotients in the sense of mumford s geometric invariant theory 14 of nonsingular complex projective varieties by complex reductive group actions see 8 9 10 similarly the ideas applied in this article to finding generators and relations for the cohomology rings of the moduli spaces i n d also lead to an analogous method of obtaining generators and relations for the rational cohomology rings of geometric invariant theoretic quotients the first section provides background material on the cohomology of the moduli spaces i n d in the second section mumford s conjecture is proved when the rank n is two the method could in principle be used to decide whether the conjecture is true for rank n 2 but appears to become significantly more complicated in the general case the third section gives a method for deciding when an expression in the generators is a relation and the fourth

the stability of formulae of the gohberg semencul trench type for moore penrose and group inverses of toeplitz matrices abstract we present a stability analysis of gohberg semencul trench type formulae for the moore penrose and group inverses of singular toeplitz matrices we develop a fast algorithm for the computation of the moore penrose inverse based on a gohberg semencul trench type formula and the lsqr method for the group inverse the dgmres method is used to perform the fast computation numerical tests show that the fast algorithms designed here are at least as good as the known newton iteration

comment on condition number of drazin inverse and their condition numbers of singular linear systems abstract in this paper we establish the condition number of drazin inverse of a singular matrix a where r a k r a k and k index a by the schur decomposition based on this form the spectral norm and frobenius norm of relative condition number for the drazin inverse and level 2 condition number of the drazin inverse are characterized the sensitivity for the drazin inverse solution of singular systems is also presented we obtain several formulas for the condition number of the drazin inverse by spectral norm instead of the p norm where p is a transformation matrix of the jordan canonical form of a thereby improving the earlier work y wei g wang d wang condition number of drazin inverse and their condition numbers of singular linear systems appl math comput 146 2003 455 467

a note on the group inverse of some 2 2 block matrices over skew fields abstract let k be a skew field and km n be the set of all m n matrices over k in this paper we give the necessary and sufficient conditions to the existence and the representations of the group inverse for block matrix ax yb a b 0 under some conditions where a y km n x b kn m

generalized inversion of finite rank hankel and toeplitz operators with rational matrix symbols abstract the goal of the paper is a generalized inversion of finite rank hankel operators and hankel or toeplitz operators with block matrices having finitely many rows to attain it a left coprime fractional factorization of a strictly proper rational matrix function and the bezout equation are used generalized inverses of these operators and generating functions for the inverses are explicitly constructed in terms of the fractional factorization

inequalities for numerical invariants of sets of matrices we prove three inequalities relating some invariants of sets of matrices such as the joint spectral radius one of the inequalities in which proof we use geometric invariant theory has the generalized spectral radius theorem of berger and wang as an immediate corollary

drazin inverse of partitioned matrices in terms of banachiewicz schur forms abstract let m a b c d be a partitioned matrix where a and d are square matrices denote the drazin inverse of a by a d the purpose of this paper is twofold firstly we develop conditions under which the drazin inverse of m having generalized schur complement s d ca d b group invertible can be expressed in terms of a matrix in the banachiewicz schur form and its powers secondly we deal with partitioned matrices satisfying rank m rank a d rank s d and give conditions under which the group inverse of m exists and a formula for its computation

generalized invertibility of hankel and toeplitz matrices abstract a result for the computation of the moore penrose inverse of hankel matrices over the field of the complex numbers is given by heinig and rost in this note some methods of characterizing and computing generalized inverses of hankel and toeplitz matrices over fields and over rings with the extended rao condition are presented the results were obtained by combining classical theory on the special structure of those classes of matrices with recent theory on generalized invertibility

on group inverse of singular toeplitz matrices in this paper we show that the group inverse of a real singular toeplitz matrix can be represented as the sum of products of lower and upper triangular toeplitz matrices such a matrix representation generalizes gohberg semencul formula in the literature

group inverses for a class of 2 2 block matrices over rings in this paper the sufficient and necessary conditions for the existence and the representations of the group inverses of the block matrix aab0 are given over an associative ring with unity 1 utilizing this result corresponding conclusions of the block matrix are obtained over a right ore domain under the condition rank b rank a over a right ore domain and under the condition a 2 a over an associative ring with unity 1 respectively

