the novikov conjecture for algebraic k theory of the group algebra over the ring of schatten class operators abstract in this paper we prove the algebraic k theory novikov conjecture for group algebras over the ring of schatten class operators the main technical tool in the proof is an explicit construction of the connes chern character

on degree constrained analytic interpolation with interpolation points close to the boundary in the recent article a theory for complexity constrained interpolation of contractive functions is developed in particular it is shown that any such interpolant may be obtained as the unique minimizer of a convex weighted entropy gain in this technical note we study this optimization problem in detail and describe how the minimizer depends on weight selection and on interpolation conditions we first show that if for a sequence of interpolants the values of the entropy gain of the interpolants converge to the optimum then the interpolants converge in h 2 but not in h infin this result is then used to describe the asymptotic behavior of the interpolant as an interpolation point approaches the boundary of the domain of analyticity for loop shaping to specifications in control design it might at first seem natural to place strategically additional interpolation points close to the boundary however our results indicate that such a strategy will have little effect on the shape another consequence of our results relates to model reduction based on minimum entropy principles where one should avoid placing interpolation points too close to the boundary

the borel conjecture for hyperbolic and cat 0 groups we prove the borel conjecture for a class of groups containing word hyperbolic groups and groups acting properly isometrically and cocompactly on a finite dimensional cat 0 space

operator norm localization for linear groups and its application to k theory abstract we prove the operator norm localization property for linear groups as an application we prove the coarse novikov conjecture for box spaces of a linear group

the integral k theoretic novikov conjecture for groups with finite asymptotic dimension the integral assembly map in algebraic k theory is split injective for any geometrically finite discrete group with finite asymptotic dimension

the coarse baum connes conjecture for spaces which admit a uniform embedding into hilbert space corollary 1 2 let be a finitely generated group if as a metric space with a word length metric admits a uniform embedding into hilbert space and its classifying space b has the homotopy type of a finite cw complex then the strong novikov conjecture holds for i e the index map from k b to k c r is injective corollary 1 2 follows from theorem 1 1 and the descent principle 23 by index theory the strong novikov conjecture implies the novikov conjecture on the homotopy invariance of higher signatures cf 8 for an excellent

large riemannian manifolds which are flexible for each k z we construct a uniformly contractible metric on euclidean space which is not mod k hypereuclidean we also construct a pair of uniformly contractible riemannian metrics on r n n 11 so that the resulting manifolds z and z are bounded homotopy equivalent by a homotopy equivalence which is not boundedly close to a homeomorphism we show that for these spaces the c algebra assembly map k lf z k c z from locally finite k homology to the k theory of the bounded propagation algebra is not a monomorphism this shows that an integral version of the coarse novikov conjecture fails for real operator algebras if we allow a single cone like singularity a similar construction yields a counterexample for complex c algebras

the inverse problem of analytic interpolation with degree constraint and weight selection for control synthesis the minimizers of certain weighted entropy functionals are the solutions to an analytic interpolation problem with a degree constraint and all solutions to this interpolation problem arise in this way by a suitable choice of weights selecting appropriate weights is pertinent to feedback control synthesis where interpolants represent closed loop transfer functions in this paper we consider the correspondence between weights and interpolants in order to systematize feedback control synthesis with a constraint on the degree there are two basic issues that we address we first characterize admissible shapes of minimizers by studying the corresponding inverse problem and then we develop effective ways of shaping minimizers via suitable choices of weights this leads to a new procedure for feedback control synthesis

an interpolation theory approach to h controller degree bounds abstract we derive upper bounds for the mcmillan degree of all h optimal controllers associated with design problems which may be embedded in a certain generalized regular configuration our analysis is confined to problems of the first kind which are characterized by the assumption that both p12 s and p21 s are square but not necessarily of the same size this paper which uses interpolation theory complements a previous paper which addresses the same problem through an approach based on approximation theory we demonstrate that the interpolation theory approach is more direct and circumvents a number of the technical difficulties in the previous method the final outcome is a much shorter proof as a by product we achieve a new result on the degree of an optimal solution of the matrix nevanlinna pick problem

constructions preserving hilbert space uniform embeddability of discrete groups uniform embeddability in a hilbert space introduced by gromov is a geometric property of metric spaces as applied to countable discrete groups it has important consequences for the novikov conjecture exactness introduced and studied extensively by kirchberg and wassermann is a functional analytic property of locally compact groups recently it has become apparent that as properties of countable discrete groups uniform embeddability and exactness are closely related we further develop the parallel between these classes by proving that the class of uniformly embeddable groups shares a number of permanence properties with the class of exact groups in particular we prove that it is closed under direct and free products with and without amalgam inductive limits and certain extensions

