on centers of blocks with one simple module abstract let g be a finite group and let b be a non nilpotent block of g with respect to an algebraically closed field of characteristic 2 suppose that b has an elementary abelian defect group of order 16 and only one simple module the main result of this paper describes the algebra structure of the center of b this is motivated by a similar analysis of a certain 3 block of defect 2 in kessar 2012 15

introduction to matrix analytic methods in stochastic modeling preface part i quasi birth and death processes 1 examples part ii the method of phases 2 ph distributions 3 markovian point processes part iii the matrix geometric distribution 4 birth and death processes 5 processes under a taboo 6 homogeneous qbds 7 stability condition part iv algorithms 8 algorithms for the rate matrix 9 spectral analysis 10 finite qbds 11 first passage times part v beyond simple qbds 12 nonhomogeneous qbds 13 processes skip free in one direction 14 tree processes 15 product form networks 16 nondenumerable states bibliography index

an accelerated technique for solving one type of discrete time algebraic riccati equations abstract algebraic riccati equations are encountered in many applications of control and engineering problems e g lqg problems and h control theory in this work we study the properties of one type of discrete time algebraic riccati equations our contribution is twofold first we present sufficient conditions for the existence of a unique positive definite solution second we propose an accelerated algorithm to obtain the positive definite solution with the rate of convergence of any desired order numerical experiments strongly support that our approach performs extremely well even in the almost critical case as a byproduct we show that this method is capable of computing the unique negative definite solution once it exists

generalized reynolds ideals and derived equivalences for algebras of dihedral and semidihedral type abstract generalized reynolds ideals are ideals of the center of a symmetric algebra over a field of positive characteristic they have been shown by the second author to be invariant under derived equivalences in this paper we determine the generalized reynolds ideals of algebras of dihedral and semidihedral type as defined by erdmann in characteristic 2 in this way we solve some open problems about scalars occurring in the derived equivalence classification of these algebras

the structure preserving doubling algorithm and convergence analysis for a nonlinear matrix equation abstract in this paper applying special properties of doubling transformation a structure preserving doubling algorithm is developed for computing the positive definite solutions for a nonlinear matrix equation further by mathematical induction we establish the convergence theory of the structure preserving doubling algorithm finally we offer corresponding numerical examples to illustrate the effectiveness of the derived algorithm

newton s method for solving a class of nonlinear matrix equations in this paper a class of nonlinear matrix equations x q a h i x c 1a is considered an equivalent form of this equation is derived and the properties of the solutions are studied then newton s iterative method is applied to solve this equivalent one finally two numerical examples are given to illustrate the efficiency of newton s method

on the matrix equation x a tx 12ma i in this paper we study iterative methods for finding the extremal positive definite solutions of the matrix equation x a tx 1ma i first a condition on the existence of a positive definite solution of this matrix equation is given then the existence as well as the rate of convergence of some proposed algorithms to obtain the extremal positive definite solutions of this equation is presented moreover a generalization of computationally simple and efficient known algorithm is applied for obtaining the extremal positive definite solutions in addition both the necessary and sufficient conditions for this matrix equation to have positive definite solution are presented numerical examples are given to illustrate the performance and the effectiveness of the algorithms

generalized reynolds ideals for non symmetric algebras abstract we show how to extend the theory of generalized reynolds ideals as introduced by b kulshammer from symmetric algebras to arbitrary finite dimensional algebras over fields with positive characteristic this provides new invariants of the derived module categories of finite dimensional algebras extending recent results of the third author for symmetric algebras

some interlacing properties of the schur complement of a hermitian matrix abstract for a hermitian matrix h with nonsingular principal submatrix a it is shown that the eigenvalues of the moore penrose inverse of the schur complement h a of a in h interlace the eigenvalues of the moore penrose inverse of h moreover if h is semidefinite it is shown that the eigenvalues of h a interlace the eigenvalues of h

new convergence results on functional iteration techniques for the numerical solution of m g 1 type markov chains by performing an accurate analysis of the convergence we give a complete theoretical explanation of the experimental behaviour of functional iteration techniques for the computation of the minimal nonnegative solution g of the matrix equation x sum i 0 infty x ia i arising in the numerical solution of m g 1 type markov chains here the a i s are nonnegative k times k matrices such that the matrix sum i 0 infty a i is column stochastic moreover we introduce a general class of functional iteration methods which includes the standard methods and we give an optimality convergence result in this class

