homotopical stable ranks for banach algebras abstract the connected stable rank and the general stable rank are homotopy invariants for banach algebras whereas the bass stable rank and the topological stable rank should be thought of as dimensional invariants this paper studies the two homotopical stable ranks viz their general properties as well as specific examples and computations the picture that emerges is that of a strong affinity between the homotopical stable ranks and a marked contrast with the dimensional ones

on the convergence of additive and multiplicative splitting iterations for systems of linear equations we study convergence conditions for the additive and the multiplicative splitting iteration methods i e two generalizations of the additive and the multiplicative schwarz iterations for hermitian and non hermitian systems of linear equations under an algebraic setting theoretical analyses show that when the coefficient and the splitting matrices are hermitian or non hermitian but diagonalizable satisfying mild conditions both additive and multiplicative splitting iteration methods are convergent even if the coefficient matrix is indefinite

logarithmic lyapunov functions for direct adaptive stabilization with normalized adaptive laws the problem of discrete time and continuous time adaptive stabilization under full state feedback control is considered in the discrete time case the main result is based on a gain update law involving a step size function the formulation generalizes and unifies prior results based on quadratic and logarithmic lyapunov functions in the continuous time case adaptive stabilization under full state feedback using a normalized gradient algorithm is considered and lyapunov stability is demonstrated

connectedness of the invertibles in certain nest algebras of order type ω we solve the connectedness problem for a class of nests of order type with finite dimensional atoms

stable ranks of split extensions of banach algebras in this paper we estimate the stable ranks of a banach algebra in terms of the stable ranks of its quotient algebra and ideal under the assumption that the quotient map splits as an application several results about the bass and the connected stable ranks of nest algebras are obtained

convergence conditions for splitting iteration methods for non hermitian linear systems abstract necessary and sufficient convergence conditions are studied for splitting iteration methods for non hermitian system of linear equations when the coefficient matrix is nonsingular when this theory is specialized to the generalized saddle point problem we obtain convergence theorem for a class of modified accelerated overrelaxation iteration methods which include the uzawa and the inexact uzawa methods as special cases moreover we apply this theory to the two stage iteration methods for non hermitian positive definite linear systems and obtain sufficient conditions for guaranteeing the convergence of these methods

on the convergence of a new splitting iterative method for non hermitian positive definite linear systems in this paper we present a new splitting method for solving a linear systems with non hermitian positive definite coefficient matrix this splitting overcomes the computation complexity of hss the spectral radius and some norm properties of the iteration matrix are discussed with the results obtained we study the reasonable choices of the parameter and introduce a preconditioner moreover an accelerated algorithm is proposed finally the numerical examples show the new method is much more efficient than the hss or the nss and the pss iteration method

block triangular and skew hermitian splitting methods for positive definite linear systems by further generalizing the concept of hermitian or normal and skew hermitian splitting for a non hermitian and positive definite matrix we introduce a new splitting called positive definite and skew hermitian splitting pss and then establish a class of pss methods similar to the hermitian or normal and skew hermitian splitting hss or nss method for iteratively solving the positive definite systems of linear equations theoretical analysis shows that the pss method converges unconditionally to the exact solution of the linear system with the upper bound of its convergence factor dependent only on the spectrum of the positive definite splitting matrix and independent of the spectrum of the skew hermitian splitting matrix as well as the eigenvectors of all matrices involved when we specialize the pss to block triangular or triangular and skew hermitian splitting btss or tss the pss method naturally leads to a btss or tss iteration method which may be more practical and efficient than the hss and nss iteration methods applications of the btss method to the linear systems of block two by two structures are discussed in detail numerical experiments further show the effectiveness of our new methods

preconditioned hermitian and skew hermitian splitting methods for non hermitian positive semidefinite linear systems for the positive semidefinite system of linear equations of a block two by two structure by making use of the hermitian skew hermitian splitting iteration technique we establish a class of preconditioned hermitian skew hermitian splitting iteration methods theoretical analysis shows that the new method converges unconditionally to the unique solution of the linear system moreover the optimal choice of the involved iteration parameter and the corresponding asymptotic convergence rate are computed exactly numerical examples further confirm the correctness of the theory and the effectiveness of the method

topological stable rank of nest algebras we establish a general result about extending a right invertible row over a banach algebra to an invertible matrix this is applied to the computation of right topological stable rank of a split exact sequence we also introduce a quantitative measure of stable rank these results are applied to compute the right left topological stable rank for all nest algebras this value is either 2 or infinity and rtsr t n 2 occurs only when n is of ordinal type less than 2 and the dimensions of the atoms grows sufficiently quickly we introduce general results on partial matrix algebras over a banach algebra this is used to obtain an inequality akin to rieffel s formula for matrix algebras over a banach algebra this is used to give further insight into the nest case

