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

the classification of extensions of c algebras where k k h denotes the compact operators on a separable infinite dimensional hubert space h the addition operation comes from associating to 1 1 its busby invariant 4 theorem 4 3 a q b 9 where 1 b denotes the outer multiplier algebra of b k and then using an injection of 1 b 1 b into 1 b if a is nuclear and b has a countable approximate unit ext ay b is actually a group the identity of the group corresponds to those extensions r which are stably split i e such that r r is split see 4 proposition 5 3 for some split extension r study of groups which classify extensions of the form 1 1 was initiated by brown douglas and fillmore bdf 3 in the case b c the complex numbers they showed that when a c x 9 x a compact metric space then unitary equivalence classes of unital essential extensions of the form 1 1 form a group ext a furthermore ext generates a homology theory on compact metric spaces which satisfies bott periodicity and other useful conditions l g brown 1 also proved the following universal coefficient theorem uct there is a natural short exact sequence

cuntz krieger algebras of infinite graphs and matrices we show that the cuntz krieger algebras of infinite graphs and infinite 0 1 matrices can be approximated by those of finite graphs we then use these approximations to deduce the main uniqueness theorems for cuntz krieger algebras and to compute their k theory since the finite approximating graphs have sinks we have to calculate the k theory of cuntz krieger algebras of graphs with sinks and the direct methods we use to do this should be of independent interest

controlling chaos and deterministic directed transport in inertia ratchets using backstepping control we address the problem of controlling chaotic motion and deterministic directed transport in inertia ratchets we employ a recursive backstepping nonlinear control technique to control intermittent chaos and then track a desired trajectory by means of the same technique for the parameter regime where two non identical attractors coexist in phase space we propose a new backstepping control scheme that is capable of controlling the directed transport exhibited by these attractors numerical simulations show that the controllers are singularity free and the closed loop systems are globally stable

rieffel deformation via crossed products we start from rieffel data a where a is a c algebra is an action of an abelian group on a and is a 2 cocycle on the dual group using landstad theory of crossed product we get a deformed c algebra a in the case of rn we obtain a very simple proof of invariance of k groups under the deformation in the general case we also get a very simple proof that nuclearity is preserved under the deformation we show how our approach leads to quantum groups and investigate their duality the general theory is illustrated by an example of the deformation of sl 2 c a description of it in terms of noncommutative coordinates is given

the stable rank of topological algebras and a problem of r g swan abstract various notions of stable ranks are studied for topological algebras some partial answers to r g swan s problem have two banach or good frechet algebras as in the density theorem in k theory the same stable rank are obtained for example a frechet dense subalgebra a of a c algebra b closed under c functional calculus of self adjoint elements has the same bass stable rank as b

gap labelling and the pressure on the boundary in quantum systems described by covariant families of 1 particle schrodinger operators on half spaces the pressure on the boundary per unit energy is topologically quantised if the fermi energy lies in a gap of the bulk spectrum its relation with the integrated density of states can be expressed in an integrated version of streda s formula this leads also to a gap labelling theorem for systems with constant magnetic field the proof uses non commutative topology

boundary maps for c crossed products with r with an application to the quantum hall effect the boundary map in k theory arising from the wiener hopf extension of a crossed product algebra with r is the connes thom isomorphism in this article the wiener hopf extension is combined with the heisenberg group algebra to provide an elementary construction of a corresponding map on higher traces and cyclic cohomol ogy it then follows directly from a non commutative stokes theorem that this map is dual w r t connes pairing of cyclic cohomology with k theory as an application we prove equality of quantized bulk and edge conductivities for the integer quantum hall effect described by continuous magnetic schrodinger operators 1 motivation and main result in a commonly used approach to study aperiodic solids particles in the bulk of the medium are described by covariant families of one particle schrodinger operators h whereis the probability space of configurations furnished with an ergodic action of space translations crossed product algebras provide a natural framework for such families be86 in particular their bounded functions are represented by elements of a c crossed product the so called bulk algebra the non commutative topology of the c algebra is a useful tool to construct topological invariants resulting from pairings between k group elements and higher traces some of these invariants may be physi cally interpreted as topologically quantised quantities the quantised hall conductivity is such an example the physics near a boundary of the solid can also be described by a c algebra the so called edge algebra the bulk algebra being essentially a crossed product of the edge algebra with r or with z in the tight binding approximation krs02 both algebras are tied together in the wiener hopf extension or respectively the toeplitz extension this extension gives rise to boundary maps between the k groups and the higher traces of the bulk and edge algebra which allow one to equate bulk and edge in variants this topological relation and its physical interpretations is our main objective we discuss one prominent physical example of this the quantum hall effect where the

