on the perturbation of the reduced minimum modulus of bounded linear operators let x and y be two banach spaces and let t x y be a bounded linear operator we study the perturbation problem of the reduced minimum modulus of t and in particular we correct a mistake in a recent paper

perturbation bounds for least squares problems with equality constraints we present some new results on the perturbation analysis for least squares problems with equality constraints based on an equivalent formation of the problem and the optimal perturbation result for general least squares problems

on a generalised connes hochschild kostant rosenberg theorem the central result of this paper is an explicit computation of the hochschild and cyclic homologies of a natural smooth subalgebra of stable continuous trace algebras having smooth manifolds x as their spectrum more precisely the hochschild homology is identified with the space of differential forms on x and the periodic cyclic homology with the twisted de rham cohomology of x thereby generalising some fundamental results of connes and hochschild kostant rosenberg the connes chern character is also identified here with the twisted chern character

rank revealing qr factorizations abstract an algorithm is presented for computing a column permutation pi and a qr factorization a pi qr of an m by n m or n matrix a such that a possible rank deficiency of a will be revealed in the triangular factor r having a small lower right block for low rank deficient matrices the algorithm is guaranteed to reveal the rank of a and the cost is only slightly more than the cost of one regular qr factorization a posteriori upper and lower bounds on the singular values of a are derived and can be used to infer the numerical rank of a keywords qr factorization rank deficient matrices least squares computation subset selection rank singular values

bi relative algebraic k theory and topological cyclic homology it is well known that algebraic k theory preserves products of rings however in general algebraic k theory does not preserve fiber products of rings and one defines bi relative algebraic k theory so as to measure the deviation it was proved recently by cortinas 6 that rationally bi relative algebraic k theory and bi relative negative cyclic homology agree in this paper we show that with finite coefficients bi relative algebraic k theory and bi relative topological cyclic homology agree as an application we show that for a possibly singular curve over a perfect field k of positive characteristic p the cyclotomic trace map induces an isomorphism of the p adic algebraic k groups and the p adic topological cyclic homology groups in non negative degrees as a further application we show that the difference between the p adic k groups of the integral group ring of a finite group and the p adic k groups of a maximal z order in the rational group algebra can be expressed entirely in terms of topological cyclic homology let f be a functor that to a unital associative ring a associates a symmetric spectrum f a if i a is a two sided ideal the relative term f a i is defined to be the homotopy fiber of the map f a f a i induced from the canonical projection if further f a b is a ring homomorphism such that f i f i is an isomorphism onto a two sided ideal of b there is an induced map f a i f b f i and the bi relative term f a b i is defined to be the homotopy fiber of this map then there is a distinguished triangle in the stable homotopy category

optimized chaos control with simple limiters we present an elementary derivation of chaos control with simple limiters using the logistic map and the henon map as examples this derivation provides conditions for optimal stabilization of unstable periodic orbits of a chaotic attractor

the expression of the generalized inverse of the perturbed operator under type i perturbation in hilbert spaces let h1 h2 be two hilbert spaces over the complex field c and let t h1 h2 be a bounded linear operator with the generalized inverse t let t t t be a bounded linear operator with t t 1 suppose that dim ker t dim ker t or r t r t 0 then t has the generalized inverse t i i t t 1 i t t i t i t t 1 1 1 tt 1t x i tt tt tt i tt 1 i tt ast 1tt i tt i 1 with t t 1 t t this result gives an analogue of theorem 3 9 of m z nashed generalized inverses and applications academic press new york 1976 in hilbert spaces

cyclic cohomology and the family lefschetz theorem any closed current on the base of a compact fibration gives rise to a cyclic cocycle on the smooth convolution algebra we prove that such cocycle furnishes additive maps from the vertically equivariant k theory to the scalars this enables to associate to any closed current on the base of the fibration a lefschetz formula for fiber preserving isometries using geometric operators on the base we deduce the integrality of some characteristic numbers

upper bound and stability of scaled pseudoinverses for given matrices x and d where d is positive definite diagonal a weighed pseudoinverse of x is defined by x d x rm h d 2 x x rm h d 2 and an oblique projection of x is defined by p d xx d when x is of full column rank stewart 3 and o leary 2 found sharp upper bound of oblique projections p d which is independent of d and an upper bound of weighed pseudoinverse x d by using the bound of p d in this paper we discuss the sharp upper bound of x d over a set d of positive diagonal matrices which does not depend on the upper bound of p d and the stability of x d over d

