on the group of homotopy equivalences of a manifold we consider the group of homotopy equivalences s m of a simply connected manifold m which is part of the fundamental extension of groups due to barcus barratt we show that the kernel of this extension is always a finite group and we compute this kernel for various examples this leads to computations of the group s m for special manifolds m for example if m is a connected sum of products s x sm of spheres in particular the group s s x s is determined completely also the connection of s m with the group of isotopy classes of diffeomorphisms of m is studied the group x of homotopy equivalences of a space x is the set of homotopy classes of homotopy equivalences x x the group structure is induced by mapcomposition the group x i e the group of automorphisms of the homotopy type of x can be regarded as the homotopy symmetry group of the space x in the literature there has been a lot of interest in the computation of such groups compare for example the excellent survey article of m arkowitz 2 this paper is concerned with the structure of m in case m is a closed compact oriented manifold or more generally a poincare complex the computation of this group is an important step for the diffeomorphism classification of manifolds by surgery 40 the group m is also important for cooke s theory 17 of replacing homotopy actions by actions for a differential manifold m the group m is comparable with the group hodiff m of isotopy classes of diffeomorphisms of m in fact via the j homomorphism there is a striking similarity between these groups as is shown in 10 below still there is little known on the group m in the literature only very specific examples are computed see 2 this paper contains on the one hand general results on the structure of the group m see 1 5 on the other hand our methods are used for explicit computations see 6 10 let e be a small open cell of the simply connected m dimensional manifold m and let m m e be the complement the inclusion f sm 1 c m of the boundary is the attaching map for m we consider the fundamental extension of groups due to barcus barratt 42 m i m m 3 s m f here m i m is the subgroup of m consisting of all elements which can be represented by an orientation preserving map m m under m this group is received by the editors august 17 1994 1991 mathematics subject classification primary 55010 57050 1996 american mathematical society 4737 this content downloaded from 157 55 39 197 on wed 26 apr 2017 17 53 44 utc all use subject to http about jstor org terms

algebraic k theory of group rings and the cyclotomic trace map we prove that the farrell jones assembly map for connective alge braic k theory is rationally injective under mild homological niteness condi tions on the group and assuming that a weak version of the leopoldt schneider conjecture holds for cyclotomic elds this generalizes a result of bokstedt hsiang and madsen and leads to a concrete description of a large direct summand of kn z g z q in terms of group homology in many cases the number theoretic conjectures are true so we obtain rational injectivity results about assembly maps in particular for whitehead groups under niteness assumptions on the group only the proof uses the cyclotomic trace map to topological cyclic homology bokstedt hsiang madsen s functor c and new general injectivity results about the assembly maps for topological hochschild homology and c

ends of maps ii versions of the finiteness obstruction and simple homotopy theory within e overx are developed this provides a setting for obstructions to the map analogs of the end ands cobordism theorems for manifolds these are applied to study equivariant mapping cylinder neighborhoods in topological group actions triangulations of locally triangulable spaces and block bundle structures on approximate fibrations

the stable mapping class group of simply connected 4 manifolds we consider mapping class groups m 0diff m fix m of smooth compact simply connected oriented 4 manifolds m bounded by a collection of 3 spheres we show that if m contains cp 2 or cp 2 as a connected summand then all dehn twists around 3 spheres are trivial and furthermore m is independent of the number of boundary components by repackaging classical results in surgery and handlebody theory from wall kreck and quinn we show that the natural homomor phism from the mapping class group to the group of automorphisms of the intersection form becomes an isomorphism after stabilization with respect to connected sum with cp 2 cp 2 we next consider the 3 1 dimensional cobordism 2 category c of 3 spheres 4 manifolds as above and enriched with isotopy classes of diffeomorphisms as 2 morphisms we identify the homotopy type of the classifying space of this cat egory as the hermitian algebraic k theory of the integers we also comment on versions of these results for simply connected spin 4 manifolds finally we observe that a related 4 manifold operad detects infinite loop spaces