theory and multipliers of stable algebras the main theorem is that if a is a c algebra with a countable approximate identity consisting of projections then the unitary group of m a k is contractible this gives a realization via the index map of ko a as components in the set of fredholm operators on ha for a c algebra a let ha ai i ai e a and eai ai converges in norm then ha becomes a right a module under the action ai b aib with the a valued inner product ai b1 ea bi ha is a hilbert a module y ha is the c algebra of operators on ha that have an adjoint k ha is an ideal of y ha called the compact operators on ha set 11 2 definition 4 operators in ha invertible modulo 3f ha are called fredholm and have an index that takes its values in ko a the paper is divided into two parts in part 1 we show that index a ko a is an isomorphism where t denotes the set of path components of the set of fredholm operators in part 2 we show that the unitary group of m a k is contractible when a has a countable approximate identity consisting of projections as 2 ha m a k this in particular implies that the unitary group of s ha is connected when a is unital this latter result is used to prove the isomorphism mentioned above the results of this paper were announced in 14 see remark 2 5 2 this research is based on the author s doctoral work done at dalhousie university under the supervision of professor p a fillmore the author also wishes to thank w j phillips for many useful suggestions part 1 the index of a fredholm operator 1 1 we describe the index map from fredholm operators on hilbert c modules to the corresponding k groups the principal ingredients of this construction have been discussed by several authors kasparov 12 miscenko and fomenko 15 and pimsner popa and voiculescu 17 however as we shall need refinements and variations of these constructions we will give a detailed account of the index map throughout this paper k or y h will denote the c algebra of compact operators on a separable infinite dimensional hilbert space h for any two c algebras a and b a b will denote the completion of the algebraic tensor product aob in the spatial or minimal c norm see e g effros and lance 8 2 received by the editors july 28 1983 and in revised form december 27 1985 1980 mathematics subject classification 1985 revision primary 46l05 46m20 01987 american mathematical society 0002 9947 87 1 00 25 per page

relatively spectral morphisms and applications to k theory abstract spectral morphisms between banach algebras are useful for comparing their k theory and their noncommutative dimensions as expressed by various notions of stable ranks in practice one often encounters situations where the spectral information is only known over a dense subalgebra we investigate such relatively spectral morphisms we prove a relative version of the density theorem regarding isomorphism in k theory we also solve swan s problem for the connected stable rank in fact for an entire hierarchy of higher connected stable ranks that we introduce

the floquet theory and the state density of quasi periodic schrodinger operators abstract we study the floquet solutions of quasi periodic schrodinger operators on flows which satisfy a diophantine condition using these solutions we show that the resolvent of such an operator is smooth with respect to certain derivations in the c algebra associated with the flow and that every projection corresponding to a spectral gap has a finite localization length based on these results we derive a formula which quantizes the integral components of the integrated density of states for the operator on each spectral gap

gap labelling conjecture with nonzero magnetic field abstract given a constant magnetic field on euclidean space r p determined by a skew symmetric p p matrix and a z p invariant probability measure on the disorder set which is by hypothesis a cantor set where the action is assumed to be minimal the corresponding integrated density of states of any self adjoint operator affiliated to the twisted crossed product algebra c z p where is the multiplier on z p associated to takes on values on spectral gaps in the magnetic gap labelling group the magnetic frequency group is defined as an explicit countable subgroup of r involving pfaffians of and its sub matrices we conjecture that the magnetic gap labelling group is a subgroup of the magnetic frequency group we give evidence for the validity of our conjecture in 2d 3d the jordan block diagonal case and the periodic case in all dimensions

the ext class of an approximately inner automorphism let a be a simple unital at algebra of real rank zero it is shown below that the range of the natural map from the approximately inner automorphism group to kk a a coincides with the kernel of the map kk a a l 1 0 hom ki a ki a 1 introduction and preliminaries 1 1 an automorphism of a unital c algebra a is said to be an approximately inner automorphism if there is a sequence of unitaries un 2 a such that a limadun a for all a 2 a it follows that the induced map on k a is the identity map there is however an invariant of k theoretical interest which can occur nontrivial extensions may arise in the six term periodic sequence for the k theory of the mapping torus we show below that every extension does arise if a is a simple unital at algebra of real rank zero as an immediate corollary we obtain a stronger form of elliott s classification theorem for such algebras an invertible kk element that preserves positivity and the class of the unit lifts to an isomorphism 1 2 recall that a unital c algebra a is said to be a unital at algebra if it is expressible as the inductive limit of finite direct sums of algebras of the form mn c t with unital embeddings let a be a unital at algebra and let be an approximately inner automorphism of a the mapping torus of is the c algebra

