analytical bounds for block approximate factorization methods abstract we derive simple analytical upper bounds on the spectral condition number associated with the preconditioning of large sparse linear systems by the so called modified block incomplete factorizations the results prove to handle a large class of discrete second order elliptic partial differential equations and support the conclusion that sparse approximate block matrix factorizations provide powerful preconditioners

conditioning analysis of positive definite matrices by approximate factorizations abstract the conditioning analysis of positive definite matrices by approximate lu factorizations is usually reduced to that of stieltjes matrices or even to more specific classes of matrices by means of perturbation arguments like spectral equivalence we show in the present work that a wider class which we call almost stieltjes matrices can be used as reference class and that it has decisive advantages for the conditioning analysis of finite element approximations of large multidimensional steady state diffusion problems

on sparse block factorization iterative methods a synthetic formalism is proposed for the description of sparse block factorization iterative methods it is used to develop existence convergence and comparison theorems and to compare block factorization schemes against point factorization schemes

exact model matching of linear systems using generalized sampled data hold functions a new technique is presented for the solution of the exact model matching problem of linear time invariant systems using generalized sampled data hold functions this technique reduces the problem to that of solving a nonhomogeneous algebraic system of equations on the basis of this system of equations the necessary and sufficient conditions are established and the expressions of the controller matrices are derived

uniform exact model matching for a class of linear time varying analytic systems abstract the problem of uniform exact model matching is studied via state and output feedback for a class of linear time varying analytic systems the following two major issues are resolved the nessecary and sufficient conditions for the problems to have a solution and the general analytical expressions for the controller matrices a major feature of the proposed approach is that it reduces the uniform exact model matching problem to that of solving a linear nonhomogeneous algebraic matrix equation

a general incomplete block matrix factorization method abstract a general method for incomplete factorization of m matrices in block matrix form is presented the method generalizes previous methods for pointwise incomplete factorization methods of varga 1960 of wo nicki 1973 and of meijerink and van der vorst 1977 as well as a method for tridiagonal block matrices also we present a new and more general result on the existence of so called modified incomplete factorizations which satisfy a generalized rowsum criterion

polynomial acceleration of iterative schemes associated with subproper splittings a subproper splitting of a matrix a is a decomposition a b c such that the kernel of a includes that of b while the range of b includes that of a our purpose in the present work is to extend the convergence analysis of polynomial acceleration to the case of iterative schemes associated with subproper splittings in the case of hermitian matrices and consistent systems briefly stated our conclusions show that the regular theory extends to the subproper case provided that convergence to the solution of ax b is understood as convergence to a solution of ax b while b 1 a is understood as b a 0 where b is the moore penrose inverse of b

comparison of eigenvalue ratios in artificial boundary perturbation and jacobi preconditioning for solving poisson equation abstract the shortley weller method is a standard finite difference method for solving the poisson equation with dirichlet boundary condition unless the domain is rectangular the method meets an inevitable problem that some of the neighboring nodes may be outside the domain in this case an usual treatment is to extrapolate the function values at outside nodes by quadratic polynomial the extrapolation may become unstable in the sense that some of the extrapolation coefficients increase rapidly when the grid nodes are getting closer to the boundary a practical remedy which we call artificial perturbation is to treat grid nodes very near the boundary as boundary points the aim of this paper is to reveal the adverse effects of the artificial perturbation on solving the linear system and the convergence of the solution we show that the matrix is nearly symmetric so that the ratio of its minimum and maximum eigenvalues is an important factor in solving the linear system our analysis shows that the artificial perturbation results in a small enhancement of the eigenvalue ratio from o 1 h h m i n to o h 3 and triggers an oscillatory order of convergence instead we suggest using jacobi or ilu type preconditioner on the matrix without applying the artificial perturbation according to our analysis the preconditioning not only reduces the eigenvalue ratio from o 1 h h m i n to o h 2 but also keeps the sharp second order convergence

on the stable exact model matching problem abstract the stable exact model matching problem semmp is investigated we state and prove a number of equivalent necessary and sufficient conditions for the existence of proper solutions to the exact model matching problem that are also stable i e have no poles inside a symmetric forbidden subset of the finite complex plane c these results can be viewed as the counterpart of the results in 3 and 9 for the case of the ring of proper and stable rational functions

