a sampling theorem on homogeneous manifolds we consider a generalization of entire functions of spherical exponential type and lagrangian splines on manifolds an analog of the paleywiener theorem is given we also show that every spectral entire functionl on a manifold is uniquely determined by its values on some discrete sets of points the main result of the paper is a formula for reconstruction of spectral entire functions from their values on discrete sets using lagrangian splines

banach strong novikov conjecture for polynomially contractible groups abstract we prove the banach strong novikov conjecture for groups having polynomially bounded higher order combinatorial functions this includes all automatic groups

infinitely many solutions for p laplacian equation involving critical sobolev growth in this paper we will prove the existence of infinitely many solutions for the following elliptic problem with critical sobolev growth pu u p 2u u p 2uin u 0on provided n p2 p where p is the p laplacian operator 1 0 and is an open bounded domain in rn

construction of hadamard states by pseudo differential calculus we give a new construction based on pseudo differential calculus of quasi free hadamard states for klein gordon equations on a class of space times whose metric is well behaved at spatial infinity in particular on this class of space times we construct all pure hadamard states whose two point function expressed in terms of cauchy data on a cauchy surface is a matrix of pseudo differential operators we also study their covariance under symplectic transformations

almost periodic higher order statistic estimation in this paper stochastic processes with higher order statistical functions decomposable into an almost periodic function plus a residual term not containing finite strength additive sinewave components are considered these processes arise in mobile communications when almost cyclostationary acs processes pass through time varying channels they include as special case the generalized almost cyclostationary processes which in turn include the acs processes in the paper the problem of estimating the fourier coefficients of the generalized fourier series expansion of the almost periodic component of higher order statistical functions is addressed estimators are proposed for cyclic temporal cross moment and cumulants they are proved to be mean square consistent and asymptotically complex normal under mild assumptions on the memory of the processes expressed in terms of summability of cross cumulants numerical results confirm the theoretical results and the derived rate of convergence to zero of bias and standard deviation of the estimators

free vibrations for a semilinear wave equation abstract existence and regularity of nontrivial time periodic solutions are proved for semilinear wave equations under mild smoothness monotonicity and superlinearity assumptions the forced vibration case is also treated

infinitely many sign changing solutions for system of p laplace equations in rn abstract in this paper we consider the system of p laplacian equations in r n p u j j x u j p 2 u j i 1 k i j u i q u j q 2 u j in r n u j x 0 as x j 1 k where 1 p n max 2 p 2 q p n p n p i j are constants i j 1 k j j 0 j 1 2 k i j 0 i j i j 1 k j j 1 k are the potential functions and satisfy suitable decay assumptions the existence of infinitely many sign changing solutions is proved by the truncation method and by the concentration compactness analysis on the approximating solutions

a c algebra of geometric operators on self similar cw complexes novikov shubin and l2 betti numbers abstract a class of cw complexes called self similar complexes is introduced together with c algebras a j of operators endowed with a finite trace acting on square summable cellular j chains since the laplacian j belongs to a j l 2 betti numbers and novikov shubin numbers are defined for such complexes in terms of the trace in particular a relation involving the euler poincare characteristic is proved l 2 betti and novikov shubin numbers are computed for some self similar complexes arising from self similar fractals

radial solutions and phase separation in a system of two coupled schrodinger equations we consider the nonlinear elliptic system left begin aligned delta u u u 3 beta v 2u 0 quad rm in mathbb b delta v v v 3 beta u 2v 0 quad rm in mathbb b u v 0 quad rm in mathbb b quad u v 0 quad rm on partial mathbb b end aligned right where n leqq 3 and mathbb b subset mathbb r n is the unit ball we show that for every beta leqq 1 and k in mathbb n the above problem admits a radially symmetric solution u v such that u v changes sign precisely k times in the radial variable furthermore as beta to infty after passing to a subsequence u w and v w uniformly in mathbb b where w w w has precisely k nodal domains and is a radially symmetric solution of the scalar equation w w w3 0 in mathbb b w 0 on partial mathbb b within a hartree fock approximation the result provides a theoretical indication of phase separation into many nodal domains for bose einstein double condensates with strong repulsion

some mixed hodge structures on l 2 cohomology groups of coverings of kahler manifolds we give methods to compute l 2 cohomology groups of a covering manifold obtained by removing the pullback of a normal crossing divisor to a covering of a compact kahler manifold we prove that in suitable quotient categories these groups admit natural mixed hodge structure whose graded pieces are given by the expected gysin maps