on the existence of a positive definite solution of the matrix equation x atx 1a i the question is raised under which conditions on the real square matrix a the matrix equation x atx 1a i has a real symmetric positive definite solution x both necessary and sufficient solvability conditions on a are derived moreover we give an algorithm to calculate the solution for a number of special cases we also present an analytic solution

efficient computation of the extreme solutions of x a x 1 a q and x a x 1 a q we propose a new quadratically convergent algorithm having a low computational cost per step and good numerical stability properties which allows the simultaneous approximation of the extreme solutions of the matrix equations x a x 1 a q and x a x 1 a q the algorithm is based on the cyclic reduction method

low memory and low complexity iterative schemes for a nonsymmetric algebraic riccati equation arising from transport theory abstract we reconsider newton s method and two fixed point methods for finding the minimal positive solution of a nonsymmetric algebraic riccati equation arising from transport theory we rewrite the subproblem of the newton and fixed point iterative schemes into an equivalent form with some special structure by the use of the particular structure of the subproblem we then present low memory and low complexity versions of these iterative methods with a factored alternating direction implicit iteration some properties of eigenvalues for iterative coefficient matrices in solving the subproblem are derived and the convergence of the proposed methods is established numerical experiments show that the new iterative schemes are highly efficient to obtain the minimal positive solution the proposed low memory and low complexity newton s method is particularly efficient for solving large scale riccati equation arising from transport theory

computations with infinite toeplitz matrices and polynomials we relate polynomial computations with operations involving infinite band toeplitz matrices and show applications to the numerical solution of markov chains of nonlinear matrix equations to spectral factorizations and to the solution of finite toeplitz systems in particular two matrix versions of graeffe s iteration are introduced and their convergence properties are analyzed correlations between graeffe s iteration for matrix polynomials and cyclic reduction for block toeplitz matrices are pointed out the paper contains a systematic treatment of known topics and presentation of new results improvements and extensions

structure constants related to symmetric hopf algebras abstract let k be an algebraically closed field of characteristic 0 in this paper we continue our study of structure constants for semisimple hopf algebras h whose character algebra is commutative and for non semisimple factorizable ribbon hopf algebras this is done from the point of view of symmetric algebras such as group algebras in particular we consider general fusion rules which are structure constants associated to products of irreducible characters and structure constants associated to generalizations of class sums and conjugacy classes our methods are reminiscent on one hand of the methods employed when analyzing tilting modules for certain quantum groups and on the other hand of splitting modules for certain drinfeld doubles the family of irreducible characters is divided in two according to the vanishing of their quantum dimension the fusion rules on the character algebra are computed with respect to this division

complex symmetric stabilizing solution of the matrix equation x a x 1a q abstract we study the matrix equation x a x 1 a q where a is a complex square matrix and q is complex symmetric special cases of this equation appear in green s function calculation in nano research and also in the vibration analysis of fast trains in those applications the existence of a unique complex symmetric stabilizing solution has been proved using advanced results on linear operators the stabilizing solution is the solution of practical interest in this paper we provide an elementary proof of the existence for the general matrix equation under an assumption that is satisfied for the two special applications moreover our new approach here reveals that the unique complex symmetric stabilizing solution has a positive definite imaginary part the unique stabilizing solution can be computed efficiently by the doubling algorithm

a solution of gaussian optimizer conjecture for quantum channels the long standing conjectures of the optimality of gaussian inputs and additivity are solved for a broad class of gauge covariant or contravariant bosonic gaussian channels which includes in particular thermal additive classical noise and amplifier channels restricting to the class of states with finite second moments we show that the vacuum is the input state which minimizes the entropy at the output of such channels this allows us to show also that the classical capacity of these channels under the input energy constraint is additive and is achieved by gaussian encodings

solving large scale continuous time algebraic riccati equations by doubling we consider the solution of large scale algebraic riccati equations with numerically low ranked solutions for the discrete time case the structure preserving doubling algorithm has been adapted with the iterates for a not explicitly computed but in the recursive form a k a k 1 2 d k 1 s k 1 d k 2 with d k 1 and d k 2 being low ranked and s k 1 being small in dimension for the continuous time case the algebraic riccati equation will be first treated with the cayley transform before doubling is applied with n being the dimension of the algebraic equations the resulting algorithms are of an efficient o n computational complexity per iteration without the need for any inner iterations and essentially converge quadratically some numerical results will be presented for instance in section 5 2 example 3 of dimension n 20209 with 204 million variables in the solution x was solved using matlab on a macbook pro within 45 s to a machine accuracy of o 10 1 6

