analysis of the solution of the sylvester equation using low rank adi with exact shifts abstract the solution to a general sylvester equation a x x b g f with a low rank right hand side is analyzed quantitatively through the low rank alternating directional implicit method lr adi with exact shifts new bounds and perturbation bounds on x are obtained a distinguished feature of these bounds is that they reflect the interplay between the eigenvalue decompositions of a and b and the right hand side factors g and f numerical examples suggest that because of this inclusion of details new perturbation bounds are much sharper than the existing ones

geometric structure of dimension functions of certain continuous fields abstract for a continuous field of c algebras a we give a criterion to ensure that the stable rank of a is one in the particular case of a trivial field this leads to a characterization of stable rank one completing accomplishments by nagisa osaka and phillips further for certain continuous fields of c algebras we study when the cuntz semigroup satisfies the riesz interpolation property and we also analyze the structure of its functionals as an application we obtain a positive answer to a conjecture posed by blackadar and handelman in a variety of situations

recovering the elliott invariant from the cuntz semigroup let a be a simple separable c algebra of stable rank one we prove that the cuntz semigroup of c t a is determined by its murray von neumann semigroup of projections and a certain semigroup of lower semicontinuous functions with values in the cuntz semigroup of a this result has two consequences first specializing to the case that a is simple finite separable and z stable this yields a description of the cuntz semigroup of c t a in terms of the elliott invariant of a second suitably interpreted it shows that the elliott functor and the functor de fined by the cuntz semigroup of the tensor product with the algebra of continuous functions on the circle are naturally equivalent

model order reduction via real schur form decomposition abstract a new algorithm is presented for computing moore s reduced order transfer function matrix without calculating the balancing transformation which tends to be ill conditioned especially when the original system is non minimal or when it has very nearly uncontrollable or unobservable modes the algorithm is based on finding the eigenspaces associated with large eigenvalues of the cross gramian matrix wco using the real schur form decomposition the algorithm does not require a minimal model to start with the state space realization obtained by this method is related to the balanced realization by a non singular matrix an example is presented to illustrate the proposed algorithm

computation of system balancing transformations and other applications of simultaneous diagonalization algorithms an algorithm is presented in this paper for computing state space balancing transformations directly from a state space realization the algorithm requires no squaring up or unnecessary matrix products various algorithmic aspects are discussed in detail a key feature of the algorithm is the determination of a contragredient transformation through computing the singular value decomposition of a certain product of matrices without explicitly forming the product other contragredient transformation applications are also described it is further shown that a similar approach may be taken involving the generalized singular value decomposition to the classical simultaneous diagonalization problem these svd based simultaneous diagonalization algorithms provide a computational alternative to existing methods for solving certain classes of symmetric positive definite generalized eigenvalue problems

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

the cuntz semigroup the elliott conjecture and dimension functions on c algebras we prove that the cuntz semigroup is recovered functorially from the elliott invariant for a large class of c algebras in particular our re sults apply to the largest class of simple c algebras for which k theoretic classification can be hoped for this work has three significant consequences first it provides new conceptual insight into elliott s classification program proving that the usual form of the elliott conjecture is equivalent among z stable algebras to a conjecture which is in general substantially weaker and for which there are no known counterexamples second and third it resolves for the class of algebras above two conjectures of blackadar and handelman concerning the basic structure of dimension functions on c algebras we also prove in passing that the cuntz pedersen semigroup is recovered functorially from the elliott invariant for a large class of simple unital c algebras

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

notes on submanifolds in warped products in this paper we study submanifolds in the warped product m fr where m is a riemannian manifold f is the warping function we investigate the graphs with positive gaussian curvature in m2 r the minimal graphs in m fr and the totally geodesic submanifolds in m fr

a real schur form method for modeling singularly perturbed systems a method is presented for modeling a two time scale system in the singularly perturbed form the method uses an ordered real schur decomposition which can be efficiently computed using standard subroutines from eispack three results are given first it is shown that any two time scale system can be modeled in the singularly perturbed form by a transformation into an ordered real schur form followed by balancing second under some conditions on the ordered real schur decomposition a procedure is given to achieve the modeling task with all fast variables chosen from the original state variables third necessary and sufficient conditions are given to achieve modeling by permutation of the original state variables

optimal chained aggregation for reduced order modeling in this paper a theory is proposed for a new optimal process to perform chained aggregation in reduced order modelling notions of almost invariant subspaces and approximate aggregation are introduced in the proposed theory and are shown to be useful concepts in model reduction to derive an exact aggregated model from the approximately aggregated model obtained by optimal chained aggregation an algorithm is presented which is a variant of the well known shifted qr algorithm examples are given to illustrate the theory and its superiority over a previous approach

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

mathcal c _ 0 x algebras stability and strongly self absorbing mathcal c algebras we study permanence properties of the classes of stable and so called mathcal d stable mathcal c algebras respectively more precisely we show that a mathcal c 0 x algebra a is stable if all its fibres are provided that the underlying compact metrizable space x has finite covering dimension or that the cuntz semigroup of a is almost unperforated a condition which is automatically satisfied for mathcal c algebras absorbing the jiang su algebra mathcal z tensorially furthermore we prove that if mathcal d is a k 1 injective strongly self absorbing mathcal c algebra then a absorbs mathcal d tensorially if and only if all its fibres do again provided that x is finite dimensional this latter statement generalizes results of blanchard and kirchberg we also show that the condition on the dimension of x cannot be dropped along the way we obtain a useful characterization of when a mathcal c algebra with weakly unperforated cuntz semigroup is stable which allows us to show that stability passes to extensions of mathcal z absorbing mathcal c algebras