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

rescaling entropy and divergence rates based on rescaling by some suitable sequence instead of the number of time units the usual notion of divergence rate is here extended to define and determine meaningful generalized divergence rates rescaling entropy rates appears as a special case suitable rescaling is naturally induced by the asymptotic behavior of the marginal divergences closed form formulas are obtained as soon as the marginal divergences behave like powers of some analytical functions a wide class of countable markov chains is proved to satisfy this property most divergence and entropy functionals defined in the literature are concerned e g the classical shannon kullback leibler renyi and tsallis for illustration purposes ferreri or basu harris hjort jones among others are also considered

analyticity of entropy rate of hidden markov chains with continuous alphabet we first prove that under certain mild assumptions the entropy rate of a hidden markov chain observed when passing a finite state stationary markov chain through a discrete time continuous output channel is analytic with respect to the input markov chain parameters we then further prove under strengthened assumptions on the channel that the entropy rate is jointly analytic as a function of both the input markov chain parameters and the channel parameters in particular the main theorems establish the analyticity of the entropy rate for two representative channels 1 cauchy and 2 gaussian the analyticity results obtained are expected to be helpful in computation estimation of entropy rate of hidden markov chains and capacity of finite state channels with continuous output alphabet

non negative pinching moduli spaces and bundles with infinitely many souls we show that in each dimension n 10 there exist infinite sequences of homotopy equivalent but mutually non homeomorphic closed simply connected riemannian n manifolds with 0 sec 1 positive ricci curvature and uniformly bounded diameter we also construct open manifolds of fixed diffeomorphism type which admit infinitely many complete nonnegatively pinched metrics with souls of bounded diameter such that the souls are mutually non homeomorphic finally we construct examples of noncompact manifolds whose moduli spaces of complete metrics with sec 0 have infinitely many connected components

obstructions to smooth group actions on 4 manifolds from families seiberg witten theory abstract let x be a smooth compact oriented 4 manifold building upon work of li liu ruberman nakamura and konno we consider a families version of seiberg witten theory and obtain obstructions to the existence of certain group actions on x by diffeomorphisms the obstructions show that certain group actions on h 2 x z preserving the intersection form can not be lifted to an action of the same group on x by diffeomorphisms using our obstructions we construct numerous examples of group actions which can be realised continuously but can not be realised smoothly for any differentiable structure for example we construct compact simply connected 4 manifolds x and involutions f h 2 x z h 2 x z such that f can be realised by a continuous involution on x or by a diffeomorphism but not by an involutive diffeomorphism for any smooth structure on x

log determinant relaxation for approximate inference in discrete markov random fields graphical models are well suited to capture the complex and non gaussian statistical dependencies that arise in many real world signals a fundamental problem common to any signal processing application of a graphical model is that of computing approximate marginal probabilities over subsets of nodes this paper proposes a novel method applicable to discrete valued markov random fields mrfs on arbitrary graphs for approximately solving this marginalization problem the foundation of our method is a reformulation of the marginalization problem as the solution of a low dimensional convex optimization problem over the marginal polytope exactly solving this problem for general graphs is intractable for binary markov random fields we describe how to relax it by using a gaussian bound on the discrete entropy and a semidefinite outer bound on the marginal polytope this combination leads to a log determinant maximization problem that can be solved efficiently by interior point methods thereby providing approximations to the exact marginals we show how a slightly weakened log determinant relaxation can be solved even more efficiently by a dual reformulation when applied to denoising problems in a coupled mixture of gaussian model defined on a binary mrf with cycles we find that the performance of this log determinant relaxation is comparable or superior to the widely used sum product algorithm over a range of experimental conditions

generalized lagrangian duals and sums of squares relaxations of sparse polynomial optimization problems sequences of generalized lagrangian duals and their sums of squares sos of polynomials relaxations for a polynomial optimization problem pop are introduced the sparsity of polynomials in the pop is used to reduce the sizes of the lagrangian duals and their sos relaxations it is proved that the optimal values of the lagrangian duals in the sequence converge to the optimal value of the pop using a method from the penalty function approach the sequence of sos relaxations is transformed into a sequence of semidefinite programing sdp relaxations of the pop which correspond to duals of modification and generalization of sdp relaxations given by lasserre for the pop

constructing approximate fibrations in this paper two results concerning the construction of approximate fibrations are established the first shows that there are approximate fibrations p m s2 which are homotopic to bundle maps but which cannot be approximated by bundle maps here m can be a compact q manifold or some topological n manifold n 5 the second shows how to construct approximate fibrations p m b whose fibers do not have finite homotopy type for any b of euler characteristic zero here m can be a compact q manifold and b only has to be an anr or m can be an n manifold n 6 and b must then also be a topological manifold