berezin quantization and k homology the e theory defined by connes and higson provides a realization of k homology the generalized homology theory dual to k theory based on the notion of asymptotic homomorphisms with this realization it becomes possible to associate a k homology element to a quantization scheme in this article we associate an asymptotic homomorphism and k homology element to the berezin quantization of a bounded symmetric domain further we identify this element with the element of k homology defined by the dolbeault operator of the domain

multiple solutions for quasilinear elliptic equations with a finite potential well abstract we consider multiplicity of solutions for a class of quasilinear problems which has received considerable attention in the past including the so called modified nonlinear schrodinger equations by combining a new variational approach via q laplacian regularization and the compactness arguments from 4 we establish infinitely many bound state solutions for the quasilinear schrodinger type equations extending the earlier work of 4 for semilinear equations

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

on the upper estimate of the heat kernel of a complete riemannian manifold let m be a complete non compact riemannian manifold whose sectional curvature is bounded between two constants k and k then one expects that the heat diffusion in such a manifold behaves like the heat diffusion in euclidean space the purpose of this paper is to give a justification of such a statement in 5 j cheeger and the third author found a lower estimate of the heat kernel by comparing it with the heat kernel of the space form whose curvature is the lower bound of the curvature of the manifold this lower estimate is sharp if we insist the dependence should be on the lower bound of the ricci curvature alone it remains to give an upper estimate of the heat kernel one does not however expect to have a comparison theorem for the upper bound because it is more sensitive to the geometry of the manifold in fact the heat kernel of the upper hialf space and the heat kernel of the complete hyperbolic manifold with finite volume have quite different behavior this is reflected by the fact that the laplacian has no discrete spectrum in the first case while infinite number of discrete eigenvalues may exist in the latter case what we will prove here is that in any case the heat kernel has to decay in a manner similar to the euclidean heat kernel thus we will prove that for any constant c 4 there exists c1 depending on c t the bound of the curvature of m and x so that for all t e 0 t

the massive feynman propagator on asymptotically minkowski spacetimes we consider the massive klein gordon equation on asymptoti cally minkowski spacetimes in the sense that the manifold is r 1 d and the metric approaches that of minkowski space at infinity in a short range way jointly in time and space variables in this setup we define feynman and anti feynman scattering data and prove the fredholm property of the klein gordon operator with the associated atiyah patodi singer type boundary conditions at infinite times we then construct a parametrix with compact remainder terms for the fredholm problem and prove that it is also a feynman parametrix in the sense of duistermaat and hormander

a quasi local characterisation of lp roe algebras abstract very recently spakula and tikuisis provide a new characterisation of uniform roe algebras via quasi locality when the underlying metric spaces have straight finite decomposition complexity in this paper we improve their method to deal with the l p version of uniform roe algebras for any p 1 due to the lack of reflexivity on l 1 spaces some extra work is required for the case of p 1

essential spectrum and heat kernel abstract upper bounds are derived for the large time behavior of the heat kernel on riemannian manifolds whose essential spectrum has a positive lower bound applications include a relative signature theorem and the construction of bounded harmonic functions

spectral invariant subalgebras of reduced crossed product c algebras abstractletg beadiscretegroupwithaproperlengthfunctionl inthispaper weprovethattheschwartz space s l2 g a is a spectral invariant dense subalgebra of the reduced crossedproduct c nr g a for all commutative c n algebra a with a g action if and only if g haspolynomialgrowthwithrespecttol r 2002elsevierscience usa allrightsreserved keywords reducedcrossedproductc n algebras spectralinvariantsubalgebras property mrd 0 introductionletg beadiscretegroupwithalengthfunctionl in 9 jolissaintprovedthatthe schwartzspace s l2 g is a spectralinvariant dense subalgebra of the reducedgroupc n algebrac n r g ifg haspolynomialgrowthorg isthefundamentalgroupofanegativelycurvedcompactmanifold in 4 delaharpegeneralizedjolissaint stheorem to all hyperbolic groups jolissaint and de la harpe s result plays animportantroleintheworkofconnes moscovicionthenovikovconjecture 2 andtheworkoflafforgueonthebaum connesconjecture 10 inthispaper wegiveanecessaryandsuf cientconditionfortheschwartzspaces