towards a generalized singular value decomposition we suggest a form for and give a constructive derivation of the generalized singular value decomposition of any two matrices having the same number of columns we outline its desirable characteristics and compare it to an earlier suggestion by van loan siam j numer anal 13 1976 pp 76 83 the present form largely follows from the work of van loan but is slightly more general and computationally more amenable than that in the paper cited we also prove a useful extension of a theorem of stewart siam rev 19 1977 pp 634 662 on unitary decompositions of submatrices of a unitary matrix

controlling chaos in unidimensional maps using constant feedback we demonstrate that a constant feedback can control chaotic oscillations in one dimensional discrete maps this is a simple method of controlling chaos as the feedback signal does not require a priori knowledge of the dynamics of the system and it generates the desired behavior by simply varying the strength of the feedback we illustrate this method with an application to the quadratic and exponential logistic maps

on bounds for scaled projections and pseudoinverses abstract let x be a matrix of full column rank and let d be a positive definite diagonal matrix in a recent paper stewart considered the weighted pseudoinverse x d x t dx 1 x t d and the associated oblique projection p d xx d and gave bounds independent of d for the norms of these matrices in this note we answer a question he raised by showing that the bounds are computable

the obstruction to excision in k theory and in cyclic homology let f a b be a ring homomorphism of not necessarily unital rings and i triangleleft a an ideal which is mapped by f isomorphically to an ideal of b the obstruction to excision in k theory is the failure of the map between relative k groups k a i k b f i to be an isomorphism it is measured by the birelative groups k a b i similarly the groups hn a b i measure the obstruction to excision in negative cyclic homology we show that the rational jones goodwillie chern character induces an isomorphism ch k a b i otimes mathbb q overset sim to hn a otimes mathbb q b otimes mathbb q i otimes mathbb q

chern character for twisted k theory of orbifolds for an orbifold x and h 3 x z we introduce the twisted cohomology h c x and prove that the non commutative chern character of connes karoubi establishes an isomorphism between the twisted k groups k x c and the twisted cohomology h

on stable perturbations for outer inverses of linear operators in banach spaces abstract in this paper we investigate stable perturbations and their characterizations for various types of outer inverses such as generalized 2 3 2 4 2 5 1 2 3 1 2 4 moore penrose group drazin and generalized drazin inverses some known results are improved and extended

perturbation analysis for the projection of a point to an affine set abstract we present a perturbation analysis for the projection of a point to an affine set in r n the matrix involved is not assumed to be full rank and the perturbation is arbitrary error bounds for the solution under the rank preserving and rank nondecreasing perturbations are obtained the results given here appear to be the most general up to now

index theory for quasi crystals i computation of the gap label group abstract in this paper we give a complete solution to the gap labelling conjecture for quasi crystals the method adopted relies on the index theory for laminations and the main tools are the connes skandalis longitudinal k theory index morphism together with the measured index formula

scaling properties of simple limiter control simple limiter control of chaotic systems is analytically and numerically investigated proceeding from the one dimensional case to higher dimensions the properties of the control method are fully described by the one parameter one dimensional flat top map family implying that orbits are stabilized in exponential time independent of the periodicity and without the need for targeting fine tuning of the control is limited by superexponential scaling in the control space where orbits of the uncontrolled system are obtained for a set of zero lebesgue measure in higher dimensions simple limiter control is a highly efficient control method provided that the proper limiter form and placement are chosen

finite dimensional approximation settings for infinite dimensional moore penrose inverses after a brief survey of basic results about finite dimensional approximation settings such as mathematical frameworks of various projection methods including the least squares method the dual least squares method and the galerkin method for infinite dimensional moore penrose inverses this paper proceeds to a detailed study from the following aspects for projection methods we investigate convergence and weak convergence of their approximation setting to develop a unified theory on projection methods for infinite dimensional moore penrose inverses this investigation yields a fundamental convergence theorem theorem 2 2 from which the criterion for convergence the criterion for weak convergence and the generalized dual least squares method are derived we also derive general results on the least squares method by which two flaws in groestch s results are corrected for nonprojection methods whose approximation setting is a more general framework we investigate weak perfect convergence of their approximation setting and provide a necessary and sufficient condition of such convergence holding theorem 3 2 several examples are proposed as counterexamples to illustrate the differences between some important concepts or as concrete algorithms to show how the present work can help to analyze their behavior

new perturbation results for equality constrained least squares problems abstract we present some perturbation results for least squares problems with equality constraints relative errors are obtained on perturbed solutions and lagrange multipliers of the problem based on the equivalence of the problem to a consistent system of linear equations