a new splitting and preconditioner for iteratively solving non hermitian positive definite systems a new iteration method for solving a linear system with coefficient matrix being non hermitian positive definite is presented in this note we study the spectral radius and contraction properties of the iteration matrix and then analyze the best possible choice of the parameter with the results obtained we show that the new method is convergent for a non hermitian positive definite linear system and propose a preconditioner to improve the condition number of the system the numerical examples show that the new method is much more efficient than the hss or pss iteration method

c algebras of real rank zero abstract the concept of real rank of a c algebra is introduced as a non commutative analogue of dimension it is shown that real rank zero is equivalent to the previously defined conditions fs and hp and that it is invariant under strong morita equivalence in particular under stable isomorphism real rank zero is also invariant under inductive limits and split extensions and the class may well be regarded as the conceptual completion of the af algebras in some cases notably when the algebra is matroid it is shown that the multiplier algebra also has real rank zero although that is not true in general by a result of g j murphy this implies a weyl von neumann type result for self adjoint multiplier elements in these cases

on the topological stable rank of non selfadjoint operator algebras we provide a negative solution to a question of m rieffel who asked if the right and left topological stable ranks of a banach algebra must always agree our example is found amongst a class of nest algebras we show that for many other nest algebras both the left and right topological stable ranks are infinite we extend this latter result to popescu s non commutative disc algebras and to free semigroup algebras as well

convergence of non stationary parallel multisplitting methods for hermitian positive definite matrices non stationary multisplitting algorithms for the solution of linear systems are studied convergence of these algorithms is analyzed when the coefficient matrix of the linear system is hermitian positive definite asynchronous versions of these algorithms are considered and their convergence investigated

dimension and stable rank in the k theory of c algebras in topological k theory which can be viewed as the algebraic side of the theory of vector bundles some of the interesting properties which one investigates are for example the conditions under which bundles must possess trivial direct summands or the extent to which the cancellation property for direct whitney sums of bundles holds such properties turn out to be controlled in part by the dimension of the base space and results describing the nature of this control are among what are frequently called stability results 17 during the past few years there has been a sudden flowering of a theory for c algebras see surveys 35 36 37 since c algebras are profitably thought of as non commutative locally compact spaces with the finitely generated projective modules being the appropriate generalization of vector bundles according to swan s theorem 40 it would be natural to look for stability results for c algebras but until now the theory has been focussed on the c groups of c algebras and there has been little discussion of stability properties presumably in part for lack of an appropriate concept of dimension for c algebras one of the main objectives of this paper is to introduce a concept of dimension for c algebras which directly generalizes the classical concept of dimension for compact spaces 16 20 22 and to develop some techniques for calculating it notably for c algebras which are obtained as crossed product algebras for an action of the group of integers in algebraic k theory which is the generalization of topological c theory to rings a substantial number of stability properties have been discovered 2 39 42 48 within that theory the concept which has played a role most analogous to that of dimension is the bass stable rank but despite its many successes the bass stable rank has often been difficult to calculate even for relatively uncomplicated rings another main objective of the present paper is to make accessible for application to c algebras some of the stability results from algebraic c theory this comes about from the fact that the concept of dimension which we introduce dominates the bass stable rank i was led to seek stability results for c algebras by my desire to determine whether the cancellation property for projective modules holds for the irrational rotation c algebras discussed in 30 which as elliott 11 has suggested can appropriately be called 2 dimensional non commutative tori in more concrete terms the question is whether two projections in such an algebra which have the same trace must be unitarily equivalent the present paper provides the general theory which is needed for the affirmative answer which i obtained in a subsequent paper i will derive the

global lyapunov stability and exponential convergence of direct adaptive control the stability properties of discrete time model reference adaptive control systems are examined lyapunov functions and strong convergence results are given for some classes of adaptive systems we show l 2 stability lyapunov stability and finite time convergence for a set of tracking error states

on strong stabilization for linear continuous time time varying systems this paper deals with the strong stabilization problem for linear time varying continuous time systems the main result is that for such systems internally stabilizable systems may not be strongly stabilizable this follows from the fact that the algebra of stable systems has infinite bass stable rank

stable rank and real rank for some classes of group c ast algebras we investigate the real and stable rank of the c algebras of locally compact groups with relatively compact conjugacy classes or finite dimensional irreducible representations estimates and formulae are given in terms of the group theoretic rank