exact and efficient isogeometric reanalysis of accurate shape and boundary modifications abstract in the traditional design through analysis pipeline geometric models and modifications are approximately represented and transformed with computational models besides after each modification the latest design often needs to be completely analyzed again leading to reanalysis these procedures produce many errors and are extremely time consuming therefore in this paper we propose a novel and an exact and efficient isogeometric reanalysis methodology of accurate shape and boundary modifications that improves the totality of integration of design and analysis greatly geometric models and shape modifications are exactly represented and transformed and the corresponding computational models will change simultaneously when the geometric models are modified thereby reducing error and time in model representation and transformation furthermore we extend and propose the isogeometric based exact reanalysis method termed indirect factorization updating ifu with the combination of isogeometric based reanalysis the method can efficiently obtain the exact solution of the modified structure without solving the complete set of modified equations of the new structure it is also applicable to all techniques for representing the cad geometry model and complex problems several examples illustrate and verify the accuracy and efficiency of this proposed method and furthermore the larger the scale of the problem the more advantageous the end result will be

toeplitz operators on flows abstract let r act continuously on a compact hausdorff space x giving rise to a flow on x let c x and let t x denote the toeplitz operator on h 2 r determined by the function x on r defined by x t x t in this paper we investigate the relation between the spectral properties of t x the dynamical properties of the flow and the value distribution theory of the analysis proceeds by imbedding t x in a type ii factor and computing the real valued index of the operator a la connes our sharpest invertibility result asserts that if the flow is strictly ergodic and if the asymptotic cycle determined by the flow is injective on h 1 x z then t x is invertible if and only if does not vanish on x and determines the zero element in h 1 x z this generalizes the classical result of gohberg and krein and its extension to toeplitz operators with almost periodic symbols due to coburn douglas schaeffer and singer when is analytic in the sense that x belongs to h r for all x we relate the ii index of the toeplitz operator determined by with the density of the zeros of x in the upper half plane much of our efforts to achieve this result are devoted to generalizing to arbitrary flows the value distribution theory of analytic almost periodic functions developed by bohr jessen and tornehave and others

current reversal with type i intermittency in deterministic inertia ratchets the intermittency is investigated when the current reversal occurs in a deterministic inertia ratchet system to determine which type the intermittency belongs to we obtain the return map of velocities of particle by using stroboscopic recordings and by numerically calculating the distribution of the average laminar length the distribution follows the scaling law of proportional to epsilon 1 2 the characteristic relation of type i intermittency

forcing inertial brownian motors efficiency and negative differential mobility abstract the noise assisted directed transport in a one dimensional dissipative inertial brownian motor of the rocking type that is exposed to an external bias is investigated we demonstrate that the velocity load characteristics is distinctly non monotonic possessing regimes with a negative differential mobility in addition we evaluate several possible efficiency quantifiers which are compared among each other these quantifiers characterize the mutual interplay between the viscous drag and the external load differently weighing the inherent rectification features from different physical perspectives

convergence and stability analyses for some vector extrapolation methods on the presence of defective iteration matrices abstract in two previous papers 10 11 convergence and stability results for the following vector extrapolation methods were presented minimal polynomial extrapolation reduced rank extrapolation modified minimal polynomial extrapolation and topological epsilon algorithm the analyses were carried out for vector sequences that include those arising from iterative methods for linear systems of equations having diagonalizable iteration matrices in this paper the analyses of 10 11 are extended to vector sequences that include those arising from iterative methods for linear systems having defective iteration matrices the results are illustrated with numerical examples the analyses above naturally suggest some old and some new extensions of the well known power method enabling one to obtain estimates for several dominant eigenvalues of a general matrix