error analysis and implementation aspects of deferred correction for equality constrained least squares problems the deferred correction procedure recently proposed by van loan on the method of weighting for equality constrained least squares problems siam j numer anal 22 1985 pp 851 864 solves the problem of minimizing f ex 2 subject to cx g by repeatedly solving the related least squares problem of minimizing left left begin array 20 c tau delta g delta f end array right left begin array 20 c tau c e end array right delta x right 2 for some large weight tau the convergence behavior of the procedure is shown to be greatly dependent upon error analysis considerations an analysis of its convergence with an approximation factorization of left begin array 20 c tau c e end array right is given an implementation improvement to the procedure that makes it more robust is also given van loan s conjecture that the method converges for all but ill conditioned problems is shown to be true

controllability of spatiotemporal systems using constant pinnings abstract most natural spatiotemporal systems are an organized ensemble of dynamical subsystems whose behaviour is regulated by nonlinearly coupled multi variable processes the response of each of these variables and or combinations of them to any single or composite external perturbation can influence its spatiotemporal behaviour in a complex and non intuitive manner this paper attempts to study the dynamical response of two coupled map lattice systems having local dynamics described by a coupled discrete maps and b coupled differential equations to external perturbation or pinning applied to each variable individually or simultaneously we show that the response of different variables to external pinning is quite different our results indicate that though this pinning approach is useful in controlling complex dynamics both globally and locally enhancing complexity in dynamics anti control is variable dependent thus complete controllability i e control and anti control of dynamics in these spatially extended systems having coupled multi variable processes does not only depend on the sign and strength of the perturbation but also on the specific variable being pinned this also implies that external manipulation of state variables in natural systems can lead to quite different responses depending at the level of action of the perturbation

cyclic homology cdh cohomology and negative k theory we prove a blow up formula for cyclic homology which we use to show that infinitesimal k theory satisfies cdh descent combining that result with some computations of the cdh cohomology of the sheaf of regular functions we verify a conjecture of weibel predicting the vanishing of algebraic k theory of a scheme in degrees less than minus the dimension of the scheme for schemes essentially of finite type over a field of characteristic zero

controlling one dimensional unimodal population maps by harvesting at a constant rate controlling chaos and periodic oscillations in dynamical systems is a well known problem it has been studied by several types of algorithms some of which were only demonstrated numerically with specific examples we study here a class of one dimensional discrete maps of the form x n 1 f x n where f x is a unimodal function satisfying a few natural conditions as a model for the dynamics of a single species population for managing the population we seek to suppress any possible chaotic or periodic behavior that may emerge the paper proposes a simple rigorously proved algorithm for controlling unimodal maps implemented by harvesting the population at a constant rate this forces the orbits of the map to converge to a limit that we can compute a priori the result holds for any initial conditions within an interval that we specify our control algorithm is easy to implement requires no updated information on the population and no changes in the parameters of the system which in general are fixed by the properties of the population it can therefore be useful for exploiting a population while maintaining it at a fixed density

perturbation theory for the rank deficient equality constrained least squares problem this paper presents perturbation theory for the equality constrained least squares problem min f in s kf g 2 with s f lf h 2 min y in c n ly h 2 lse in which l and begin array 20 c l k end array may not be of full rank the analysis generalizes the results of elden 4 perturbation theorems for the lse problem are proved numerical experiments are also given to verify the error bounds

equivalent formulae for the supremum and stability of weighted pseudoinverses during recent decades there have been a great number of research articles studying interior point methods for solving problems in mathematical programming and constrained optimization stewart and o leary obtained an upper bound for scaled pseudoinverses sup w1 2x w1 2 2 of a matrix x w p where p is a set of diagonal positive definite matrices we improved their results to obtain the supremum of scaled pseudoinverses and derived the stability property of scaled pseudoinverses forsgren further generalized these results to derive the supremum of weighted pseudoinverses sup w1 2x w1 2 2 where w p p is a set of diagonally dominant positive semidefinite matrices by using a signature decomposition of weighting matrices w and by applying the binet cauchy formula and cramer s rule for determinants the results are also extended to equality constrained linear least squares problems in this paper we extend forsgren s results to a general complex matrix x to establish several equivalent formulae for sup w1 2x w1 2 2 where p is a w p set of diagonally dominant positive semidefinite matrices or a set of weighting matrices arising from solving equality constrained least squares problems we also discuss the stability property of these weighted pseudoinverses

a survey of condition number estimation for triangular matrices we survey and compare a wide variety of techniques for estimating the condition number of a triangular matrix and make recommendations concerning the use of the estimates in applications each of the methods is shown to bound the condition number the bounds can broadly be categorised as upper bounds from matrix theory and lower bounds from heuristic or probabilistic algorithms for each bound we examine by how much at worst it can overestimate or underestimate the condition number numerical experiments are presented in order to illustrate and compare the practical performance of the condition estimators