dynamically relaxed block incomplete factorizations for solving two and three dimensional problems to efficiently solve second order discrete elliptic pdes by krylov subspace like methods one needs to use some robust preconditioning techniques relaxed incomplete factorizations rilu are powerful candidates unfortunately their efficiency critically depends on the choice of the relaxation parameter omega whose optimal value is not only hard to estimate but also strongly varies from one problem to another these methods interpolate between the popular ilu and its modified variant milu concerning the pointwise schemes a new variant of rilu that dynamically computes variable omega omega i has been introduced recently like its ancestor rilu and unlike standard methods it is robust with respect to both existence and performance on top of that it breaks the problem dependence of omega opt a block version of this dynamically relaxed method is proposed and compared with classical pointwise and blockwise methods as well as with some existing dynamic variants showing that with the new blockwise preconditioning technique anisotropies are handled more effectively

auslander gorenstein algebras and precluster tilting abstract we generalize the notions of n cluster tilting subcategories and selfinjective algebras into n precluster tilting subcategories and n selfinjective algebras where we show that a subcategory naturally associated to n precluster tilting subcategories has a higher auslander reiten theory furthermore we give a bijection between n precluster tilting subcategories and n minimal auslander gorenstein algebras which is a higher dimensional analog of auslander solberg correspondence 8 as well as a gorenstein analog of n auslander correspondence 22 the auslander reiten theory associated to an n precluster tilting subcategory is used to classify the n minimal auslander gorenstein algebras into four disjoint classes our method is based on relative homological algebra due to auslander solberg

on the conditioning analysis of block approximate factorization methods abstract the paper is devoted to the conditioning analysis of modified block incomplete factorizations of a given stieltjes matrix we obtain new results improve other theories and compare all existing upper bounds through numerical experiments applied to discrete elliptic pdes our results show that an o h 1 spectral bound can be achieved for a large class of problems

taking advantage of the potentialities of dynamically modified block incomplete factorizations recently modified block incomplete factorizations with dynamic diagonal perturbations have been introduced as preconditioning techniques to solve large linear systems and were successfully tested on isotropic and moderately anisotropic two dimensional partial differential equations pdes an analytic study is performed on the basis of improved versions of results published elsewhere displaying why dynamic methods should be preferred to standard block methods in particular the empirical observation that the optimal choice of the involved parameter does not significantly vary from one problem to another is theoretically confirmed it is also explained why in the case of strong anisotropy lack of attention when adding diagonal perturbations may result in very poor performance rules to surmount this inconvenience are discussed and tested on discretized two dimensional pdes

almost split sequences in subcategories let r be a commutative artin ring and let i be an r algebra which is a finitely generated r module in 5 auslander and reiten introduced the notion of an almost split sequence in mod a the category of finitely generated modules over a and the existence and uniqueness of such short exact sequences were established there the theory of almost split sequences developed further and similar results for certain subcategories of mod i were proved by bautista and martinez 8 and roggenkamp 9 the main purpose of this paper is to develop a more general theory for subcategories of mod i having almost split sequences which has the previous examples as special cases notions closely related to that of almost split sequences are those of minimal left and right almost split morphisms these notions were introduced in 61 for mod i and further developed in 7 for subcategories of mod ii this paper is based upon 7 and we therefore use the same notations and conventions as used there we now give some definitions and then proceed to describe the content of the paper section by section let c be a subcategory of mod ii then a morphism g b c in c is said to be a right almost split morphism in c if i g is not a splittable epimorphism and ii whenever there is a nonsplittable epimorphism h c c in c there exists an h c b such that gh h dually a morphismfi a f b in c is said to be left almost split if i f is not a splittable monomorphism and ii whenever there is a nonsplittable monomorphism h a a there exists an h b a such that h f h we say that c has right almost split morphisms if for all indecomposable objects

model matching and decoupling for continuous and discrete time linear time varying systems solutions to the exact model matching and block decoupling problems for both continuous and discrete time linear time varying systems are presented the parametrisation of the whole class of proper solutions is given for the decoupling the minimal delay problem is also considered in a time varying setting the approach is algebraic and based on the smith macmillan form at infinity of a transfer matrix of a time varying system which has been recently introduced in systems theory this avoids the difficulties related to the inversion of the transfer matrices with entries in non commutative fields over which the determinants of dieudonne or ore type are much more complicated the solutions presented here involve only standard matrix computations excluding direct matrix inversions and are thus easy to implement in practice examples are treated in detail to illustrate the theoretical results and the way in which the computations are done and a physical example is also shown