discrete time adaptive command following and disturbance rejection with unknown exogenous dynamics we present an adaptive controller that requires limited model information for stabilization command following and disturbance rejection for mult input multi output minimum phase discrete time systems specifically the controller requires knowledge of the open loop system s relative degree as well as a bound on the first nonzero markov parameter notably the controller does not require knowledge of the command or the disturbance spectrum as long as the command and disturbance signals are generated by a lyapunov stable linear system thus the command and disturbance signals are combinations of discrete time sinusoids and steps in addition the markov parameter based adaptive controller uses feedback action only and thus does not require a direct measurement of the command or disturbance signals using a logarithmic lyapunov function we prove global asymptotic convergence for command following and disturbance rejection as well as lyapunov stability of the adaptive system when the open loop system is asymptotically stable

adaptive neural network nonparametric identifier with normalized learning laws this paper addresses the design of a normalized convergent learning law for neural networks nns with continuous dynamics the nn is used here to obtain a nonparametric model for uncertain systems described by a set of ordinary differential equations the source of uncertainties is the presence of some external perturbations and poor knowledge of the nonlinear function describing the system dynamics a new adaptive algorithm based on normalized algorithms was used to adjust the weights of the nn the adaptive algorithm was derived by means of a nonstandard logarithmic lyapunov function llf two identifiers were designed using two variations of llfs leading to a normalized learning law for the first identifier and a variable gain normalized learning law in the case of the second identifier the inclusion of normalized learning laws yields to reduce the size of the convergence region obtained as solution of the practical stability analysis on the other hand the velocity of convergence for the learning laws depends on the norm of errors in inverse form this fact avoids the peaking transient behavior in the time evolution of weights that accelerates the convergence of identification error a numerical example demonstrates the improvements achieved by the algorithm introduced in this paper compared with classical schemes with no normalized continuous learning methods a comparison of the identification performance achieved by the no normalized identifier and the ones developed in this paper shows the benefits of the learning law proposed in this paper

convergence theorems for parallel multisplitting two stage iterative methods for mildly nonlinear systems abstract for the system of mildly nonlinear equations ax f x where a r n n is an n by n sparse real matrix and f r n r n a general nonlinear mapping both local and global convergence properties of the multisplitting two stage iterative method numer algorithms 15 1997 347 are further studied in depth when a r n n is symmetric positive definite and its multiple splittings are symmetric p regular

a unified framework for the construction of various matrix multisplitting iterative methods for large sparse system of linear equations abstract a unified framework for the construction of various synchronous and asynchronous parallel matrix multisplitting iterative methods suitable to the simd and mimd multiprocessor systems respectively is presented and its convergence theory is established under rather weak conditions these afford general method models and systematical convergence criterions for studying the parallel iterations in the sense of matrix multisplitting in addition how the known parallel matrix multisplitting iterative methods can be classified into this new framework and what novel ones can be generated by it are shown in detail

an efficient method for constructing an ilu preconditioner for solving large sparse nonsymmetric linear systems by the gmres method abstract the main idea of this paper is in determination of the pattern of nonzero elements of the lu factors of a given matrix a the idea is based on taking the powers of the boolean matrix derived from a this powers of a boolean matrix strategy pbs is an efficient effective and inexpensive approach construction of an ilu preconditioner using pbs is described and used in solving large nonsymmetric sparse linear systems effectiveness of the proposed ilu preconditioner in solving large nonsymmetric sparse linear systems by the gmres method is also shown numerical experiments are performed which show that it is possible to considerably reduce the number of gmres iterations when the ilu preconditioner constructed here is used in numerical examples the influence of k the dimension of the krylov subspace on the performance of the gmres method using an ilu preconditioner is tested for all the tests carried out the best value for k is found to be 10

sufficient conditions for the convergent splittings of non hermitian positive definite matrices abstract we present sufficient conditions for the convergent splitting of a non hermitian positive definite matrix these results are applicable to identify the convergence of iterative methods for solving large sparse system of linear equations

product type skew hermitian triangular splitting iteration methods for strongly non hermitian positive definite linear systems by further generalizing the modified skew hermitian triangular splitting iteration methods studied in l wang z z bai skew hermitian triangular splitting iteration methods for non hermitian positive definite linear systems of strong skew hermitian parts bit numer math 44 2004 363 386 in this paper we present a new iteration scheme called the product type skew hermitian triangular splitting iteration method for solving the strongly non hermitian systems of linear equations with positive definite coefficient matrices we discuss the convergence property and the optimal parameters of this method moreover when it is applied to precondition the krylov subspace methods the preconditioning property of the product type skew hermitian triangular splitting iteration is analyzed in detail numerical results show that the product type skew hermitian triangular splitting iteration method can produce high quality preconditioners for the krylov subspace methods for solving large sparse positive definite systems of linear equations of strong skew hermitian parts