the periodic cyclic homology of crossed products of finite type algebras we study the periodic cyclic homology groups of the cross product of a finite type algebra a by a discrete group in case a is commutative and the group is finite our results are complete and given in terms of the singular cohomology of the sets of fixed points these groups identify our cyclic homology groups with the orbifold cohomology of the underlying algebraic orbifold the proof is based on a careful study of localization at fixed points and of the resulting koszul complexes this is achieved by extending to our class of noncommutative algebras the concept of an infinitesimal neighborhood that plays such an important role in commutative algebra we provide examples of azumaya algebras for which this identification is however no longer valid as an example we discuss some affine weyl groups

characterization of spatial networklike patterns from junction geometry we propose a method for quantitative characterization of spatial networklike patterns with loops such as surface fracture patterns leaf vein networks and patterns of urban streets such patterns are not well characterized by purely topological estimators also patterns that both look different and result from different morphogenetic processes can have similar topology a local geometric cue the angles formed by the different branches at junctions can complement topological information and allow the quantification of the large scale spatial coherence of the pattern for patterns that grow over time such as fracture lines on the surface of ceramics the rank assigned by our method to each individual segment of the pattern approximates the order of appearance of that segment we apply the method to various networklike patterns and find a continuous but sharp dichotomy between two classes of spatial networks hierarchical and homogeneous the former class results from a sequential growth process and presents large scale organization and the latter presents local but not global organization

on the continuity of generalized inverses of linear operators in hilbert spaces abstract let x and y be hilbert spaces and let t x y be a bounded linear operator with closed range we study the continuity problem of the generalized inverse of t and related least squares solutions to the operator equation tx y

bivariant algebraic k theory we show how methods from k theory of operator algebras can be applied in a completely algebraic setting to define a bivariant m1 stable homotopy invariant excisive k theory of algebras over a fixed unital ground ring h a b 7 kk a b which is universal in the sense that it maps uniquely to any other such theory it turns out kk is related to c weibel s homotopy algebraic k theory kh we prove that if h is commutative and a is central as an h bimodule then kk h a kh a we show further that some calculations from operator algebra kk theory such as the exact sequence of pimsner voiculescu carry over to algebraic kk

on the perturbation of the least squares solutions in hilbert spaces abstract let h1 and h2 be hilbert spaces and let t h1 h2 be a bounded linear operator with closed range we present some results of the perturbation analysis for the least squares solutions to the operator equation tx y

on perturbations for oblique projection generalized inverses of closed linear operators in banach spaces the main concern of this paper is the perturbation problem for oblique projection generalized inverses of closed linear operators in banach spaces we provide a new stability characterization of oblique projection generalized inverses of closed linear operators under t bounded perturbations which improves some well known results in the case of the closed linear operators under the bounded perturbation or that the perturbation does not change the null space

controlling chaos in unidimensional maps using macroevolutionary algorithms we introduce a simple search algorithm that explores the parameter of periodically perturbed discrete maps in order to find desired orbits through chaos control the method has been applied to one dimensional maps but is easily extendable to higher dimensional systems here we consider two types of chaos control involving proportional pulses in the system variables phys rev lett 72 1455 1994 and constant feedback phys rev e 51 6239 1995 the first case being presented in detail it is shown that our method allows a rapid exploration of parameter space and the finding of high fitness i e controlled solutions close to the target orbits even when high periodicities are required

perturbation of generalized inverses of linear operators in hilbert spaces abstract let x and y be hilbert spaces and let t x y be a bounded linear operator with closed range in this paper we present some new results of the perturbation analysis on the generalized inverse of t and the related least squares solution to the operator equation tx y

comparison between constant feedback and limiter controllers using symbolic dynamics of the one dimensional unimodal map the chaos stabilization mechanics of the feedback and limiter control schemes are considered for feedback control it is found that the control strength can be efficiently obtained from the superstable parameter of the embedded periodic orbits and the scaling of the control space period doubling bifurcation cascade still obeys the feigenbaum law for sarkovskii orbits the scaling is also consistent with that of the original chaotic system for limiter control a single critical point in the unimodal map is extended to a superstable periodic window and a simple approach for determining the value of the control plateau is found the scaling in the control space of the period doubling bifurcation cascade is indeed superexponential a different scaling for the fine structure of the sarkovskii sequence is also found simple one dimensional unimodal maps can also be used to generate maximum length shift register sequences