structure preserving arnoldi type algorithm for solving eigenvalue problems in leaky surface wave propagation we study the generalized eigenvalue problems geps that arise from modeling leaky surface wave propagation in an acoustic resonator with an infinite amount of periodically arranged interdigital transducers the constitutive equations are discretized by finite element methods with mesh refinements along the electrode interfaces and corners the nonzero eigenvalues of the resulting gep appear in reciprocal pairs l 1 l we transform the gep into a t palindromic quadratic eigenvalue problem tpqep to reveal the important reciprocal relationships of the eigenvalues the tpqep is then solved by a structure preserving algorithm incorporating a generalized t skew hamiltonian implicitly restarted arnoldi method so that the reciprocal relationship of the eigenvalues may be automatically preserved compared with applying the arnoldi method to solve the geps our numerical results show that the eigenpairs produced by the proposed structure preserving method not only preserve the reciprocal property but also possess high efficiency and accuracy

vibration of fast trains palindromic eigenvalue problems and structure preserving doubling algorithms the vibration of fast trains is governed by a quadratic palindromic eigenvalue problem l 2a 1 email protected 0 a 1 x 0 where a 0 a email protected c n x n and a 0 t a 0 accurate and efficient solution can only be obtained using algorithms which preserve the structure of the eigenvalue problem this paper reports on the successful application of the structure preserving doubling algorithms

coherent information of one mode gaussian channels the general case of non zero added classical noise we prove that whenever the coherent information of a one mode gaussian omg channel is non zero its supremum is achieved for the infinite input power this is a well established fact for zero added classical noise whereas the non zero case has not been studied in detail the presented analysis fills the gap for three canonical classes of omg channels the lossy amplifying and additive classical noise channel class for the remaining omg channel classes the coherent information is known to vanish

computing the extremal positive definite solutions of a matrix equation an efficient and numerically stable implementation of a known algorithm is suggested for finding the extremal positive definite solutions of the matrix equation x a x 1 a i if such solutions exist the convergence rate is analyzed a new algorithm that avoids matrix inversion is presented numerical examples are given to illustrate the effectiveness of the algorithms

hermitian solutions of the equation x q nx 1n we consider the matrix equation x q nx 1n its hermitian solutions are parametrized in terms of the generalized lagrangian eigenspaces of a certain matrix pencil we show that the equation admits both a largest and a smallest solution the largest solution corresponds to the unique positive definite solution the smallest solution is the unique negative definite solution if and only if n is nonsingular if n is singular no negative definite solution exists an interesting relation between the given equation and a standard algebraic riccati equation of kalman filtering theory is also obtained finally we present an algorithm which converges to the positive definite solution for a wide range of initial conditions

the matrix equation x a tx 1 a q and its application in nano research the matrix equation x a tx 1 a q has been studied extensively when a and q are real square matrices and q is symmetric positive definite the equation has positive definite solutions under suitable conditions and in that case the solution of interest is the maximal positive definite solution the same matrix equation plays an important role in green s function calculations in nano research but the matrix q there is usually indefinite so the matrix equation has no positive definite solutions and one is interested in the case where the matrix equation has no positive definite solutions even when q is positive definite the solution of interest in this nano application is a special weakly stabilizing complex symmetric solution in this paper we show how a doubling algorithm can be used to find good approximations to the desired solution efficiently and reliably

iterative solution of two matrix equations we study iterative methods for finding the maximal hermitian positive definite solutions of the matrix equations x a x 1 a q and x a x 1 a q where q is hermitian positive definite general convergence results are given for the basic fixed point iteration for both equations newton s method and inversion free variants of the basic fixed point iteration are discussed in some detail for the first equation numerical results are reported to illustrate the convergence behaviour of various algorithms

numerical solution of nonlinear matrix equations arising from green s function calculations in nano research the green s function approach for treating quantum transport in nano devices requires the solution of nonlinear matrix equations of the form x c i hd x 1 c i hd r i hp where r and p are hermitian p ld l 1d is positive definite for all l on the unit circle and h 0 for each fixed h 0 we show that the required solution is the unique stabilizing solution x h then x lim h 0 x h is a particular weakly stabilizing solution of the matrix equation x c x 1c r in nano applications the matrices r and c are dependent on a parameter which is the system energy e in practice one is mainly interested in those values of e for which the equation x c x 1c r has no stabilizing solutions or equivalently the quadratic matrix polynomial p l l 2c lr c has eigenvalues on the unit circle we point out that a doubling algorithm can be used to compute x h efficiently even for very small values of h thus providing good approximations to x we also explain how the solution x can be computed directly using subspace methods such as the qz algorithm by determining which unimodular eigenvalues of p l should be included in the computation in some applications the matrices c d r p have very special sparsity structures we show how these special structures can be exploited to drastically reduce the complexity of the doubling algorithm for computing x h