heat kernel on smooth metric measure spaces with nonnegative curvature we derive a local gaussian upper bound for the f heat kernel on complete smooth metric measure space m g e f dv with nonnegative bakry emery ricci curvature as applications we obtain a sharp l f 1 liouville theorem for f subharmonic functions and an l f 1 uniqueness property for nonnegative solutions of the f heat equation assuming f is of at most quadratic growth in particular any l f 1 integrable f subharmonic function on gradient shrinking and steady ricci solitons must be constant we also provide explicit f heat kernel for gaussian solitons

multirate processing of time series exhibiting higher order cyclostationarity the paper investigates the effects of multirate systems on the higher order wide sense cyclostationarity wscs properties of discrete time series to this end definitions and properties of the cyclic higher order statistics which were introduced for continuous time series in the fraction of time probability framework are first extended to discrete time series then starting from the consideration of the basic multirate building blocks viz m fold decimators and l fold interpolators results for typical interconnections are derived the problem of eliminating the images in the cyclic higher order spectra of an interpolated time series is addressed moreover the problem of avoiding aliasing in both cycle and spectral frequency domains in the cyclic higher order spectra of a decimated time series is considered finally a sufficient condition to avoid both aliasing and imaging effects in the cyclic higher order spectra of a time series decimated by a fractional factor is derived

on smooth cauchy hypersurfaces and geroch s splitting theorem given a globally hyperbolic spacetime m we show the existence of a smooth spacelike cauchy hypersurface s and thus a global diffeomorphism between m and r s

positive solutions for a weakly coupled nonlinear schrodinger system abstract existence of a nontrivial solution is established via variational methods for a system of weakly coupled nonlinear schrodinger equations the main goal is to obtain a positive solution of minimal action if possible with all vector components not identically zero generalizations for nonautonomous systems are considered

transversal dirac families in riemannian foliations we describe a family of differential operators parametrized by the transversal vector potentials of a riemannian foliation relative to the clifford algebra of the foliation this family is non elliptic but in certain ways behaves like a standard dirac family in the absolute case as a result of its elliptic like regularity properties the analytic and topological indices of this family are defined as elements of k theory in the parameter space we indicate how the cohomology of the parameter space is described via suitable maps to fredholm operators we outline the proof of a theorem of vafa witten type on uniform bounds for the eigenvalues of this family using a spectral flow argument a determinant operator is also defined with the appropriate zeta function regularization dependent on the codimension of the foliation with respect to a generalized coupled dirac yang mills system we indicate how chiral anomalies are located relative to the foliation

least energy solitary waves for a system of nonlinear schrodinger equations in mathbb r n in this paper we consider systems of coupled schrodinger equations which appear in nonlinear optics the problem has been considered mostly in the one dimensional case here we make a rigorous study of the existence of least energy standing waves solitons in higher dimensions we give conditions on the parameters of the system under which it possesses a solution with least energy among all multi component solutions conditions under which the system does not have positive solutions and the associated energy functional cannot be minimized on the natural set where the solutions lie

eta invariants of dirac operators on foliated manifolds we define the eta function of dirac operators on foliated manifolds we show that the eta functions are regular at the origin thus defining corresponding eta invariants of foliated manifolds under the hypothesis of invertibility of the operator in question we prove the atiyah singer relative index theorem for dirac operators on foliated manifolds then we discuss the homotopy invariance of the index with respect to secondary data

the longitudinal cocycle and the index of toeplitz operators abstract let d be a self adjoint leafwise elliptic operator on a foliated manifold compressing multiplication operators to the range of the positive spectral projection for d yields the class of leafwise toeplitz operators the extension generated by these operators is constructed a topological formula for the index of a toeplitz operator with invertible symbol is given this index can also be obtained by pairing the k theory class of the symbol with a certain cyclic cocycle if one lifts an elliptic operator on a closed manifold to a leafwise elliptic operator on an associated flat foliated principal bundle then this cocycle can be used to obtain refined invariants of the original operator

the cumulant theory of cyclostationary time series ii development and applications for pt i see ibid vol 42 no 12 p 3387 3408 1994 the development of the theory of nonlinear processing of cyclostationary time series that is initiated in part i is continued a new type of cumulant for complex valued variables is introduced and used to generalize the temporal and spectral moments and cumulants for cyclostationary time series from real valued to complex valued time series the relations between the temporal and spectral moments and cumulants at the inputs and outputs of several signal processing operations are determined formulas for the temporal and spectral cumulants of complex valued pulse amplitude modulated time series are derived estimators for the temporal moments and cumulants and for the cyclic polyspectra are presented and their properties are discussed the performance of these estimators is illustrated by several computer simulation examples for pulse amplitude modulated time series the theory is applied to the problems of weak signal detection and interference tolerant time delay estimation