singular value analysis of nonlinear symmetric systems in this paper we introduce the notions of state space symmetry for nonlinear systems and of the cross operator as a nontrivial natural extension of the linear symmetric case in terms of the controllability and observability operators associated to it we give a characterization of a symmetric nonlinear system in terms of the cross operator and a coordinate transformation then we analyze the use of the cross operator for solving the hankel singular value problem of the system the result is a new and simpler characterization of the solutions of this problem in terms of the cross operator and a metric

pullbacks c x algebras and their cuntz semigroup abstract in this paper we analyse the structure of the cuntz semigroup of certain c x algebras for compact spaces of low dimension that have no k 1 obstruction in their fibres in a strong sense the techniques developed yield computations of the cuntz semigroup of some surjective pullbacks of c algebras as a consequence this allows us to give a complete description in terms of semigroup valued lower semicontinuous functions of the cuntz semigroup of c x a where a is a not necessarily simple c algebra of stable rank one and vanishing k 1 for each closed two sided ideal we apply our results to study a variety of examples

model reduction via balanced state space representations a model reduction procedure based on balanced state space representations is studied in this paper the reduced order model is examined from the point of view of stability controllability and observability both continuous time and discrete time systems are considered

an algebraic approach to the radius of comparison the radius of comparison is an invariant for unital c algebras which extends the theory of covering dimension to noncommutative spaces we extend its definition to general c algebras and give an algebraic as opposed to functional theoretic reformulation this yields new permanence properties for the radius of comparison which strengthen its analogy with covering dimension for commutative spaces we then give several applications of these results new examples of c algebras with finite radius of comparison are given and the question of when the cuntz classes of finitely generated hilbert modules form a hereditary subset of the cuntz semigroup is addressed most interestingly perhaps we treat the question of when a full hereditary subalgebra b of a stable c algebra a is itself stable giving a characterization in terms of the radius of comparison we also use the radius of comparison to quantify the least n for which a c algebra d without bounded 2 quasitraces or unital quotients has the property that mn d is stable

on the structure of balanced and other principal representations of siso systems a new matrix w co which can be considered as a cross gramian matrix which contains information about both controllability and observability is defined for single input single output linear systems using this matrix the structural properties of linear systems are studied in the context of principal component analysis the matrix w co can be used in obtaining balanced and other principal representations without computation of the controllability and the observability gramians the importance of this matrix in model order reduction is highlighted

continuous fields of c algebras over finite dimensional spaces abstract let x be a finite dimensional compact metrizable space we study a technique which employs semiprojectivity as a tool to produce approximations of c x algebras by c x subalgebras with controlled complexity the following applications are given all unital separable continuous fields of c algebras over x with fibers isomorphic to a fixed cuntz algebra o n n 2 3 are locally trivial they are trivial if n 2 or n for n 3 finite such a field is trivial if and only if n 1 1 a 0 in k 0 a where a is the c algebra of continuous sections of the field we give a complete list of the kirchberg algebras d satisfying the uct and having finitely generated k theory groups for which every unital separable continuous field over x with fibers isomorphic to d is automatically locally trivial or trivial in a more general context we show that a separable unital continuous field over x with fibers isomorphic to a kk semiprojective kirchberg c algebra is trivial if and only if it satisfies a k theoretical fell type condition

recasting the elliott conjecture let a be a simple unital finite and exact c algebra which absorbs the jiang su algebra mathcal z tensorially we prove that the cuntz semigroup of a admits a complete order embedding into an ordered semigroup which is obtained from the elliott invariant in a functorial manner we conjecture that this embedding is an isomorphism and prove the conjecture in several cases in these same cases mathcal z stable algebras all we prove that the elliott conjecture in its strongest form is equivalent to a conjecture which appears much weaker outside the class of mathcal z stable c algebras this weaker conjecture has no known counterexamples and it is plausible that none exist thus we reconcile the still intact principle of elliott s classification conjecture that mathrm k theoretic invariants will classify separable and nuclear c algebras with the recent appearance of counterexamples to its strongest concrete form

one parameter continuous fields of kirchberg algebras we prove that all unital separable continuous fields of c algebras over 0 1 with fibers isomorphic to the cuntz algebra mathcal o n 2 leq n leq infty are trivial more generally we show that if a is a separable unital or stable continuous field over 0 1 of kirchberg c algebras satisfying the uct and having finitely generated k theory groups then a is isomorphic to a trivial field if and only if the associated k theory presheaf is trivial for fixed d in 0 1 we also show that under the additional assumption that the fibers have torsion free k d group and trivial k d 1 group the k d sheaf is a complete invariant for separable stable continuous fields of kirchberg algebras

stable minimal surfaces in m ℝ in 4 we developed the theory of properly embedded minimal surfaces in m r where m is a compact riemannian surface one of the first results in 4 is that a properly embedded noncompact minimal surface in m r of bounded gaussian curvature is quasiperiodic in the following sense given any sequence of vertical translates n of a subsequence of the n converges on compact subsets of m r to another properly embedded minimal surface by the curvature estimates of schoen 8 every properly embedded stable minimal surface in m r has bounded curvature therefore every properly embedded noncompact stable minimal surface in m r is quasiperiodic this quasiperiodicity property will be essential in proving the next theorem throughout this paper m denotes a compact riemannian surface

on the stable rank of algebras of operator fields over metric spaces let be a finitely generated torsion free two step nilpotent group let c denote the universal c algebra of we show that sr c sr c 1 where for a unital c algebra a sr a is the stable rank of a and where 1 is the space of one dimensional representations of in process we give a stable rank estimate for maximal full algebras of operator fields over metric spaces