extensions and renormalized traces it has been shown by nistor doc math j dmv 2 263 295 1997 that given any extension of associative algebras over mathbb c the connecting morphism in periodic cyclic homology is compatible under the chern connes character with the index morphism in lower algebraic k theory the proof relies on the abstract properties of cyclic theory essentially excision which does not provide explicit formulas a priori avoiding the use of excision we explain in this article how to get explicit formulas in a wide range of situations the method is connected to the renormalization procedure introduced in our previous work on the bivariant chern character for quasihomomorphisms perrot j geom phys 60 1441 1473 2010 leading to local index formulas in the sense of non commutative geometry we illustrate these principles with the example of the classical family index theorem we find that the characteristic numbers of the index bundle associated to a family of elliptic pseudodifferential operators are expressed in terms of the fiberwise wodzicki residue

topology of technology graphs small world patterns in electronic circuits recent theoretical studies and extensive data analyses have revealed a common feature displayed by biological social and technological networks the presence of small world patterns here we analyze this problem by using several graphs obtained from one of the most common technological systems electronic circuits it is shown that both analogic and digital circuits exhibit small world behavior we conjecture that the small world pattern arises from the compact design in which many elements share a small close physical neighborhood plus the fact that the system must define a single connected component which requires shortcuts connecting different integrated clusters the degree distributions displayed are consistent with a conjecture concerning the sharp cutoffs associated to the presence of costly connections amaral et al proc natl acad sci usa 97 11 149 2000 thus providing a limit case for the classes of universality of small world patterns from real artificial networks the consequences for circuit design are outlined

algebra extensions and nonsingularity this paper is concerned with a notion of nonsingularity for noncommutative algebras which arises naturally in connection with cyclic homology let us consider associative unital algebras over the complex numbers we call an algebra a quasi free when it behaves like a free algebra with respect to nilpotent extensions in the sense that any homomorphism a r i where i is a nilpotent ideal in r can be lifted to a homomorphism a r if we restrict to the category of finitely generated commutative algebras then this lifting property characterizes smooth algebras the ones corresponding to nonsingular affine varieties in this way quasi free algebras appear as noncommutative analogues of smooth algebras stretching the analogy we can even regard quasi free algebras as analogues of manifolds one of the aims of this paper is to develop the analogy further by showing that quasi free algebras provide a natural setting for noncommutative versions of certain aspects of manifolds to give an example let us consider the analogue of an embedding an extension a r i where a and r are quasi free algebras playing the role of the submanifold and ambient manifold respectively in the manifold situation i i2 is the module of linear functions on the nor2 mal bundle and the symmetric algebra sa iii is the algebra of polynomial functions now in passing from commutative to noncommutative algebras the symmetric algebra of a module is replaced by the tensor algebra of a bimodule

a bivariant chern character for families of spectral triples in this paper we construct a bivariant chern character defined on families of spectral triples such families should be viewed as a version of unbounded kasparov bimodules adapted to the category of bornological algebras the chern character then takes its values in the bivariant entire cyclic cohomology of meyer the basic idea is to work within quillen s algebra cochains formalism and construct the chern character from the exponential of the curvature of a superconnection leading to a heat kernel regularization of traces the obtained formula is a bivariant generalization of the jlo cocycle

perturbation analysis of the least squares solution in hilbert spaces abstract let h 1 h 2 be two hilbert spaces over the same field and let t h 1 h 2 be a bounded linear operator with closed range we give a complete description of the perturbation analysis for the least squares solution to the operator equation t x y where x h 1 y h 2

on the error estimate for the projection of a point onto a linear manifold abstract suppose that the linear system a h x b is consistent the problem of finding the solution x to min a h x b 1 2 x p 2 2 is studied where a c n m b c m x p c n and a b p are given when a b and p are perturbed an error bound for the solution x is obtained roundoff error bounds for the singular value decomposition method the column pivoting method and the rank revealing method are obtained the results in this paper extend those of arioli laratta and menchi to the case when a is not of full rank and either m n or m n

pro unitality and pro excision in algebraic k theory and cyclic homology the purpose of this paper is to study pro excision in algebraic k theory and cyclic homology after suslin wodzicki cuntz quillen cortinas and geisser hesselholt as well as continuity properties of andre quillen and hochschild homology a key tool is first to establish the equivalence of various pro tor vanishing conditions which appear in the literature this allows us to prove that all ideals of commutative noetherian rings are pro unital in a suitable sense we show moreover that such pro unital ideals satisfy pro excision in derived hochschild and cyclic homology it follows hence and from the suslin wodzicki criterion that ideals of commutative noetherian rings satisfy pro excision in derived hochschild and cyclic homology and in algebraic k theory in addition our techniques yield a strong form of the pro hochschild kostant rosenberg theorem an extension to general base rings of the cuntz quillen excision theorem in periodic cyclic homology a generalisation of the f igin tsygan theorem a short proof of pro excision in topological hochschild and cyclic homology and new artin rees and continuity statements in andre quillen and hochschild homology msc 19d55 primary 16e40 13d03 secondary