sensitivity shaping with degree constraint by nonlinear least squares optimization this paper presents a new approach to shaping of the frequency response of the sensitivity function in this approach a desired frequency response is assumed to be specified at a finite number of frequency points a sensitivity shaping problem is formulated as an approximation problem to the desired frequency response with a function in a class of sensitivity functions with a degree bound the sensitivity shaping problem is reduced to a finite dimensional constrained nonlinear least squares optimization problem to solve the optimization problem numerically standard algorithms for an unconstrained version of nonlinear least squares problems are modified to incorporate the constraint numerical examples illustrate how these design parameters are tuned in an intuitive manner as well as how the design proceeds in actual control problems

a new approach to spectral estimation a tunable high resolution spectral estimator traditional maximum entropy spectral estimation determines a power spectrum from covariance estimates here we present a new approach to spectral estimation which is based on the use of filter banks as a means of obtaining spectral interpolation data such data replaces standard covariance estimates a computational procedure for obtaining suitable pole zero arma models from such data is presented the choice of the zeros ma part of the model is completely arbitrary by suitable choices of filter bank poles and spectral zeros the estimator can be tuned to exhibit high resolution in targeted regions of the spectrum

color edge extraction using orthogonal polynomials based zero crossings scheme abstract a new color edge extraction scheme based on identifying the zero crossings in the second directional derivatives in terms of orthogonal polynomials has been proposed in this paper the proposed scheme supports a complete set of finite difference operators which are easily configured from orthogonal polynomials a color image region in presence of gaussian noise is represented as a linear combination of these operators towards color edges and noise statistical hypothesis testing procedures are employed to separate out the responses towards noise from those towards color edges the usage of the proposed framework is highlighted in devising a color edge detector based on identifying the zero crossings in the second directional derivatives of the color image region the proposed color edge extraction scheme is also compared with two existing color edge extraction schemes

groups generated by reflections and aspherical manifolds not covered by euclidean space a coxeter system r v is a group r a coxeter group together with a set of generators v such that each element of v has order two and such that all relations in r are consequences of relations of the form vw m v w 1 where v w e v and m v w denotes the order of vw the m v w s which are positive integers or oo obviously determine the coxeter system up to isomorphism the list of finite coxeter groups is short and well known however general coxeter groups are much more flexible in fact the set v and the m v w s can be specified arbitrarily at least if v is finite subject only to the conditions m v v 1 and m v w m w v 2 if v w suppose that r v is a coxeter system that x is a hausdorff space and that xv v v is a locally finite family of closed subspaces indexed by v the xv are called the panels of x 2 there is a classical method for constructing a transformation group from these data for each x e x let v x denote the set of v in v such that x e xv for each subset s of v let rs be the subgroup generated by s and let xs be the face of x defined by

weight selection in feedback design with degree constraints we present an approach for feedback design which is based on recent developments in analytic interpolation with a degree constraint performance is cast as an interpolation problem with bounded analytic functions minimizers of a certain weighted entropy functional provide interpolants having degree less than the number of constraints the choice of weight parameterizes all such bounded degree solutions however the relationship between the weights and the shape of corresponding transfer functions is not direct thus in this paper we develop a formalism that guides weight selection

matrix valued nevanlinna pick interpolation with complexity constraint an optimization approach over the last several years a new theory of nevanlinna pick interpolation with complexity constraint has been developed for scalar interpolants in this paper we generalize this theory to the matrix valued case also allowing for multiple interpolation points we parameterize a class of interpolants consisting of most interpolants of no higher degree than the central solution in terms of spectral zeros this is a complete parameterization and for each choice of interpolant we provide a convex optimization problem for determining it this is derived in the context of duality theory of mathematical programming to solve the convex optimization problem we employ a homotopy continuation technique previously developed for the scalar case these results can be applied to many classes of engineering problems and to illustrate this we provide some examples in particular we apply our method to a benchmark problem in multivariate robust control by constructing a controller satisfying all design specifications but having only half the mcmillan degree of conventional h sup spl infin controllers we demonstrate the advantage of the proposed method

a generalized entropy criterion for nevanlinna pick interpolation with degree constraint we present a generalized entropy criterion for solving the rational nevanlinna pick problem for n 1 interpolating conditions and the degree of interpolants bounded by n the primal problem of maximizing this entropy gain has a very well behaved dual problem this dual is a convex optimization problem in a finite dimensional space and gives rise to an algorithm for finding all interpolants which are positive real and rational of degree at most n the criterion requires a selection of a monic schur polynomial of degree n it follows that this class of monic polynomials completely parameterizes all such rational interpolants and it therefore provides a set of design parameters for specifying such interpolants the algorithm is implemented in a state space form and applied to several illustrative problems in systems and control namely sensitivity minimization maximal power transfer and spectral estimation