the stable topological hyperbolic space form problem for complete manifolds of finite volume about fifteen years ago a borel posed the following conjecture let m be a closed aspherical manifold i e rcim o for i 1 if g n m is a homotopy equivalence where n is another manifold then g is homotopic to a homeomorphism a stable version of this conjecture for m r a closed nonpositively curved manifold was verified in 4 corollary b namely if m is a closed non positively curved manifold and g n m is a homotopy equivalence where n is a manifold then

approximation of approximate fibrations by bundle maps in this paper we give some conditions under which approximate fibrations can be approximated by locally trivial bundle maps

semidefinite programming for min max problems and games we consider two min max problems 1 minimizing the supremum of finitely many rational functions over a compact basic semi algebraic set and 2 solving a 2 player zero sum polynomial game in randomized strategies with compact basic semi algebraic sets of pure strategies in both problems the optimal value can be approximated by solving a hierarchy of semidefinite relaxations in the spirit of the moment approach developed in lasserre siam j optim 11 796 817 2001 math program b 112 65 92 2008 this provides a unified approach and a class of algorithms to compute nash equilibria and min max strategies of several static and dynamic games each semidefinite relaxation can be solved in time which is polynomial in its input size and practice on a sample of experiments reveals that few relaxations are needed for a good approximation and sometimes even for finite convergence a behavior similar to what was observed in polynomial optimization

cones and gauges in complex spaces spectral gaps and complex perron frobenius theory we introduce complex cones and associated projective gauges generalizing a real birkhoff cone and its hilbert metric to complex vector spaces we deduce a variety of spectral gap theorems in complex banach spaces we prove a dominated complex cone contraction theorem and use it to extend the classical perron frobenius theorem to complex matrices jentzsch s theorem to complex integral operators a krein rutman theorem to compact and quasi compact complex operators and a ruelle perron frobenius theorem to complex transfer operators in dynamical systems in the simplest case of a complex n by n matrix a m n we have the following statement suppose that 0 c is such that im a ij mm c re a ij mn for all indices then a has a spectral gap

on the higher whitehead groups of a bieberbach group let t be a bieberbach group i e the fundamental group of a compact flat riemannian manifold in this paper we show that if p 2 is a prime then the p torsion subgroup of wh r vanishes for 0 i 2p 2 where wh r is the th higher whitehead group of t the proof involves farrell and hsiang s structure theorem for bieberbach groups parametrized surgery pseudoisotopy and waldhausen s algebraic if theory of spaces

negatively curved manifolds with exotic smooth structures let m denote a compact real hyperbolic manifold with dimension m 5 and sectional curvature k 1 and let y be an exotic sphere of dimension m given any small number j 0 we show that there is a finite covering space m of m satisfying the following properties the connected sum m x is not diffeomorphic to m but it is homeomorphic to m m x supports a riemannian metric having all of its sectional curvature values in the interval 1 j 1 5 thus there are compact riemannian manifolds of strictly negative sectional curvature which are not diffeomorphic but whose fundamental groups are isomorphic this answers problem 12 of the list com piled by yau 22 i e it gives counterexamples to the lawson yau conjecture note that mostow s rigidity theorem 17 implies that m x does not support a riemannian metric whose sectional curvature is identically 1 in fact it is not diffeomorphic to any locally symmetric space thus the manifold m x supports a riemannian metric with sectional curvature arbitrarily close to 1 but it does not support a riemannian metric whose sectional curvature is iden tically 1 more complicated examples of manifolds satisfying the properties of the previous sentence were first constructed by gromov and thurston 11 department of mathematics columbia university new york new york 10027 department of mathematics suny at stony brook stony brook new york 11794 this content downloaded from 207 46 13 71 on wed 27 apr 2016 07 05 48 utc all use subject to http about jstor org terms