k theoretic invariants for c algebras associated to transformations and induced flows abstract we discuss a technique of studying the k theory of a unital c algebra associated to a homomorphism on a compact metric space y z by examining the non unital c algebra associated to the induced topological flow ind z r y r the thom isomorphism of connes and the schwartzman asymptotic cycle are used to calculate the range of the trace corresponding to an invariant measure on the k0 group of c x r for a continuous flow on a compact metric space x r under certain conditions projections in c x r with trace r corresponding to cross sections to the flow can be constructed for every positive real number r in this range again by combining techniques of connes and schwartzman applications to the calculation of the tracial range of k0 c x r are discussed in particular this invariant is calculated for minimal affine actions of z on n tori which have quasi discrete spectrum and for minimal actions of z on compact abelian groups with topologically discrete spectrum in both cases this tracial invariant is shown to be the preimage of the eigenvalues for y z under the natural projection o r r z s1

nuclear dimension mathcal z stability and algebraic simplicity for stably projectionless c algebras the main result here is that a simple separable c algebra is mathcal z stable where mathcal z denotes the jiang su algebra if i it has finite nuclear dimension or ii it is approximately subhomogeneous with slow dimension growth this generalizes the main results of toms invent math 183 2 225 244 2011 and winter invent math 187 2 259 342 2012 to the nonunital setting as a consequence finite nuclear dimension implies mathcal z stability even in the case of a separable c algebra with finitely many ideals algebraic simplicity is established as a fruitful weakening of being simple and unital and the proof of the main result makes heavy use of this concept

on the existence and structure of ψ algebras of totally characteristic operators on compact manifolds with boundary as a contribution to the pseudodifferential analysis on manifolds with singularities we construct for each smooth compact manifold x with boundary a algebra a b x b 1 2 l bl2 x b 1 2 containing the algebra 0b cl x b 1 2 of totally characteristic pseudodifferential operators introduced by melrose 25 in 1981 as a dense subalgebra further there is a homomorphism b a a b x b 1 2 q b characterizing the fredholm property of a a b x b 1 2 by means of the invertibility of b a a q b where q b is an algebra of c symbols reflecting the smooth structure of the manifold x the fredholm inverses of fredholm operators in a b x b 1 2 are again in the algebra a b x r 1 2 and we have elliptic regularity corresponding to the scale bhmb x b 1 2 of b sobolev spaces naturally associated to x localized to the interior of x we recover the ordinary pseudodifferential calculus finally spectrum jacobson topology and the relationship of certain closed ideals in the algebra a b x b 1 2 are described explicitly

t duality for torus bundles with h fluxes via noncommutative topology it is known that the t dual of a circle bundle with h flux given by a neveu schwarz 3 form is the t dual circle bundle with dual h flux however it is also known that torus bundles with h flux do not necessarily have a t dual which is a torus bundle a big puzzle has been to explain these mysterious missing t duals here we show that this problem is resolved using noncommutative topology it turns out that every principal t2 bundle with h flux does indeed have a t dual but in the missing cases which we characterize the t dual is non classical and is a bundle of noncommutative tori the duality comes with an isomorphism of twisted k theories just as in the classical case the isomorphism of twisted cohomology which one gets in the classical case is replaced by an isomorphism of twisted cyclic homology

exact and accurate solutions in the approximate reanalysis of structures combined approximations ca is an efe cient method for reanalysis of structures where binomial series terms are used as basis vectors in reduced basis approximations in previous studies high quality approximations have been achieved for large changes in the design but the reasons for the high accuracy were not fully understood in this work some typical cases where exact and accurate solutions are achieved by the method are presented and discussed exact solutions are obtained when a basis vector is a linear combination of the previous vectors such solutions are obtained also for low rank modie cations to structures or scaling of the initial stiffness matrix in general the ca method provides approximate solutions but the results presented explain the high accuracy achieved with only a small number of basis vectors accurate solutions are achieved in many cases where the basis vectors come close to being linearly dependent such solutions are achieved also for changes in a small number of elements or when the angle between the two vectors representing the initial design and modie ed design is small numerical examples of various changes in cross sections of elements and in the layout of the structure show that accurate results are achieved even in cases where the series of basis vectors diverges i introduction m ultiplerepeatedanalysesareneeded invariousdesign and optimization problems in general the structural response cannot be expressed explicitly in terms of the structure properties and structural analysis involves the solution of a set ofsimultaneous equations reanalysis methods are intended to efe ciently analyze structures modie ed due to changes in the design approximate reanalysis methods have been used extensively in structural optimization to reduce the number of exact analyses and the overall computational cost during the solution process the combined approximations ca method developed recently is considered in this paper the method combines several concepts and methods such as reduced basis series approximations matrix factorization and gram schmidt orthonormalization these and other methods are used to achieve effective solution procedures the effectivenessofthemethodinvariousoptimizationproblemshasbeen demonstrated in previous studies 1i5 initially the ca method was used only for linear reanalysis models recently the method has been used successfully also in eigenvalue 6 and nonlinear analysis 7 problems applications of the method in a large variety ofproblems are discussed elsewhere 8i11 high quality approximations of the structural response for large changes in the design have been achieved in previous studies but the reasons for the high accuracy were not fully understood in this paper some typical cases where exact and accurate solutions are achieved by the camethod are presented anddiscussed in general the ca method provides exact solutions but the results presented in the paper explain the high accuracy achieved with only a small number of basis vectors the solution procedure isbriee y described in sec ii three typical cases where exact solutions are achieved by the ca method are introduced and discussed in sec iii exact solutions are obtained when a basis vector is a linear combination of the previous vectors such solutions are obtained also for lowrank modie cations to structures or scaling of the initial stiffness matrix various cases of accurate solutions are discussed in sec iv convergence properties of the series of basis vectors and the series of the ca terms are presented and criteria intended to evaluate