analyses on the finite difference method by gibou et al for poisson equation gibou et al in 4 introduced a finite difference method for solving the poisson equation in irregular domains with the dirichlet boundary condition contrary to its great importance its properties have not been mathematically analyzed but have just been numerically observed in this article we present two analyses for the method one proves that its solution is second order accurate and the other estimates the condition number of its linear system according to our estimation the condition number of the unpreconditioned linear system is of size o 1 h h min and each of jacobi sgs and ilu preconditioned systems is of size o h 2 furthermore our analysis shows that the condition number of milu is of size o h 1 the most successful one

conditioning analysis of modified block incomplete factorizations abstract we improve the conditioning analysis of modified block incomplete factorizations of stieltjes matrices letting n denote the number of diagonal blocks our results show that the spectral condition number is bounded by n for a large class of two dimensional pdes

model reference adaptive and nonadaptive control of linear time varying plants we address the adaptive and nonadaptive model reference control problem for a class of multivariable linear time varying plants namely index invariant ones we show that under appropriate controllability and observability conditions this class of plants admits a fractional description in terms of polynomial differential operators and as such allows for a polynomial equation based controller design we also show that for a model reference control objective the controller can be designed by solving a set of algebraic equations further when the plant parameters are only partially known we employ a gradient based adaptive law with projection and normalization to update the controller parameters and establish the stability and tracking properties of adaptive closed loop plant finally we present a simple example to illustrate the design and realization of both the adaptive and nonadaptive control laws

on approximate factorization methods for block matrices suitable for vector and parallel processors abstract some existence results for methods based on the approximate factorization of block matrices are proven these methods are based on recursive computations of diagonal block matrices and the approximation of their inverses to preserve sparsity we also discuss a recently proposed 1 inverse free factorization method and present some numerical tests for it

auslander reiten sequences and t structures on the homotopy category of an abelian category we use t structures on the homotopy category kb r mod for an artin algebra r and watts representability theorem to give an existence proof for auslander reiten sequences of r modules this framework naturally leads to a notion of generalized or higher auslander reiten sequences

higher auslander reiten sequences and t structures abstract let r be an artin algebra and c an additive subcategory of mod r we construct a t structure on the homotopy category k c and argue that its heart h c is a natural domain for higher auslander reiten ar theory in the paper 5 we showed that k mod r is a natural domain for classical ar theory here we show that the abelian categories h mod r and h c interact via various functors if c is functorially finite then h c is a quotient category of h mod r we illustrate our theory with two examples when c is a maximal n orthogonal subcategory iyama developed a higher ar theory see 10 in this case we show that the simple objects of h c correspond to iyama s higher ar sequences and derive his higher ar duality from the existence of a serre functor on the derived category d b h c the category o of a complex semi simple lie algebra fits into higher ar theory in the situation when r is the coinvariant algebra of the weyl group

the hodge theory of soergel bimodules we prove soergel s conjecture on the characters of indecomposable soergel bimodules we deduce that kazhdan lusztig polynomials have positive coecients for arbitrary coxeter systems using results of soergel one may deduce an algebraic proof of the kazhdan lusztig conjecture

conditioning analysis of block incomplete factorizations and its application to elliptic equations the paper deals with eigenvalue estimates for block incomplete fac torization methods for symmetric matrices first some previous results on upper bounds for the maximum eigenvalue of preconditioned matrices are generalized to each eigenvalue second upper bounds for the maximum eigenvalue of the preconditioned matrix are further estimated which presents a substantial im provement of earlier results finally the results are used to estimate bounds for every eigenvalue of the preconditioned matrices in particular for the maximum eigenvalue when a modified block incomplete factorization is used to solve an elliptic equation with variable coefficients in two dimensions the analysis yields a new upper bound of type h 1 for the condition number of the preconditioned matrix and shows clearly how the coefficients of the differential equation influ ence the positive constant