uniform k theory and poincare duality for uniform k homology abstract we revisit spakula s uniform k homology construct the external product for it and use this to deduce homotopy invariance of uniform k homology we define uniform k theory and on manifolds of bounded geometry we give an interpretation of it via vector bundles of bounded geometry we further construct a cap product with uniform k homology and prove poincare duality between uniform k theory and uniform k homology on spinc manifolds of bounded geometry

almost sure convergence analysis of mixed time averages and kth order cyclic statistics the strong law of large numbers for general processes is established under certain conditions and the convergence rate of time averages is given these results are used to obtain the convergence rates and the almost sure convergence properties of mixed time averages and kth order cyclic statistics

hodge decomposition along the leaves of a riemannian foliation abstract a hodge decomposition theorem is proved for the leafwise laplacian of a riemannian foliation on a closed riemannian manifold

nonparametric polyspectral estimators for kth order almost cyclostationary processes second and higher order almost cyclostationary processes are random signals with almost periodically time varying statistics the class includes stationary and cyclostationary processes as well as many real life signals of interest cyclic and time varying cumulants and polyspectra are defined for discrete time real kth order cyclostationary processes and their interrelationships are explored smoothed polyperiodograms are proposed for cyclic polyspectral estimation and are shown to be consistent and asymptotically normal asymptotic covariance expressions are derived along with their computable forms higher than second order cyclic cumulants and polyspectra convey time varying phase information and are theoretically insensitive to any stationary for nonzero cycles as well as additive cyclostationary gaussian noise for all cycles

characteristic classes and distortion of diffeomorphisms let m and n be manifolds of bounded geometry that is the absolute value of their curvatures as well as their covariant derivatives are bounded from above and their injectivity radii are bounded from below a map f m n has bounded d stortion if there is an upper bound on the norm of the derivative d f a bounded distortion diffeomorphism cb d d is assumed to have an inverse which also has bounded distortion similarly homotopies are assumed to have bounded distortion note that this implies that homotopic maps are a finite distance apart we prove the following theorem that describes when characteristic classes of a compact manifold obstruct the existence of bounded distortion diffeomorphisms of universal covers theorem there are compact manifolds a and b with fundamental group r and a map f a b with jcpcb i pca for some rational pontrjagin class p whose universal cover is b d homotopic to a b d diffeomorphism if and only if r is nonamenable

sign changing solutions for coupled nonlinear schrodinger equations with critical growth abstract we consider the following nonlinear schrodinger system with critical growth u j j u j i 1 k i j u i 2 2 u j 2 2 2 u j in u j 0 on j 1 k where is a bounded smooth domain in r n 2 2 n n 2 0 j 1 j 1 k 1 is the first eigenvalue of with zero dirichlet boundary condition we consider the repulsive case namely j j 0 j 1 k i j j i 0 i j i j 1 k the existence of infinitely many sign changing solutions as bound states is proved provided n 7 by approximations of systems with subcritical growth and by the concentration analysis on approximating solutions

a trace on fractal graphs and the ihara zeta function starting with ihara s work in 1968 there has been a growing interest in the study of zeta functions of finite graphs by sunada hashimoto bass stark and terras mizuno and sato to name just a few authors then clair and mokhtari sharghi studied zeta functions for infinite graphs acted upon by a discrete group of automorphisms the main formula in all these treatments establishes a connection between the zeta function originally defined as an infinite product and the laplacian of the graph in this article we consider a different class of infinite graphs they are fractal graphs i e they enjoy a self similarity property we de ne a zeta function for these graphs and using the machinery of operator algebras we prove a determinant formula which relates the zeta function with the laplacian of the graph we also prove functional equations and a formula which allows approximation of the zeta function by the zeta functions of finite subgraphs