percolation in insect nest networks evidence for optimal wiring optimization has been shown to be a driving force for the evolution of some biological structures such as neural maps in the brain or transport networks here we show that insect networks also display characteristic traits of optimality by using a graph representation of the chamber organization of termite nests and a disordered lattice model it is found that these spatial nests are close to a percolation threshold this suggests that termites build efficient systems of galleries spanning most of the nest volume at low cost the evolutionary consequences are outlined

on the semi continuity of generalized inverses in banach algebras abstract the main purpose of this paper is to study the continuity of several kinds of generalized inverses of elements in a banach algebra with identity we first obtain a sufficient and necessary condition for the lower semi continuity of reflexive generalized inverses as set valued mappings based on this result we characterize the continuity of the moore penrose inverse in a c algebra and therefore derive some new and well known criteria in operator theory

the kadison kaplansky conjecture for word hyperbolic groups in this paper we prove the kadison kaplansky idempotent conjecture for torsion free word hyperbolic groups the conjecture asserts that the following equivalent statements hold for a torsion free discrete group the reduced group c algebra c r contains no idempotents except 0 and 1 the spectrum of every element of the reduced group c algebra is connected the canonical trace on c r takes integer values on idempotents the last assertion can be viewed as a statement about the pairing between the k theory and the local cyclic cohomology of the group c algebra it is in this setting that we will prove the conjecture our proof is based on a partial analysis of the assembly maps in k theory and local cyclic homology we compare these assembly maps by means of an equivariant bivariant chern connes character before going into details we recall some previous work on the conjecture the first progress was achieved by pimsner and voiculescu pv who proved the kadison kaplansky conjecture for free groups as a consequence of their computation of the k theory of the group c algebra subsequently it was realized that more generally the kadison kaplansky conjecture was a consequence of the baum connes conjecture which gives a geometric description of k c r for any torsion free discrete group in fact the baum connes conjecture states that the k theoretic assembly map

excision in cyclic homology theories in this paper we establish excision for z 2z graded cyclic homology theories including periodic entire asymptotic and local bivariant cyclic cohomology our work is a modification of the approach of cuntz quillen cq to excision in bivariant periodic cyclic homology and makes also use of the work of wodzicki wo we study the dimension shift of periodic cyclic cohomology under the boundary map associated to an extension of algebras and obtain the possible values of this dimension shift excision in entire cyclic cohomology is used to calculate these groups in a number of hitherto unknown cases finally a bivariant and multiplicative chern connes character on kasparov s bivariant k theory ka with values in bivariant local cyclic cohomology is constructed to put our approach into perspective we recall the previous work on excision in cyclic homology ever since the invention of cyclic homology by connes and independently by tsygan the problem of excision played a central role in the theory it is concerned with the question of to what extent an extension

windows in one dimensional maps abstract the window structure in the chaotic part of the parameter space of one dimensional maps is described in detail explicit expressions are derived for both the internal structure and the width of the periodic windows by considering a universal local submap both features are found to depend only on the order of the extremum of this submap moreover the windows can be ordered in trees the various branches of these trees define accumulating families of windows for which exact scaling relations can be derived finally we discuss an extension of our description of window phenomena into two dimensions

on the method of weighting for equality constrained least squares problems the generalized singular value decomposition is used to analyze the problem of minimizing ax b 2 subject to the constraint bx d a byproduct of the analysis is a new iterative procedure that can be used to improve an approximate solution obtained via the method of weights all that is required to implement the procedure is a single qr factorization these developments turn out to be of interest when a and b are sparse and for the case when systolic architectures are used to carry out the computations