spaces of tilings finite telescopic approximations and gap labeling the continuous hull of a repetitive tiling t in d with the finite pattern condition fpc inherits a minimal d lamination structure with flat leaves and a transversal open image in new window which is a cantor set this class of tiling includes the penrose the amman benkker ones in 2d as well as the icosahedral tilings in 3d we show that the continuous hull with its canonical d action can be seen as the projective limit of a suitable sequence of branched oriented and flat compact d manifolds as a consequence the longitudinal cohomology and the k theory of the corresponding c algebra open image in new window are obtained as direct limits of cohomology and k theory of ordinary manifolds moreover the space of invariant finite positive measures can be identified with a cone in the d th homology group canonically associated with the orientation of d at last the gap labeling theorem holds given an invariant ergodic probability measure on the hull the corresponding integrated density of states ids of any selfadjoint operators affiliated to open image in new window takes on values on spectral gaps in the module generated by the occurrence probabilities of finite patches in the tiling

on the k theory of some c algebras of toeplitz and singular integral operators abstract we study the c algebras gj x r are i x r of singular integral operators and toeplitz operators respectively associated with a strictly ergodic flow x r we show that the commutator ideals of these algebras cgj x r and cl x r are simple and are closely related to the transformation group c algebra c x r we calculate the k theory of gj x r l x r and their commutator ideals the main results of this calculation corollary 3 8 4 and theorem 4 1 1 assert that c x r is contained in cgj x r and if j denotes the inclusion map then j k 0 c x r k 0 cgj x r is an order isomorphism and there is a short exact sequence 0 k 1 c r i i k 1 c r j i k 1 cgj x r 0 where i is the canonical imbedding of c r into c x r we show also that up to a change of scale there is a unique trace on each of the commutator ideals the key ingredient of our analysis is theorem 3 1 1 which asserts a bijective correspondence between silov representations of the algebra of analytic functions on the flow and c representations of gj x r and l x r this simultaneously generalizes coburn s theorem on the uniqueness of the c algebra generated by an isometry and douglas s theorem on the uniqueness of the c algebra generated by an isometric representation of a dense subsemigroup of r

rokhlin dimension for flows we introduce a notion of rokhlin dimension for one parameter automorphism groups of c algebras this generalizes kishimoto s rokhlin property for flows and is analogous to the notion of rokhlin dimension for actions of the integers and other discrete groups introduced by the authors and zacharias in previous papers we show that finite nuclear dimension and absorption of a strongly self absorbing c algebra are preserved under forming crossed products by flows with finite rokhlin dimension and that these crossed products are stable furthermore we show that a flow on a commutative c algebra arising from a free topological flow has finite rokhlin dimension whenever the spectrum is a locally compact metrizable space with finite covering dimension for flows that are both free and minimal this has strong consequences for the associated crossed product c algebras those containing a non zero projection are classified by the elliott invariant for compact manifolds this consists of topological k theory together with the space of invariant probability measures and a natural pairing given by the ruelle sullivan map