a trichotomy theorem for transformation groups of locally symmetric manifolds and topological rigidity abstract let m be a locally symmetric irreducible closed manifold of dimension 3 a result of borel 6 combined with mostow rigidity imply that there exists a finite group g g m such that any finite subgroup of homeo m is isomorphic to a subgroup of g borel 6 asked if there exist m s with g m trivial and if the number of conjugacy classes of finite subgroups of homeo m is finite we answer both questions 1 for every finite group g there exist m s with g m g and 2 the number of maximal subgroups of homeo m can be either one countably many or continuum and we determine at least for dim m 4 when each case occurs our detailed analysis of 2 also gives a complete characterization of the topological local rigidity and topological strong rigidity for dim m 4 of proper discontinuous actions of uniform lattices in semisimple lie groups on the associated symmetric spaces

topological rigidity for hyperbolic manifolds soit m une variete de riemann complete ayant une courbure sectionnelle constante 1 et un volume fini soit m sa compactification variete de gromov margulis et on suppose que la dimension de m est 5 on montre qu une equivalence d homotopie h n n m m ou n est une variete compacte est homotopique a un homeomorphisme

optimality conditions and finite convergence of lasserre s hierarchy lasserre s hierarchy is a sequence of semidefinite relaxations for solving polynomial optimization problems globally this paper studies the relationship between optimality conditions in nonlinear programming theory and finite convergence of lasserre s hierarchy our main results are i lasserre s hierarchy has finite convergence when the constraint qualification strict complementarity and second order sufficiency conditions hold at every global minimizer under the standard archimedean condition the proof uses a result of marshall on boundary hessian conditions ii these optimality conditions are all satisfied at every local minimizer if a finite set of polynomials which are in the coefficients of input polynomials do not vanish at the input data i e they hold in a zariski open set this implies that under archimedeanness lasserre s hierarchy has finite convergence generically

on the asymptotic moments and edgeworth expansions for some processes in random dynamical environment we prove that certain asymptotic moments exist for some random distance expanding dynamical systems and markov chains in random dynamical environment and compute them in terms of the derivatives at 0 of an appropriate pressure function it will follow that these moments satisfy the relations that the asymptotic moments gamma k lim n rightarrow infty n frac k 2 mathbb e sum i 1 n x i k of sums of independent and identically distributed centered random variables satisfy under certain mixing conditions we will also estimate the convergence rate towards these limits the arguments in the proof of these results yield that the partial sums generated by the random ruelle perron frobenius triplets and all of their parametric derivatives considered as functions on the base corresponding to appropriate random transfer or markov operators satisfy several probabilistic limit theorems such as the central limit theorem we will also obtain certain edgeworth asymptotic expansions related to the central limit theorem for such processes our proofs rely on a parametric random complex ruelle perron frobenius theorem which replaces some of the spectral techniques used in literature in order to obtain limit theorems for deterministic dynamical systems and markov chains

tight relaxations for polynomial optimization and lagrange multiplier expressions this paper proposes tight semidefinite relaxations for polynomial optimization the optimality conditions are investigated we show that generally lagrange multipliers can be expressed as polynomial functions in decision variables over the set of critical points the polynomial expressions are determined by linear equations based on these expressions new lasserre type semidefinite relaxations are constructed for solving the polynomial optimization we show that the hierarchy of new relaxations has finite convergence or equivalently the new relaxations are tight for a finite relaxation order

differentiability of thermodynamical quantities in non uniformly expanding dynamics abstract in this paper we study the ergodic theory of a robust non uniformly expanding maps where no markov assumption is required we prove that the topological pressure is differentiable as a function of the dynamics and analytic with respect to the potential moreover we not only prove the continuity of the equilibrium states and their metric entropy as well as the differentiability of the maximal entropy measure and extremal lyapunov exponents with respect to the dynamics we also prove a local large deviations principle and central limit theorem and show that the rate function mean and variance vary continuously with respect to observables potentials and dynamics finally we show that the correlation function associated to the maximal entropy measure is differentiable with respect to the dynamics and it is c 1 convergent to zero in addition precise formulas for the derivatives of thermodynamical quantities are given

on the diffeomorphism groups of elliptic surfaces in this paper we determine for relatively minimal elliptic surfaces with positive euler number the image of the natural representation of the group of orientation preserving self diffeomorphisms on h the second homology group reduced modulo torsion to this end we construct as many embedded spheres of square 2 such that an isometry not induced from any combination of reflections at such spheres or from complex conjugation can be shown not to be induced from some diffeomorphism at all this is done with the help of seiberg witten invariants