excision in cyclic homology and in rational algebraic k theory for a precise definition see 1 below by replacing everywhere k by k q one obtains the corresponding notion in rational algebraic k theory the above definition has an obvious counterpart for cyclic homology and algebras over a fixed field in the case of a general commutative ground ring k some restrictions on the class of allowable extensions seem inevitable due to the well known limitations of cyclic homology considered as a homology functor for algebras not flat over a ground ring an extension of k algebras will be called pure if it is pure as an extension of k modules in the sense of p m cohn 6 cf also appendix a 3 below the class of pure extensions contains e g i extensions which admit a k module splitting ii extensions a e r s with s flat over k in fact one among the several possible characterizations of purity says that an extension is pure if and only if the underlying extension of k modules is an inductive limit of split extensions see theorem a 4 of appendix a below everywhere in this paper the word excision used in the context of cyclic homology will mean excision with respect to the class of pure extensions the second purpose of the present paper is to give a complete characterization of the class of algebras possessing the excision property in cyclic homology before stating the corresponding result we need the following definition let us

some new properties of the equality constrained and weighted least squares problem abstract in this paper some new properties of the equality constrained and weighted least squares problem wlse min w 1 2 kx g 2 subject to lx h are obtained we derive a perturbation bound based on an unconstrained least squares problem and deduce some equivalent formulae for the projectors of this unconstrained ls problem we also present a new way to compute the minimum norm solution x wlse of the wlse problem by using the qr decomposition of the corresponding matrices and propose an algorithm to compute x wlse using the qr factorizations some numerical examples are provided to compare different methods for solving the wlse problem

a generalization of a classic theorem in the perturbation theory for linear operators let x and y be banach spaces and let t x y be a bounded linear operator with closed range in this paper we extend a classic perturbation result from the invertible case to the general case for the consistent operator equation tx y

on a bifurcation structure mimicking period adding in this work we investigate a piecewise linear discontinuous scalar map defined on three partitions this map is specifically constructed in such a way that it shows a recently discovered bifurcation scenario in its pure form owing to its structure on the one hand and the similarities to the nested period adding scenario on the other hand we denoted the new bifurcation scenario as nested period incrementing bifurcation scenario the new bifurcation scenario occurs in several physical and electronical systems but usually not isolated which makes the description complicated by isolating the scenario and using a suitable symbolic description for the asymptotically stable periodic orbits we derive a set of rules in the space of symbolic sequences that explain the structure of the stable periodic domain in the parameter space entirely hence the presented work is a necessary step for the understanding of the more complicated bifurcation scenarios mentioned above

perturbation of the least squares problem consider the least squares solutions to ax b ax b where a a a and b b b are perturbed from a b respectively in an earlier paper the author deduced the perturbation bounds for the solution when b r a in this paper we extend the results to cover more general cases when b r a and a is not of full rank

algebraic properties of the rank deficient equality constrained and weighted least squares problems abstract we study the algebraic properties of the solutions of the equality constrained least squares problem min f s v kf g v 2 with s f v h lf v 2 min z c n v v h lz v 2 lse and corresponding weighted least squares problem wls in which l and l k may not be of full rank general formulas for the solutions are given and the algebraic relations between the lse and the wls solutions are obtained

error analysis of algorithms for computing the projection of a point onto a linear manifold the accumulation of rounding errors in methods used to compute the projection of a point onto a linear manifold is studied the methods are based on modified gram schmidt householder and givens orthogonal factorizations

a new description of kasparov s theory of c algebra extensions in this paper we show that kasparov s ext theory for c algebras can be defined as a topological version of yoneda s ext theory for modules we also give a description of the ext theory for c algebras in terms of a universal c algebra ea associated with the c algebra a this algebra ea is the odd case analogue of the c algebra qa defined by cuntz

k regularity cdh fibrant hochschild homology and a conjecture of vorst it is a well known fact that algebraic if theory is homotopy invariant as a functor on regular schemes if x is a regular scheme then the natural map kn x kn x x 1 is an isomorphism for all n ez this is false in general for nonregular schemes and rings to express this failure bass introduced the terminology that for any contravari ant functor v defined on schemes a scheme x is called v regular if the pullback maps v x v x x ar are isomorphisms for all r 0 if x spec i we also say that r is p regular thus regular schemes are ifn regular for every n in contrast it was observed as long ago as in 2 that a nonreduced affine scheme can never be k regular in particular if a is an artinian ring that is a o dimensional noetherian ring then a is regular that is reduced if and only if a is ifi regular in 17 vorst conjectured that for an affine scheme x of finite type over a field f and of dimension d regularity and k i regularity are equivalent vorst proved this conjecture for d 1 by proving that regularity implies normality in this paper we prove vorst s conjecture in all dimensions provided the char acteristic of the ground field f is zero in fact we prove a stronger statement we say that x is regular in codimension n in x note that for all n z if a ring r is ifn regular then it is kn i regular this is proved in 17 for n 1 and in 6 4 4 for n kh x where k x is the algebraic if theory spectrum of x and kh x is the homotopy if theory of x defined in 19 we write fk r for fk spec r theorem 0 1 let r be a commutative ring which is essentially of finite type over a field f of characteristic 0 then the following hold a if tk r is n connected then r is regular in codimension khi x is an isomorphism for i n and a surjection for i n 1 so that fk x