finite group actions on c algebras with the rohlin property i basic properties of finite group actions with the rohlin property on unital c algebras are investigated a characterization of finite group actions with the rohlin property on the cuntz algebra 02 is given in terms of central sequences which may be considered as an equivariant version of e kirchberg and n c phillips s characterization of 02 a large class of symmetries on 02 are classified in terms of the fixed point algebras for conjugacy and the crossed products for cocycle conjugacy model actions of symmetries of 02 are constructed for given k theoretical invariants

minimal dynamics and the classification of c algebras let x be an infinite compact metrizable space of finite covering dimension and x x a minimal homeomorphism we prove that the crossed product c x absorbs the jiang su algebra tensorially and has finite nuclear dimension as a consequence these algebras are determined up to isomorphism by their graded ordered k theory under the necessary condition that their projections separate traces this result applies in particular to those crossed products arising from uniquely ergodic homeomorphisms

k theory of equivariant quantization abstract using an equivariant version of connes thom isomorphism we prove that equivariant k theory is invariant under strict deformation quantization for a compact lie group action

universal cycles and homological invariants of locally convex algebras abstract using an appropriate notion of locally convex kasparov modules we show how to induce isomorphisms under a large class of functors on the category of locally convex algebras examples are obtained from spectral triples our considerations are based on the action of algebraic k theory on these functors and involve compatibility properties of the induction process with this action and with kasparov type products this is based on an appropriate interpretation of the connes skandalis connection formalism as an application we prove bott periodicity and a thom isomorphism for algebras of schwartz functions as a special case this applies to the theories kk for locally convex algebras considered by cuntz

a riemann roch theorem for one dimensional complex groupoids we consider a smooth groupoid of the form where is a riemann surface and a discrete pseudogroup acting on by local conformal diffeomorphisms after defining a k cycle on the crossed product c 0 generalising the classical dolbeault complex we compute its chern character in cyclic cohomology using the index theorem of connes and moscovici this involves in particular a generalisation of the euler class constructed from the modular automorphism group of the von neumann algebra l

a reduction method for boundary element reanalysis abstract this paper is concerned with reanalysis for boundary element systems an efficient and accurate reduction method is proposed for reanalyzing such systems the new method has several merits in particular the reduced system has been uncoupled by a gram schmidt orthonormalization procedure also a computation inexpensive convergence criterion is proposed for the automatic determination of the number of basis vectors required to approximate the solution within a specified level of accuracy several example problems are illustrated to verify the accuracy and efficiency of the proposed reduction method

a bott periodicity theorem for infinite dimensional euclidean space abstract we formulate and prove an equivariant bott periodicity theorem for infinite dimensional euclidean vector spaces the main features of our argument are i the construction of a non commutative c algebra to play the role of the algebra of functions on infinite dimensional euclidean space and ii the construction of a certain index one elliptic partial differential operator which provides the basis for an inverse to the bott periodicity map these tools have applications to index theory and the novikov conjecture notably a proof of the novikov conjecture for amenable groups the applications will be considered elsewhere

reduced basis approximations of structural displacements for optimaldesign a new solution procedure for improved approximate reanalyses of structures using results of a single precise analysis is presented the proposed procedure is based on combining the computed terms of a series expansion used as high quality basis vectors with coefficients of a reduced basis expression the latter coefficients can readily be determined for each trial design by various criteria the results are measured by the errors in satisfying the analysis equations and are compared in terms of the quality of the approximations the proposed approach is suitable for various types of design variables and can be used with a general finite element model a general reanalysis procedure is introduced and the physical significance of first order approximations is demonstrated several numerical examples illustrate the effectiveness of the solution process it is shown that high quality approximations of displacements can be obtained with a small computational effort for very large changes in the design variables

geometric invariants of the quantum hall effect we study the two dimensional hall effect with a random potential the hall conductivity is identified as a geometric invariant associated with an algebra of observables using the pairing betweenk theory and cyclic cohomology theory we identify this geometric invariant with a topological index thereby giving the hall conductivity a new interpretation

representable theory of smooth crossed products by and we show that the thom isomorphism and the pimsner voiculescu exact sequence both hold for smooth crossed products of frechet algebras by r and z respectively we also obtain the same results for li crossed products of banach algebras by r and z