markov additive processes i eigenvalue properties and limit theorems on etudie d un point de vue general les proprietes d existence et de regularite des valeurs propres et des fonctions propres de l operateur de feynman kac

on the geometric and topological rigidity of hyperbolic 3 manifolds theorem 0 1 let n be a closed hyperbolic 3 manifold containing an embedded hyperbolic tube of radius log 3 2 549306 about a closed geodesic then i iff m n is a homotopy equivalence where m is an irreducible 3 manifold then f is homotopic to a homeomorphism ii if f g m s n are homotopic homeomorphisms then f is isotopic to g iii the space of hyperbolic metrics on n is path connected

the moment problem on compact semi algebraic sets in this chapter we begin the study of the multidimensional moment problem the passage to dimensions d 2 brings new difficulties and unexpected phenomena in sect 3 2 we derived solvability criteria of the moment problem on intervals in terms of positivity conditions it seems to be natural to look for similar characterizations in higher dimensions as well this leads us immediately into the realm of real algebraic geometry and to descriptions of positive polynomials on semi algebraic sets in this chapter we treat this approach for basic closed compact semi algebraic subsets of mathbb r d it turns out that for such sets there is a close interaction between the moment problem and positivstellensatze for strictly positive polynomials

sparse quasi newton updates with positive definite matrix completion quasi newton methods are powerful techniques for solving unconstrained minimization problems variable metric methods which include the bfgs and dfp methods generate dense positive definite approximations and therefore are not applicable to large scale problems to overcome this difficulty a sparse quasi newton update with positive definite matrix completion that exploits the sparsity pattern e i j nabla 2 f x ij neq 0 rm for some x in r n of the hessian is proposed the proposed method first calculates a partial approximate hessian h qn ij i j in f where f supseteq e using an existing quasi newton update formula such as the bfgs or dfp methods next a full matrix h k 1 which is a maximum determinant positive definite matrix completion of h qn ij i j in f is obtained if the sparsity pattern e or its extension f has a property related to a chordal graph then the matrix h k 1 can be expressed as products of some sparse matrices the time and space requirements of the proposed method are lower than those of the bfgs or the dfp methods in particular when the hessian matrix is tridiagonal the complexities become o n the proposed method is shown to have superlinear convergence under the usual assumptions

produits aleatoires d operateurs matrices de transfert nous etudions le comportement asymptotique de produits aleatoires d operateurs de ruelle perron frobenius nous etendons le travail de ruelle obtenu dans le cas homogene au cas aleatoire

a dynamic inequality generation scheme for polynomial programming hierarchies of semidefinite programs have been used to approximate or even solve polynomial programs this approach rapidly becomes computationally expensive and is often tractable only for problems of small size in this paper we propose a dynamic inequality generation scheme to generate valid polynomial inequalities for general polynomial programs when used iteratively this scheme improves the bounds without incurring an exponential growth in the size of the relaxation as a result the proposed scheme is in principle scalable to large general polynomial programming problems when all the variables of the problem are non negative or when all the variables are binary the general algorithm is specialized to a more efficient algorithm in the case of binary polynomial programs we show special cases for which the proposed scheme converges to the global optimal solution we also present several examples illustrating the computational behavior of the scheme and provide comparisons with lasserre s approach and for the binary linear case with the lift and project method of balas ceria and cornuejols

k theory and dynamics i un theoreme de h cobordisme avec commande feuillettee la variete associee le transfert asymptotique dynamique du flot geodesique completion de la demonstration du theoreme principal revue de la theorie de la commande structures de longues cellules minces