low rank approximation to the solution of a nonsymmetric algebraic riccati equation from transport theory abstract we consider the solution of the large scale nonsymmetric algebraic riccati equation xcx xd ax b 0 from transport theory juang 1995 with m d c b a r 2 n 2 n being a nonsingular m matrix in addition a d are rank 1 updates of diagonal matrices with the products a 1 u a u d 1 v and d v computable in o n complexity for some vectors u and v and b c are rank 1 the structure preserving doubling algorithm by guo et al 2006 is adapted with the appropriate applications of the sherman morrison woodbury formula and the sparse plus low rank representations of various iterates the resulting large scale doubling algorithm has an o n computational complexity and memory requirement per iteration and converges essentially quadratically as illustrated by the numerical examples

on the tripling algorithm for large scale nonlinear matrix equations with low rank structure for the large scale nonlinear matrix equations with low rank structure the well developed doubling algorithm in low rank form da lr is known as an efficient method to compute the stabilizing solution by further analyzing the global efficiency index constructed in this paper we propose a tripling algorithm in low rank form ta lr from two points of view the cyclic reduction and the symplectic structure preservation the new presented algorithm shares the same pre processing complexity with that of da lr but can attain the prescribed normalized residual level within less iterations by only consuming some negligible iteration time as an offset under the solvability condition the proposed algorithm is demonstrated to inherit a cubic convergence and is capable of delivering errors from the current iteration to the next with the same order numerical experiments including some from nano research show that the ta lr is highly efficient to compute the stabilizing solution of large scale nonlinear matrix equations with low rank structure

positive solutions to x a bx 1 b abstract we study the positive semidefinite solutions to the matrix equation x a bx 1 b under the assumption that a 0 we show that a positive solution exists if and only if a certain block tridiagonal operator is positive in which case the solution is given by the shorted operator or generalized schur complement of that operator

necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation x a x 1a q abstract we consider the problem of when the matrix equation x a x 1 a q has a positive definite solution here q is positive definite we study both the real and the complex case this equation plays a crucial role in solving a special case of the discrete time riccati equation we present both necessary and sufficient conditions for its solvability this result is obtained by using an analytic factorization approach moreover we present algebraic recursive algorithms to compute the largest and smallest the solution of the equation respectively finally we discuss the number of solutions

on the semigroup of standard symplectic matrices and its applications abstract a matrix z r 2n 2n is said to be in the standard symplectic form if z enjoys a block lu decomposition in the sense of a 0 h i z i g 0 a t where a is nonsingular and both g and h are symmetric and positive definite in r n n such a structure arises naturally in the discrete algebraic riccati equations this note contains two results first by means of a parameter representation it is shown that the set of all 2n 2n standard symplectic matrices is closed under multiplication and thus forms a semigroup secondly block lu decompositions of powers of z can be derived in closed form which in turn can be employed recursively to induce an effective structure preserving algorithm for solving the riccati equations the computational cost of doubling and tripling of the powers is investigated it is concluded that doubling is the better strategy

on the matrix equation x atx 1a i simple necessary and sufficient conditions for the matrix equation x atx 1a i to have a positive definite solution are given these conditions are used to explain some properties of the equation and to obtain new information finally we prove a conjecture on the extremal solutions

central ideals and cartan invariants of symmetric algebras in this paper we investigate certain ideals in the center of a symmetric algebra a over an algebraically closed field of characteristic p 0 these ideals include the higman ideal and the reynolds ideal they are closely related to the p power map on a we generalize some results concerning these ideals from group algebras to symmetric algebras and we obtain some new results as well in case p 2 these ideals detect odd diagonal entries in the cartan matrix of a in a sequel to this paper we will apply our results to group algebras

methods for estimating capacities and rates of gaussian quantum channels optimization methods aimed at estimating the capacities of a general gaussian channel are developed specifically evaluation of classical capacity as maximum of the holevo information is pursued over all possible gaussian encodings for the lossy bosonic channel but extension to other capacities and other gaussian channels seems feasible solutions for both memoryless and memory channels are presented it is first dealt with single use single mode channel where the capacity dependence on channel s parameters is analyzed providing the full classification of possible cases then it is dealt with multiple uses multimode channel where the capacity dependence on the multimode environment state is analyzed when both total environment energy and environment purity are fixed this allows a fair comparison among different environments thus understanding the role of memory intermode correlations and phenomenon like superadditivity of the capacity the developed methods are also used for deriving transmission rates with heterodyne and homodyne measurements at the channel output classical capacity and transmission rates are presented within a unique framework where the rates can be treated as logarithmic approximations of the capacity