weighted least squares solutions to general coupled sylvester matrix equations this paper is concerned with weighted least squares solutions to general coupled sylvester matrix equations gradient based iterative algorithms are proposed to solve this problem this type of iterative algorithm includes a wide class of iterative algorithms and two special cases of them are studied in detail in this paper necessary and sufficient conditions guaranteeing the convergence of the proposed algorithms are presented sufficient conditions that are easy to compute are also given the optimal step sizes such that the convergence rates of the algorithms which are properly defined in this paper are maximized and established several special cases of the weighted least squares problem such as a least squares solution to the coupled sylvester matrix equations problem solutions to the general coupled sylvester matrix equations problem and a weighted least squares solution to the linear matrix equation problem are simultaneously solved several numerical examples are given to illustrate the effectiveness of the proposed algorithms

the perturbation of consistent least squares problems abstract suppose that the linear system ax b is consistent and a a a b b b are perturbed from a b respectively where a and are m n matrices b and b are m dimensional vectors let a f e a 2 b 2 e b 2 and a 2 a 2 be the condition number of a it is shown that if e 1 then the least squares solution x of a x b is close to a solution x of ax b even when a is not of full rank and rank rank a

finding the orthogonal projection of a point onto an affine subspace a simple method is proposed to find the orthogonal projection of a given point to the solution set of a system of linear equations this is also a direct method for solving systems of linear equations the output of the method is either the projection or inconsistency of the system moreover in the process also linearly dependent equations are recognized this paper is constrained for giving theoretical foundations computational complexity and some numerical experiments with dense matrices although the method allows to employ sparsity the raw method could not compete with best software packages in solving linear equations for general matrices but it was competitive in finding projections for matrices with small number of rows relative to the number of columns

on scaled projections and pseudoinverses abstract let x be a matrix of full column rank and let d be a diagonal matrix with positive diagonal elements the weighted pseudoinverse defined by x d x t dx 1 x t d and the associated oblique projection p d xx d arise in many applications in this paper we show that the norms of both matrices are bounded by numbers that are independent of d

d branes rr fields and duality on noncommutative manifolds we develop some of the ingredients needed for string theory on noncommutative spacetimes proposing an axiomatic formulation of t duality as well as establishing a very general formula for d brane charges this formula is closely related to a noncommutative grothendieck riemann roch theorem that is proved here our approach relies on a very general form of poincare duality which is studied here in detail among the technical tools employed are calculations with iterated products in bivariant mathsf k theory and cyclic theory which are simplified using a novel diagram calculus reminiscent of feynman diagrams

comparison between algebraic and topological k theory of locally convex algebras abstract this paper is concerned with the algebraic k theory of locally convex c algebras stabilized by operator ideals and its comparison with topological k theory we show that if l is locally convex and j a frechet operator ideal then all the different variants of topological k theory agree on the completed projective tensor product l j and that the obstruction for the comparison map k l j k top l j to be an isomorphism is absolute algebraic cyclic homology we prove the existence of an exact sequence theorem 6 2 1 we show that cyclic homology vanishes in the case when j is the ideal of compact operators and l is a frechet algebra whose topology is generated by a countable family of sub multiplicative seminorms and admits an approximate right or left unit which is totally bounded with respect to that family theorem 8 3 3 this proves the generalized version of karoubi s conjecture due to mariusz wodzicki and announced in his paper m wodzicki algebraic k theory and functional analysis in first european congress of mathematics vol ii paris 1992 in progr math vol 120 birkhauser basel 1994 pp 485 496 we also consider stabilization with respect to a wider class of operator ideals called sub harmonic every frechet ideal is sub harmonic but not conversely for example the schatten ideal l p is sub harmonic for all p 0 but is frechet only if p 1 we prove a variant of the exact sequence above which essentially says that if a is a c algebra and j is sub harmonic then the obstruction for the periodicity of k a c j is again cyclic homology theorem 7 1 1 this generalizes to all algebras a result of wodzicki for h unital algebras announced in m wodzicki algebraic k theory and functional analysis in first european congress of mathematics vol ii paris 1992 in progr math vol 120 birkhauser basel 1994 pp 485 496 the main technical tools we use are the diffeotopy invariance theorem of cuntz and the second author which we generalize in theorem 6 1 6 and the excision theorem for infinitesimal k theory due to the first author