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

isomorphism conjectures in algebraic k theory duff in this paper we are concerned with the four functors y y3 if and which map from the category of topological spaces x to the category diff of q spectra the functor 3 or if maps the space x to the qspectrum of stable topological or smooth pseudoisotopies of x the spaces diff dif 3i x id9 x i 0 are hatcher s deloopings of 970 x yd x cf 29 the functor j maps a path connected space x to the algebraic ktheoretic s2 spectrum for the integral group ring z7r1x the spaces x x i 0 are the gersten wagoner deloopings of z o x cf 27 52 5 x is also defined on a nonpath connected space x to be x1 where xi i e i are the path components of x and ii xi indicates the direct limit of all finite products of the i xi i e i if x is path connected then the functor maps x to the l surgery classifying spaces for oriented surgery problems with fundamental group r i1x identified by their fourfold periodicity k 90 x k 4j x cf 35 39 44 47 in addition if x is not path connected then we set y77 x tiei5 fj xi where xi i e i are the path components of x results obtained by the authors over the past five years cf 15 25 and in particular 19 16 a 1 a 18 together with many earlier results and conjectures which are reviewed in 1 6 1 1 6 6 suggest that for any connected cwcomplex x and for 5 denoting any of the above functors the q spectrum x should be computable in a simple way from the values taken by 9 on covering spaces of x which have very simple fundamental groups in more detail we let 9 x denote the collection of all subgroups h c ir ix which are either finite or virtually infinite cyclic a group h is virtually infinite cyclic if there is a short exact sequence of groups 1 z h f 1 with f equal to a finite group and z equal to the infinite cyclic group for each such group we let xh x denote the covering space projection corresponding to h then the q spectrum 5 x should be computable in a simple way from the q spectra 5 xh h e 9 x the purpose of this paper is to formulate precisely a conjecture along these lines and to verify the conjecture for any x

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

a pl geometric study of algebraic k theory this paper manages to apply the farrell jones theory on algebraic k groups of closed negatively curved riemannian manifolds to gromov s hyperbolic group theory the paper reaches the conclusion that for any finite polyhedron k with negative curvature wh 1 k 0

solving moment problems by dimensional extension polynomial inequalities the second part of the paper uses the representation results of positive functionals on certain spaces of rational functions developed in the flrst part for decomposing a polynomial which is positive on such a semi algebraic set into a canonical sum of squares of rational functions times explicit multipliers let n 1 be a flxed integer due to the fact that for n 1 not every nonnegative polynomial in r n can be written as a sum of squares of polynomials see for instance 2 x6 3 the moment problems in n variables are more di cult than the classical one variable problems this very intriguing territory has been investigated by many authors see 2 7 12 and their references although characterizations for measures whose support lies in an arbitrary generally unbounded semi algebraic set do not seem to exist the present paper starts from an idea of the second author see 19 about solving moment problems by a change of basis via an embedding of r n into a submanifold of a higher dimensional euclidean space rougly speaking we prove that certain n 1 dimensional extensions of a moment sequence are naturally characterized by positivity conditions and moreover these extensions parametrize all possible solutions of the moment problem to be more speciflc let fi z r n x fi d x fi 2 z n

stable mapping class groups of 4 manifolds with boundary we give a complete algebraic description of the mapping class groups of compact simply connected 4 manifolds with boundary up to connected sum with copies of s 2 s 2

a copositive formulation for the stability number of infinite graphs in the last decade copositive formulations have been proposed for a variety of combinatorial optimization problems for example the stability number independence number in this paper we generalize this approach to infinite graphs and show that the stability number of an infinite graph is the optimal solution of some infinite dimensional copositive program for this we develop a duality theory between the primal convex cone of copositive kernels and the dual convex cone of completely positive measures we determine the extreme rays of the latter cone and we illustrate this theory with the help of the kissing number problem

locally flat 2 knots in with the same fundamental group we consider a locally flat 2 sphere in s x s2 representing a primitive homology class 4 which is referred to as a 2 knot in s x s2 representing 4 then for any given primitive class there exists a 2 knot in s2 x s2 representing 4 with simply connected complement in this paper we consider the classification of 2 knots in s2 x s2 whose complements have a fixed fundamental group we show that if the complement of a 2 knot s in s2 x s2 is simply connected then the ambient isotopy type of s is determined in the case of nontrivial 7r1 however we show that the ambient isotopy type of a 2 knot in s2 x s2 with nontrivial 71 is not always determined by 71

sparsity in sums of squares of polynomials representation of a given nonnegative multivariate polynomial in terms of a sum of squares of polynomials has become an essential subject in recent developments of sums of squares optimization and semidefinite programming sdp relaxation of polynomial optimization problems we discuss effective methods to obtain a simpler representation of a sparse polynomial as a sum of squares of sparse polynomials by eliminating redundancy