blocks of defect zero and products of elements of orderp abstract suppose thatgis a finite group and thatfis a field of characteristicp 0 which is a splitting field for all subgroups ofg lete0be the sum of the block idempotents of defect zero infg and let be the set of solutions togp 1 ing we show thate0 2 whenpis odd ande0 3 whenp 2 in the latter case 2 r whereris the set of real elements of 2 defect zero soe0 r r 2 we also show thate0 4 4 2 whenp 2 where 4is the set of solutions tog4 1 these results give us various criteria for the existence ofp blocks of defect zero

solving large scale nonlinear matrix equations by doubling we consider the solution of the large scale nonlinear matrix equation x bx 1a q 0 with a b q x cn n and in some applications b a or h the matrix q is assumed to be nonsingular and sparse with its structure allowing the solution of the corresponding linear system qv r in o n computational complexity furthermore b and a are respectively of ranks ra rb n the type 2 structure preserving doubling algorithm by lin and xu 2006 24 is adapted with the appropriate applications of the sherman morrison woodbury formula and the low rank updates of various iterates two resulting large scale doubling algorithms have an o ra rb 3 computational complexity per iteration after some pre processing of data in o n computational complexity and memory requirement and converge quadratically these are illustrated by the numerical examples

higman ideal stable hochschild homology and auslander reiten conjecture let a and b be two finite dimensional algebras over an algebraically closed field related to each other by a stable equivalence of morita type we prove that a and b have the same number of isomorphism classes of simple modules if and only if their 0 degree hochschild homology groups hh0 a and hh0 b have the same dimension the first of these two equivalent conditions is claimed by the auslander reiten conjecture for symmetric algebras we will show that the auslander reiten conjecture is equivalent to other dimension equalities involving the centers and the projective centers of a and b this motivates our detailed study of the projective center which now appears to contain the main obstruction to proving the auslander reiten conjecture for symmetric algebras as a by product we get several new invariants of stable equivalences of morita type

cartan invariants and central ideals of group algebras abstract in this paper we investigate general properties of cartan invariants of a finite group g in characteristic 2 one of our results shows that the cartan matrix of g in characteristic 2 contains an odd diagonal entry if and only if g contains a real element of 2 defect zero we also apply these results to 2 blocks of symmetric groups and to blocks with normal or abelian defect groups the second part of the paper deals with annihilators of certain ideals in centers of group algebras and blocks

characters and a verlinde type formula for symmetric hopf algebras abstract we study certain aspects of finite dimensional non semisimple symmetric hopf algebras h and their duals h we focus on the set i h of characters of projective h modules which is an ideal of the algebra of cocommutative elements of h this ideal corresponds via a symmetrizing form to the projective center higman ideal of h which turns out to be h where is an integral of h and is the left adjoint action of h on itself we describe h via primitive and central primitive idempotents of h we also show that it is stable under the quantum fourier transform our best results are obtained when h is a factorizable ribbon hopf algebra over an algebraically closed field of characteristic 0 in this case h is also the image of i h under a translated drinfel d map we use this fact to prove the existence of a steinberg like character the above ingredients are used to prove a verlinde type formula for h

the inverse mean problem of geometric mean and contraharmonic means abstract in this paper we solve the inverse mean problem of contraharmonic and geometric means of positive definite matrices proposed in w n anderson m e mays t d morley g e trapp the contraharmonic mean of hsd matrices siam j algebra disc meth 8 1987 674 682 a x y 2 x 1 y 1 1 b x y by proving its equivalence to the well known nonlinear matrix equation x t bx 1b where t 1 2 a a a 8 ba 1 b is the unique positive definite solution of x a 2bx 1b the inverse mean problem is solvable if and only if b a

structure preserving algorithms for periodic discrete time algebraic riccati equations in this paper we investigate structure preserving algorithms for computing the symmetric positive semi definite solutions to the periodic discrete time algebraic riccati equations p dares using a structure preserving swap and collapse procedure a single symplectic matrix pair in standard symplectic form is obtained the p dares can then be solved via a single dare using a structure preserving doubling algorithm we develop the structure preserving doubling algorithm from a new point of view and show its quadratic convergence under assumptions which are weaker than stabilizability and detectability with several numerical results the algorithm is shown to be efficient out performing other algorithms on a large set of benchmark problems