on rigidity of roe algebras roe algebras are c algebras built using large scale or coarse aspects of a metric space x d in the special case that x is a finitely generated group and d is a word metric the simplest roe algebra associated to d is isomorphic to the crossed product c algebra l 1 r roe algebras are coarse invariants in the sense that if x and y are coarsely equivalent metric spaces then their roe algebras are isomorphic motivated in part by the coarse baum connes conjecture we ask if there is a converse to the above statement that is if x and y are metric spaces with isomorphic roe algebras must x and y be coarsely equivalent we show that for very large classes of spaces the answer to this question and some related questions is yes this can be thought of as a c rigidity result it shows that the roe algebra construction preserves a large amount of information about the space and is thus surprisingly rigid as an example of our results in the group case we have that if and are finitely generated elementary amenable hyperbolic or linear groups

micro local approach to the hadamard condition in quantum field theory on curved space time for the two point distribution of a quasi free klein gordon neutral scalar quantum field on an arbitrary four dimensional globally hyperbolic curved space time we prove the equivalence of 1 the global hadamard condition 2 the property that the feynman propagator is a distinguished parametrix in the sense of duistermaat and hormander and 3 a new property referred to as the wave front set spectral condition wfssc because it is reminiscent of the spectral condition in axiomatic quantum field theory on minkowski space results in micro local analysis such as the propagation of singularities theorem and the uniqueness up toc of distinguished parametrices are employed in the proof we include a review of kay and wald s rigorous definition of the global hadamard condition and the theory of distinguished parametrices specializing to the case of the klein gordon operator on a globally hyperbolic space time as an alternative to a recent computation of the wave front set of a globally hadamard two point distribution on a globally hyperbolic curved space time given elsewhere by kohler to correct an incomplete computation in 32 we present a version of this computation that does not use a deformation argument such as that used in fulling narcowich and wald and is independent of the cauchy evolution argument of fulling sweeny and wald both of which are relied upon in kohler s proof this leads to a simple micro local proof of the preservation of hadamard form under cauchy evolution first shown by fulling sweeny and wald relying only on the propagation of singularities theorem in another paper 33 the equivalence theorem is used to prove a conjecture by kay that a locally hadamard quasi free klein gordon state on any globally hyperbolic curved space time must be globally hadamard

a renormalized index theorem for some complete asymptotically regular metrics the gauss bonnet theorem abstract we study the gauss bonnet theorem as a renormalized index theorem for edge metrics these metrics include the poincare einstein metrics of the ads cft correspondence and the asymptotically cylindrical metrics of the atiyah patodi singer index theorem we use renormalization to make sense of the curvature integral and the dimensions of the l 2 cohomology spaces as well as to carry out the heat equation proof of the index theorem for conformally compact metrics even mod x m we show that the finite time supertrace of the heat kernel on conformally compact manifolds renormalizes independently of the choice of special boundary defining function

an index for gauge invariant operators and the dixmier douady invariant let g b be a bundle of compact lie groups acting on a fiber bundle y b in this paper we introduce and study gauge equivariant k theory groups k i g y these groups satisfy the usual properties of the equivariant k theory groups but also some new phenomena arise due to the topological non triviality of the bundle g b as an application we define a gauge equivariant index for a family of elliptic operators p b b b invariant with respect to the action of g b which in this approach is an element of k 0 g b we then give another definition of the gauge equivariant index as an element of k 0 c g the k theory group of the banach algebra c g we prove that k 0 c g k 0 g g and that the two definitions of the gauge equivariant index are equivalent the algebra c g is the algebra of continuous sections of a certain field of c algebras with non trivial dixmier douady invariant the gauge equivariant k theory groups are thus examples of twisted k theory groups which have recently turned out to be useful in the study of ramond ramond fields

robust l estimation based forms of signal transforms and time frequency representations the l estimation based signal transforms and time frequency tf representations are introduced by considering the corresponding minimization problems in the huber 1981 1998 estimation theory the standard signal transforms follow as the maximum likelihood solutions for the gaussian additive noise environment for signals corrupted by an impulse noise the median based transforms produce robust estimates of the non noisy signal transforms when the input noise is a mixture of gaussian and impulse noise the l estimation based signal transforms can outperform other estimates in quadratic and higher order tf analysis the resulting noise is inherently a mixture of the gaussian input noise and an impulse noise component in this case the l estimation based signal representations can produce the best results these transforms and tf representations give the standard and the median based forms as special cases a procedure for parameter selection in the l estimation is proposed the theory is illustrated and checked numerically