on the existence of convex decompositions of partially separable functions the concept of a partially separable functionf developed in 4 is generalized to include all functionsf that can be expressed as a finite sum of element functionsf i whose hessians have nontrivial nullspacesn i such functions can be efficiently minimized by the partitioned variable metric methods described in 5 provided that each element functionf i is convex if this condition is not satisfied we attempt toconvexify the given decomposition by shifting quadratic terms among the originalf i such that the resulting modified element functions are at least locally convex to avoid tests on the numerical value of the hessian we study the totally convex case where all locally convexf with the separability structuren i 1 have a convex decomposition it is shown that total convexity only depends on the associated linear conditions on the hessian matrix in the sparse case when eachn i is spanned by cartesian basis vectors it is shown that a sparsity pattern corresponds to a totally convex structure if and only if it allows a permuted ldlt factorization without fill in

infinite loop spaces from operads with homological stability abstract motivated by the operad built from moduli spaces of riemann surfaces we consider a general class of operads in the category of spaces that satisfy certain homological stability conditions we prove that such operads are infinite loop space operads in the sense that the group completions of their algebras are infinite loop spaces the recent strong homological stability results of galatius and randal williams for moduli spaces of even dimensional manifolds can be used to construct examples of operads with homological stability as a consequence diffeomorphism groups and mapping class groups are shown to give rise to infinite loop spaces furthermore the map to k theory defined by the action of the diffeomorphisms on the middle dimensional homology is shown to be a map of infinite loop spaces

decay of correlations dedicated to micheline ishay i would like to thank p boyland l chierchia v donnay g de martino c gole j l lebowitz m lyubich m rychlik i g schwarz s vaienti and especially g gallavotti for helpful and enlightening discussions particularly warm thanks go to n chernov for carefully reading and finding a mistake in an early version the present paper benefits from several improvements due to his sharp criticism in addition i am indebted to p collet and most of all m wojtkowski for introducing me to the magical world of cones finally i thank j milnor director of the institute for mathematical sciences at stony brook university where i was visiting during part of this work and the italian c n r gnfm for providing travel funds

partitioned variable metric updates for large structured optimization problems this paper presents a minimization method based on the idea of partitioned updating of the hessian matrix in the case where the objective function can be decomposed in a sum of convex element functions this situation occurs in a large class of practical problems including nonlinear finite elements calculations some theoretical and algorithmic properties of the update are discussed and encouraging numerical results are presented

algebraic k theory of discrete subgroups of lie groups abstract let g be a lie group with finitely many connected components and be a discrete cocompact torsion free subgroup of g we rationally calculate the algebraic k theory of the integral group ring z in terms of the homology of with trivial rational coefficients

the teichmuller space of pinched negatively curved metrics on a hyperbolic manifold is not contractible for a smooth manifold m we define the teichmuller space 2t m of all riemannian metrics on m and the teichmuller space 2t m of pinched negatively curved metrics on m where 0 oo we prove that if m is hyperbolic the natural inclusion f m 2t m is in general not homotopically trivial in particular 2t m is in general not contractible

on the asymptotic geometry of abelian by cyclic groups gromov s polynomial growth theorem gro81 states that the property of having polynomial growth characterizes virtually nilpotent groups among all finitely generated groups gromov s theorem inspired the more general problem see e g gdlh91 of understanding to what extent the asymptotic geometry of a finitelygenerated solvable group determines its algebraic structure in short are solvable groups quasi isometrically rigid in general they aren t very recently a dioubina dio99 has found a solvable group which is quasi isometric to a group which is not virtually solvable these groups are finitely generated but not finitely presentable in the opposite direction first steps in identifying quasi isometrically rigid solvable groups which are not virtually nilpotent were taken for a special class of examples the solvable baumslagsolitar groups in fm98 and fm99b the goal of the present paper is to show that a much broader class of solvable groups the class of finitely presented nonpolycyclic abelian bycyclic groups is characterized among all finitely generated groups by its quasi isometry type we also give a complete quasi isometry classification of the groups in this class such a classification for nilpotent groups remains a major open question motivated by these results we offer a conjectural picture of quasi isometric classification and rigidity for polycyclic abelianby cyclic groups in 10 1

isotopy of surfaces in 4 manifolds after a single stabilization abstract any two homologous closed oriented surfaces of the same genus embedded in a smooth 4 manifold x with simply connected complement are shown to be smoothly isotopic in x s 2 s 2 if the surfaces are ordinary and in x s 2 s 2 if they are characteristic