existence and conditioning properties of sparse approximate block factorizations a particular class of sparse approximate block lu factorizations is investigated sufficient conditions are determined for the existence of b lp 1 u for a given coefficient matrix a bounds on the spectral condition number of b 1 a are obtained for symmetric b and a when applied to discrete multidimensional elliptic partial differential equations these results lead under reasonable assumptions to o h 1 spectral condition numbers

on the eigenvalue distribution of a class of preconditioning methods a class of preconditioning methods depending on a relaxation parameter is presented for the solution of large linear systems of equationax b wherea is a symmetric positive definite matrix the methods are based on an incomplete factorization of the matrixa and include both pointwise and blockwise factorization we study the dependence of the rate of convergence of the preconditioned conjugate gradient method on the distribution of eigenvalues ofc 1a wherec is the preconditioning matrix we also show graphic representations of the eigenvalues and present numerical tests of the methods

lower eigenvalue bounds for pencils of matrices abstract eigenvalue bounds are obtained for pencils of matrices a vb where a is a stieltjes matrix and b is positive definite under assumptions suitable for the estimation of asymptotic convergence rates of factorization iterative methods where b represents the approximate factorization of a the upper bounds obtained depend on the connectivity structure of the matrices involved which enters through matrix graph considerations in addition a more classical argument is used to obtain a lower bound potential applications of these results include a partial confirmation of gustafsson s conjecture concerning the nonnecessity of axelsson s perturbations

modified block approximate factorization strategies two variants of modified incomplete block matrix factorization with additive correction are proposed for the iterative solution of large linear systems of equations both rigorous theoretical support and numerical evidence are given displaying their efficiency when applied to discrete second order partial differential equations pdes even in the case of quasi singular problems

higher dimensional auslander reiten theory on maximal orthogonal subcategories abstract we introduce the concept of maximal n 1 orthogonal subcategories over artin algebras and orders and develop n 1 dimensional auslander reiten theory on them we give the n auslander reiten translation and the n auslander reiten duality then show the existence of n almost split sequences and n fundamental sequences we give some examples

derived counterparts of fusion categories of quantum groups abstract in this paper we study derived versions of the fusion category associated to lusztig s quantum group u q the categories that arise in this way are not semisimple but recovers the usual fusion ring when passing to grothendieck rings on the derived level it turns out that it is possible to define fusion for u q without using the notion of tilting modules hence we arrive at a definition of the fusion ring that makes sense in any spherical category we apply this new definition to the small quantum group and we relate it with some rings appearing in 22

d representation finite self injective algebras abstract in this paper we initiate the study of higher dimensional auslander reiten theory of self injective algebras we give a systematic construction of weakly d representation finite self injective algebras as orbit algebras of the repetitive categories of algebras of finite global dimension satisfying a certain finiteness condition for the serre functor the condition holds in particular for all fractionally calabi yau algebras of global dimension at most d this generalizes riedtmann s classical construction of representation finite self injective algebras our method is based on an adaptation of gabriel s covering theory for k linear categories to the setting of higher dimensional auslander reiten theory applications include n fold trivial extensions and classical and higher preprojective algebras which are shown to be d representation finite in many cases we also get a complete classification of all d representation finite self injective nakayama algebras for arbitrary d

a survey of some estimates of eigenvalues and condition numbers for certain preconditioned matrices eigenvalue and condition number estimates for preconditioned iteration matrices provide the information required to estimate the rate of convergence of iterative methods such as preconditioned conjugate gradient methods in recent years various estimates have been derived for perturbed modified block incomplete factorizations we survey and extend some of these and derive new estimates in particular we derive upper and lower estimates of individual eigenvalues and of condition number this includes a discussion that the condition number of preconditioned second order elliptic difference matrices is o h 1 some of the methods are applied to compute certain parameters involved in the computation of the preconditioner

linear time varying system control based on the inversion transformation this paper proposes a new state feedback control design for uniformly controllable linear time varying systems the new design is based on a nonlinear state transformation that converts the stabilization problem of the system state into a destabilization problem of the transformed state such a transformation results in a new control that has two advantages over previous controls 1 in some of the previous control designs forecast of future information of the time varying system matrices over some time interval t t is required to calculate the state feedback gain at any time instant t while in the new control design no such forecast is needed 2 in other previous control designs consecutive time differentiation on the time varying system matrices is required while in the new control design time integration instead of differentiation is applied to the system matrices the new control can therefore be applied to systems with time varying parameters that are piecewise continuous in time