unbounded derivations in at algebras letabe a simple unital at algebra of real rank zero such that it has a unique tracial state andk1 a is neither 0 norz for each hom k1 a r with dense range inrwe construct a closed derivation inawhich generates a one parameter automorphism group ofasuch that u u 2 i u for any unitaryu d furthermore we construct such an with the rohlin property which is defined in kishimoto comm math phys 179 1996 599 622 in this case the crossed producta ris a simple at algebra of real rank zero as an application we obtain that for such ac algebraathe kernel of the natural homomorphism of the groupinn a of approximately inner automorphisms intoext k1 a k0 a ext k0 a k1 a is the group hinn a of automorphisms homotopic to inner automorphisms combining with the result of kishimoto and kumjian trans amer math soc to appear inn a hinn a is isomorphic to the above direct sum as another application of the construction of derivations we show that ifais ac algebra of the above type and hinn a has the rohlin property and comes from hom k1 a r with dense range as in kishimoto and kumjian preprint then the crossed producta zis again of the same type in particulara zis an at algebra the other properties are known from kishimoto j operator theory40 1998

an analogue of the thom isomorphism for crossed products of a c algebra by an action of r abstract in this paper we show the existence and uniqueness of a natural isomorphism oj of kj a with kj 1 a r j z 2 where a r is a c dynamical r system k is the functor of topological k theory and a r is the crossed product of a by the action of r the pimsner voiculescu exact sequence is obtained as a corollary we show that given an invariant trace on a with dual trace gt one has gto 1 u 1 2 i u u for any unitary u in the domain of the derivation of a associated to the action finally we show that the crossed product of c s3 continuous functions on the 3 sphere by a minimal diffeomorphism is a simple c algebra with no nontrivial idempotent

k theory for the leaf space of foliations by reeb components abstract the k theory of the c algebra c v f associated to c foliations v f of a manifold v in the simplest non trivial case i e dim v 2 is studied since the case of the kronecker foliation was settled by pimsner and voiculescu j operator theory 4 1980 93 118 the remaining problem deals with foliations by reeb components the k theory of c v f for the reeb foliation of s3 is also computed in these cases the c algebra c v f is obtained from simpler c algebras by means of pullback diagrams and short exact sequences the k groups k c v f are computed using the associated mayer vietoris and six term exact sequences the results characterize the c algebra of the reeb foliation of t 2 uniquely as an extension of c s1 by c s1 for the foliations of t 2 it is found that the k groups count the number of reeb components separated by stable compact leaves a c foliation of t 2 such that k c t 2 f has infinite rank is also constructed finally it is proved by explicit calculation using m penington k theory and c algebras of lie groups and foliations d phil thesis oxford 1983 that the natural map k bg k c v f is an isomorphism for foliations by reeb components of t 2 and s3 in particular this proves the baum connes conjecture p baum and a connes geometric k theory for lie groups preprint 1982 a connes proc symp pure math 38 1982 521 628 when v t 2

mutual embeddability equivalence relation for rotation algebras abstract mutual embeddability is an equivalence relation in c algebras in this paper we characterize the classification of rotation algebras in the sense of mutual embeddability

crossed products by minimal homeomorphisms let x be an infinite compact metric space with finite covering dimension and let h x x be a minimal homeomorphism we show that the associated crossed product c algebra a 1 4 c z x h has tracial rank zero whenever the image of k0 a in a t a is dense as a consequence we show that these crossed product c algebras are in fact simple ah algebras with real rank zero when x is connected and h is further assumed to be uniquely ergodic then the above happens if and only if the rotation number associated to h has irrational values by applying the classification result for nuclear simple c algebras with tracial rank zero we show that two such dynamical systems have isomorphic crossed products if and only if they have isomorphic scaled ordered k theory

crossed products of the cantor set by free minimal actions of ℤd let d be a positive integer let x be the cantor set and let d act freely and minimally on x we prove that the crossed product c d x has stable rank one real rank zero and cancellation of projections and that the order on k0 c d x is determined by traces we obtain the same conclusion for the c algebras of various kinds of aperiodic tilings

connes s trace theorem for curved noncommutative tori application to scalar curvature in this paper we prove a version of connes s trace theorem for noncommutative tori of any dimension n 2 this allows us to recover and improve earlier versions of this result in dimensions n 2 and n 4 by fathizadeh and khalkhali we also recover connes s integration formula for flat noncommutative tori of mcdonald sukochev and zanin as a further application we prove a curved version of this integration formula in terms of the laplace beltrami operator defined by an arbitrary riemannian metric for the class of the so called self compatible riemannian metrics including the conformally flat metrics of connes and tretkoff this shows that connes s noncommutative integral allows us to recover the riemannian density this exhibits a neat link between this notion of noncommutative integral and noncommutative measure theory in the sense of operator algebras as an application of these results we set up a natural notion of scalar curvature for curved noncommutative tori