on the k theory of groups with finite asymptotic dimension it is proved that the assembly maps in algebraic kand l theory with respect to the family of finite subgroups is injective for groups with finite asymptotic dimension that admit a finite model for e the result also applies to certain groups that admit only a finite dimensional model for e in particular it applies to discrete subgroups of virtually connected lie groups

a notion of geometric complexity and its application to topological rigidity we introduce a geometric invariant called finite decomposition complexity fdc to study topological rigidity of manifolds we prove for instance that if the fundamental group of a compact aspherical manifold m has fdc and if n is homotopy equivalent to m then m n is homeomorphic to n n for n large enough this statement is known as the stable borel conjecture on the other hand we show that the class of fdc groups includes all countable subgroups of gl n k for any field k

a shaping limitation of rational sensitivity functions with a degree constraint this note concerns a certain shaping limitation of sensitivity functions the focus is on a frequency wise infimum of gains of rational sensitivity functions with a degree constraint an explicit infimum is derived for a special case the result is useful for determining the inability of sensitivity functions of low degrees to achieve a specification in the frequency domain and thus for motivating the use of higher degree sensitivity functions to fulfill the specification

a hurewicz type theorem for asymptotic dimension and applications to geometric group theory we prove an asymptotic analog of the classical hurewicz theorem on mappings that lower dimension this theorem allows us to find sharp upper bound estimates for the asymptotic dimension of groups acting on finite dimensional metric spaces and allows us to prove a useful extension theorem for asymptotic dimension as applications we find upper bound estimates for the asymptotic dimension of nilpotent and polycyclic groups in terms of their hirsch length we are also able to improve the known upper bounds on the asymptotic dimension of fundamental groups of complexes of groups amalgamated free products and the hyperbolization of metric spaces possessing the higson property

closed loop shaping based on nevanlinna pick interpolation with a degree bound this note presents a novel method for shaping the frequency response of a single input single output closed loop system based on the theory of nevanlinna pick interpolation with degree constraint the method imposes a degree bound on the closed loop transfer function and searches for a function with a desired frequency response numerical examples illustrate the potential of the method in designing controllers with lower degrees than the ones obtained by conventional h sup spl infin controller design methods with weighting functions

a robust solver using a continuation method for nevanlinna pick interpolation with degree constraint this note modifies a previous algorithm for solving a certain convex optimization problem introduced by byrnes georgiou and lindquist to determine any nevanlinna pick interpolant satisfying degree constraint the modified algorithm is based on a continuation method with predictor corrector steps and it turns out to be quite efficient and numerically robust

important moments in systems and control the moment problem matured from its various special forms in the late 19th and early 20th centuries to a general class of problems that continues to exert profound influence on the development of analysis and its applications to a wide variety of fields in particular the theory of systems and control is no exception where the applications have historically been to circuit theory optimal control robust control signal processing spectral estimation stochastic realization theory and the use of the moments of a probability density many of these applications are also still works in progress in this paper we consider the generalized moment problem expressed in terms of a basis of a finite dimensional subspace p of the banach space c a b and a positive sequence c but with a new wrinkle inspired by the applications to systems and control we seek to parameterize solutions which are positive rational measures in a suitably generalized sense our parameterization is given in terms of smooth objects in particular the desired solution space arises naturally as a manifold which can be shown to be diffeomorphic to a euclidean space and which is the domain of some canonically defined functions the analysis of these functions and related maps yields interesting corollaries for the moment problem and its applications which we compare to those in the recent literature and which play a crucial role in part of our proof

the novikov conjecture for groups with finite asymptotic dimension recall that the asymptotic dimension is a coarse geometric analogue of the covering dimension in topology page 28 14 more precisely the asymptotic dimension for a metric space is the smallest integer n such that for any r 0 there exists a uniformly bounded cover c ui iei of the metric space for which the r multiplicity of c is at most n 1 i e no ball of radius r in the metric space intersects more than n 1 members of c 14 the class of finitely generated discrete groups with finite asymptotic dimension is hereditary in the sense that if a finitely generated group has finite asymptotic dimension as metric space with a word length metric then its finitely generated subgroups also have finite asymptotic dimension as metric spaces with word length metrics cf section 6 this together with a result of gromov in 14 implies that finitely generated subgroups of gromov s hyperbolic groups have finite asymptotic dimension currently no example of a finitely generated group with infinite asymptotic dimension and finite classifying space is known it should also be noted that two different definitions of asymptotic dimension