sums of squares and semidefinite program relaxations for polynomial optimization problems with structured sparsity unconstrained and inequality constrained sparse polynomial optimization problems pops are considered a correlative sparsity pattern graph is defined to find a certain sparse structure in the objective and constraint polynomials of a pop based on this graph sets of the supports for sums of squares sos polynomials that lead to efficient sos and semidefinite program sdp relaxations are obtained numerical results from various test problems are included to show the improved performance of the sos and sdp relaxations

the classical moment problem as a self adjoint finite difference operator abstract this is a comprehensive exposition of the classical moment problem using methods from the theory of finite difference operators among the advantages of this approach is that the nevanlinna functions appear as elements of a transfer matrix and convergence of pade approximants appears as the strong resolvent convergence of finite matrix approximations to a jacobi matrix as a bonus of this we obtain new results on the convergence of certain pade approximants for series of hamburger

a representation theorem for certain partially ordered commutative rings let a be a commutative ring with 1 let p subset a be a preordering of higher level i e 0 1 in p 1 not in p p p subset p p cdot p subset p and a 2n subset p for some n in n and let m subset a be an archimedean p module i e 1 in m 1 not in m m m subset m p cdot m subset m and forall a in a exists n in n n a in m we endow x m varphi in rm hom a r mid varphi m subset r with the weak topology with respect to all mappings widehat a x m rightarrow r widehat a varphi varphi a and consider the representation phi m a rightarrow cal c x m r a mapsto widehat a we find that x m is a non empty compact hausdorff space further we prove that

topological triviality of smoothly knotted surfaces in 4 manifolds some generalizations of the fintushel stern rim surgery are known to produce smoothly knotted surfaces we show that if the fundamental groups of their complements are standard then these surfaces are topologically unknotted

on the farrell jones conjecture for higher algebraic k theory here b is the classifying space of the group and we denote by k r the non connective algebraic k theory spectrum of the ring r the homotopy groups of this spectrum are denoted kn r and coincide with quillen s algebraick groups of r qui73 in positive dimensions and with the negative k groups of bass bas68 in negative dimensions the homotopy groups of the spectrum x k r are denotedhn x k r they yield a generalized homology theory and in particular standard computational tools such as the atiyah hirzebruch spectral sequence apply to the left hand side of the assembly map above as a corollary of the main result of this paper we prove conjecture 1 1 in the case where is the fundamental group of a closed riemannian manifold with strictly negative sectional curvature in fact our result is more general and applies to group rings r where r is a completely arbitrary coefficient ring note that if one replaces in conjecture 1 1 the coefficient ring z by an arbitrary coefficient ringr the corresponding conjecture would be false already in the simplest non trivial case if c is the infinite cyclic group the bass heller swan formula bhs64 gra76 p 236 for kn rc kn r t 1 yields that kn rc kn 1 r kn r nkn r nkn r where nkn r is defined as the cokernel of the split inclusion kn r kn r t and does not vanish in general but since s is a model for bc one obtains on the left hand side of the assembly map only hn bc k r kn r kn 1 r

approximately fibering a manifold over an aspherical one the paper is devoted to the problem when a map from some closed connected manifold to an aspherical closed manifold approximately fibers i e is homotopic to manifold approximate fibration we define obstructions in algebraic k theory their vanishing is necessary and under certain conditions sufficient basic ingredients are quinn s thin h cobordism theorem and end theorem and knowledge about the farrell jones conjectures in algebraic k and l theory and the maf rigidity conjecture by hughes taylor williams

recognizing underlying sparsity in optimization exploiting sparsity is essential to improve the efficiency of solving large optimization problems we present a method for recognizing the underlying sparsity structure of a nonlinear partially separable problem and show how the sparsity of the hessian matrices of the problem s functions can be improved by performing a nonsingular linear transformation in the space corresponding to the vector of variables a combinatorial optimization problem is then formulated to increase the number of zeros of the hessian matrices in the resulting transformed space and a heuristic greedy algorithm is applied to this formulation the resulting method can thus be viewed as a preprocessor for converting a problem with hidden sparsity into one in which sparsity is explicit when it is combined with the sparse semidefinite programming relaxation by waki et al for polynomial optimization problems the proposed method is shown to extend the performance and applicability of this relaxation technique preliminary numerical results are presented to illustrate this claim