efficient design of cluster randomized trials with treatment dependent costs and treatment dependent unknown variances cluster randomized trials evaluate the effect of a treatment on persons nested within clusters where treatment is randomly assigned to clusters current equations for the optimal sample size at the cluster and person level assume that the outcome variances and or the study costs are known and homogeneous between treatment arms this paper presents efficient yet robust designs for cluster randomized trials with treatment dependent costs and treatment dependent unknown variances and compares these with 2 practical designs first the maximin design mmd is derived which maximizes the minimum efficiency minimizes the maximum sampling variance of the treatment effect estimator over a range of treatment to control variance ratios the mmd is then compared with the optimal design for homogeneous variances and costs balanced design and with that for homogeneous variances and treatment dependent costs cost considered design the results show that the balanced design is the mmd if the treatment to control cost ratio is the same at both design levels cluster person and within the range for the treatment to control variance ratio it still is highly efficient and better than the cost considered design if the cost ratio is within the range for the squared variance ratio outside that range the cost considered design is better and highly efficient but it is not the mmd an example shows sample size calculation for the mmd and the computer code spss and r is provided as supplementary material the mmd is recommended for trial planning if the study costs are treatment dependent and homogeneity of variances cannot be assumed

additive results for the wg drazin inverse abstract in this paper we prove the formula for the expression a b d w in terms of a b w a d w b d w assuming some conditions for a b and w here s d w denotes the generalized w weighted drazin inverse of a linear bounded operator s on a banach space

generalized inverses of hankel and toeplitz mosaic matrices abstract hankel and toeplitz mosaic matrices are block matrices with hankel or toeplitz blocks respectively it is shown that hankel and toeplitz mosaic matrices possess reflexive generalized inverses which are bezoutians furthermore the bezoutian structure of the moore penrose and group inverses is investigated

sample size formula with a continuous outcome for unequal group sizes and unequal variances guenther derived a simple formula for the accurate computation of equal sample sizes when the two sample t test with pooled variance estimate is used in the present paper the guenther formula is generalized to the case of unequal sample sizes in addition the case of unequal variances is considered

structured perturbations of group inverse and singular linear system with index one in this paper we study the normwise perturbation theory of singular linear structured system with index one the structures under investigation are toeplitz circulant hankel matrices we show that the structured condition number is smaller than unstructured condition number for specific right hand side we present the condition number for structured system

on the solution of singular linear systems of algebraic equations by semiiterative methods for a square matrixt n n where i t is possibly singular we investigate the solution of the linear fixed point problemx tx c by applying semiiterative methods sim s to the basic iterationx0 n xk tck 1 c k 1 such problems arise if one splits the coefficient matrix of a linear systemax b of algebraic equations according toa m n m nonsingular which leads tox m 1nx m 1b tx c even ifx tx c is consistent there are cases where the basic iteration fails to converge namely ift possesses eigenvalues 1 with 1 or if 1 is an eigenvalue oft with nonlinear elementary divisors in these cases and also ifx tx c is incompatible we derive necessary and sufficient conditions implying that a sim tends to a vector hat x which can be described in terms of the drazin inverse of i t we further give conditions under which hat x is a solution or a least squares solution of i t x c

some results on the generalized drazin inverse of operator matrices the generalized drazin inverse m d of a 2 2 operator matrix is considered where a x and d y are generalized drazin invertible expressions for the generalized drazin inverse m d of operator matrix m in terms of the individual blocks a b c d a d and d d are derived under some conditions

convergence properties of krylov subspace methods for singular linear systems with arbitrary index abstract krylov subspace methods have been recently considered to solve singular linear systems ax b in this paper we derive the necessary and sufficient conditions guaranteeing that a krylov subspace method converges to a vector adb px0 where ad is the drazin inverse of a and p is the projection p i a d a let k be the index of a we further show that a d b px 0 x 0 r a k 1 n a is a generalized least squares solution of ax b in r a k n a finally we present the convergence bounds for the quasi minimal residual algorithm qmr and transpose free quasi minimal residual algorithm tfqmr the index k of a in this paper can be arbitrary which extends to the main results of freund and hochbruck numer linear algebra appl 1 1994 403 420 that only considers the case k 1

representation of the drazin inverse for special block matrix abstract for a c n n the matrix x c n n is said to be the drazin inverse of a if it holds that xax x ax xa a r xa a r where r is the smallest nonnegative integer such that rank a r rank a r 1 in this paper we give the representation for the drazin inverse of the block matrix a a b 0 a b c n n a 2 a