on the structure of twisted group c algebras we first give general structural results for the twisted group algebras c g of a locally compact group g with large abelian subgroups in particular we use a theorem of williams to realise c g as the sections of a c bundle whose fibres are twisted group algebras of smaller groups and then give criteria for the simplicity of these algebras next we use a device of rosenberg to show that when is a discrete subgroup of a solvable lie group g the k groups k c are isomorphic to certain twisted k groups k g of the homogeneous space g and we discuss how the twisting class h 3 g z depends on the cocycle

the density of states for almost periodic schrodinger operators and the frequency module a counter example we exhibit an example of a one dimensional discrete schrodinger operator with almost periodic potential for which the steps of the density of states do not belong to the frequency module this example is suggested by thek theory 3

the rotation number for finite difference operators and its properties we discuss a rotation number for second order finite difference operators ifk denotes the integrated density of states thenk 2 for almost periodic operators k is proved to lie in the frequency module whenever is outside the spectrum this yields a labelling of the gaps of the spectrum

the implementation of a vector valued rational approximate method in structural reanalysis problems an efficient implementation of a vector valued rational approximation in structural reanalysis problems is presented the implementation is based on combining the power series expansion with a rational approximation the main objectives are 1 to preserve the ease of implementation of the local approximation and 2 to improve significantly the quality of the results such that the method can be used in problems with very large changes in the design variables including geometrical changes and elimination of members the method is suitable for different types of structures such as trusses frames grillages etc the computational cost is significantly reduced numerical examples have shown that excellent results can be achieved for very large changes in cross sections and in structural geometry

classifying crossed product c algebras i combine recent results in the structure theory of nuclear rm c algebras and in topological dynamics to classify certain types of crossed products in terms of their elliott invariants in particular transformation group rm c algebras associated to free minimal bbb z d actions on the cantor set with compact space of ergodic measures are classified by their ordered rm k theory in fact the respective statement holds for finite dimensional compact metrizable spaces provided that projections of the crossed products separate tracial states moreover rm c algebras associated to certain minimal homeomorphisms of spheres s 2n 1 are only determined by their spaces of invariant borel probability measures without a condition on the space of ergodic measures finally i show that for a large collection of classifiable rm c algebras crossed products by bbb z d actions are generically again classifiable

the method of moving asymptotes a new method for structural optimization a new method for non linear programming in general and structural optimization in particular is presented in each step of the iterative process a strictly convex approximating subproblem is generated and solved the generation of these subproblems is controlled by so called moving asymptotes which may both stabilize and speed up the convergence of the general process

cancellation and stable rank for direct limits of recursive subhomogeneous algebras we prove the following results for a unital simple direct limit of recursive subhomogeneous algebras with no dimension growth 1 2 the projections in satisfy cancellation if then 3 satisfies blackadar s second fundamental comparability question if are projections such that for all normalized traces on then 4 is unperforated for the strict order if and there is such that then the last three of these results hold under certain weaker dimension growth conditions and without assuming simplicity we use these results to obtain previously unknown information on the ordered k theory of the crossed product obtained from a minimal homeomorphism of a finite dimensional infinite compact metric space specifically is unperforated for the strict order and satisfies the following k theoretic version of blackadar s second fundamental comparability question if satisfies for all normalized traces on then there is a projection such that

the classification of rokhlin flows on mathrm c c algebras we study flows on mathrm c algebras with the rokhlin property we show that every kirchberg algebra carries a unique rokhlin flow up to cocycle conjugacy which confirms a long standing conjecture of kishimoto we moreover present a classification theory for rokhlin flows on mathrm c algebras satisfying certain technical properties which hold for many mathrm c algebras covered by the elliott program as a consequence we obtain the following further classification theorems for rokhlin flows firstly we extend the statement of kishimoto s conjecture to the non simple case up to cocycle conjugacy a rokhlin flow on a separable nuclear mathcal o infty absorbing mathrm c algebra is uniquely determined by its induced action on the prime ideal space secondly we give a complete classification of rokhlin flows on simple classifiable kk contractible mathrm c algebras two rokhlin flows on such a mathrm c algebra are cocycle conjugate if and only if their induced actions on the cone of lower semicontinuous traces are affinely conjugate