representations for the drazin inverse of 2 2 block operator matrix with singular schur complement abstract in this paper we present formulae for the drazin inverse of 2 2 block operator matrix with generalized schur complement being drazin invertible moreover necessary and sufficient conditions for the existence as well as the expressions for the group inverse are obtained

on moore penrose inverses of toeplitz matrices abstract the moore penrose inverses of toeplitz matrices can always be represented as a sum of products of lower and upper triangular toeplitz matrices which extends the well known theorem by gohberg and semencul regarding the inverses of toeplitz matrices

convexity properties of the moment mapping re examined abstract consider a hamiltonian action of a compact lie group on a compact symplectic manifold a theorem of kirwan s says that the image of the momentum mapping intersects the positive weyl chamber in a convex polytope i present a new proof of kirwan s theorem which gives explicit information on how the vertices of the polytope come about and on how the shape of the polytope near any point can be read off from infinitesimal data on the manifold it also applies to some interesting classes of noncompact or singular hamiltonian spaces such as cotangent bundles and complex affine varieties

explicit group inverse of an innovative patterned matrix in this paper we present an innovative patterned matrix rfpl toeplitz matrix is neither the extension of toeplitz matrix nor its special case we show that the group inverse of this new patterned matrix can be represented as the sum of products of lower and upper triangular toeplitz matrices first the explicit expression of the group inverse of an rfpl toeplitz matrix is obtained second the decomposition of the group inverse is given finally an example demonstrates availability of the two methods for the group inverse

on the perturbation of the group inverse and oblique projection for given matrices a and e which are n matrices and matrices and e we discuss upper bounds for b bb b a a and bb aa aa also a lower bound for b is derived

generalized drazin inverses of anti triangular block matrices abstract in this paper we give formulae for the generalized drazin inverse m d of an anti triangular matrix m a b c 0 in two different ways one is to express m d in terms of a d with arbitrary b and c the other is to express m d in terms of b d and c d with arbitrary a moreover the results are applied to obtain generalized drazin inverses of various structured matrices and some special cases are analyzed

sample size adjustments for varying cluster sizes in cluster randomized trials with binary outcomes analyzed with second order pql mixed logistic regression adjustments of sample size formulas are given for varying cluster sizes in cluster randomized trials with a binary outcome when testing the treatment effect with mixed effects logistic regression using second order penalized quasi likelihood estimation pql starting from first order marginal quasi likelihood mql estimation of the treatment effect the asymptotic relative efficiency of unequal versus equal cluster sizes is derived a monte carlo simulation study shows this asymptotic relative efficiency to be rather accurate for realistic sample sizes when employing second order pql an approximate simpler formula is presented to estimate the efficiency loss due to varying cluster sizes when planning a trial in many cases sampling 14 per cent more clusters is sufficient to repair the efficiency loss due to varying cluster sizes since current closed form formulas for sample size calculation are based on first order mql planning a trial also requires a conversion factor to obtain the variance of the second order pql estimator in a second monte carlo study this conversion factor turned out to be 1 25 at most

displacement structure theory and applications in this survey paper we describe how strands of work that are important in two different fields matrix theory and complex function theory have come together in some work on fast computational algorithms for matrices with what we call displacement structure in particular a fast triangularization procedure can be developed for such matrices generalizing in a striking way an algorithm presented by schur 1917 j reine angew math 147 1917 pp 205 232 in a paper on checking when a power series is bounded in the unit disc this factorization algorithm has a surprisingly wide range of significant applications going far beyond numerical linear algebra we mention among others inverse scattering analytic and unconstrained rational interpolation theory digital filter design adaptive filtering and state space least squares estimation

representations for the drazin inverse of the sum p q r s and its applications abstract let p q r and s be complex square matrices and m p q r s a quadruple p q r s is called a pseudo block decomposition of m if pq qp 0 ps sq qr rp 0 and r d s d 0 where r d and s d are the drazin inverses of r and s respectively we investigate the problem of finding formulae for the drazin inverse of m the explicit representations for the drazin inverses of m and p q r are developed under some assumptions as its application some representations are presented for 2 2 block matrices a b 0 c and a b d c where the blocks a and c are square matrices several results of this paper extend the well known representation for the drazin inverse of a b 0 c given by hartwig and shoaf meyer and rose in 1977 an illustrative example is given to verify our new representations

a note on the drazin inverse of an anti triangular matrix abstract in this paper we give formulae for the generalized drazin inverse m d of an anti triangular matrix m under some conditions moreover some particular cases of these results are also considered