median based cyclic polyspectrum estimation in this paper median based estimation methods for the cyclic polyspectrum are proposed the algorithms do not require a priori knowledge of the spl beta submanifolds that is they do not require the knowledge of all of the lower order cycle frequencies of the time series available for the estimation of the cyclic polyspectrum therefore such methods are particularly useful when the cyclostationarity of the signals under consideration is not completely known the proposed estimators converge to the theoretical values of the cyclic polyspectrum when the collect time approaches infinity and the spectral resolution becomes infinitesimal furthermore their accuracy is very nearly the same as that of the usual time and frequency smoothed cyclic periodogram methods that use a priori knowledge of lower order cycle frequencies to avoid the spl beta submanifolds

a serre swan theorem for bundles of bounded geometry abstract the serre swan theorem in differential geometry establishes an equivalence between the category of smooth vector bundles over a smooth compact manifold and the category of finitely generated projective modules over the unital ring of smooth functions this theorem is here generalized to manifolds of bounded geometry in this context it states that the category of hilbert bundles of bounded geometry is equivalent to the category of operator modules over the operator algebra of continuously differentiable functions which vanish at infinity operator modules are generalizations of hilbert c modules where the category of c algebras has been replaced by a more flexible category of involutive algebras of bounded operators the operator algebras operator modules play an important role in the study of the unbounded kasparov product

hadamard states for the linearized yang mills equation on curved spacetime we construct hadamard states for the yang mills equation linearized around a smooth space compact background solution we assume the spacetime is globally hyperbolic and its cauchy surface is compact or equal mathbb r d

an index theorem on open manifolds ii on decrit et on applique un nouveau theoreme d indice pour des operateurs elliptiques sur certaines varietes non compactes on calcule un indice a valeur reelle pour des operateurs de type dirac sur des varietes non compactes de geometrie bornee

uniform k homology theory we define a uniform version of analytic k homology theory for separable proper metric spaces furthermore we define an index map from this theory into the k theory of uniform roe c algebras analogous to the coarse assembly map from analytic k homology into the k theory of roe c algebras we show that our theory has a mayer vietoris sequence we prove that for a torsion free countable discrete group the direct limit of the uniform k homology of the rips complexes of limd k u pd is isomorphic to k top l the left hand side of the baum connes conjecture with coefficients in l in particular this provides a computation of the uniform k homology groups for some torsion free groups as an application of uniform k homology we prove a criterion for amenability in terms of vanishing of a fundamental class in spirit of similar criteria in uniformly finite homology and k theory of uniform roe algebras

aperiodic tilings positive scalar curvature and amenability of spaces the object of this paper is to begin a geometric study of noncompact spaces whose local structure has bounded complexity manifolds of this sort arise as leaves of foliations of compact manifolds and as their universal covers we shall introduce a coarse homology theory using chains of bounded complexity and study some of its first properties the most interesting result characterizes when h uf x vanishes as an analogue and strengthening of f0lner s amenability criterion for groups in terms of isoperimetric inequalities see 4 one can view this result as producing a successful infinite ponzi scheme on any nonamenable space each point with only finite resources gives to some of its neighbors some of these resources yet receives more from the remaining neighbors as one can imagine this is useful for eliminating obstructions on noncompact spaces this has a number of applications we present two of them the first produces tilings that are unbalanced on any nonamenable polyhedron unbalanced tilings are automatically aperiodic and this gives many examples of sets of tiles that tile only aperiodically unfortunately imbalance is a particularly unsubtle reason for aperiodicity so that the aperiodic tilings of euclidean space penrose tilings are necessarily not accessible to our method on the other hand most other simply connected noncompact symmetric spaces even have unbalanced tilings using our criterion the second application regards characteristic numbers of manifolds whose universal covers have positive scalar curvature we prove a converse to a theorem of roe we show that for any nonamenable group f one can find a spin manifold with fundamental group f with nonzero a genus whose universal cover has a uniformly positive scalar curvature metric of bounded geometry in the natural strict quasi isometry class

plancherel type estimates and sharp spectral multipliers abstract we study general spectral multiplier theorems for self adjoint positive definite operators on l2 x where x is any open subset of a space of homogeneous type we show that the sharp hormander type spectral multiplier theorems follow from the appropriate estimates of the l2 norm of the kernel of spectral multipliers and the gaussian bounds for the corresponding heat kernel the sharp hormander type spectral multiplier theorems are motivated and connected with sharp estimates for the critical exponent for the riesz means summability which we also study here we discuss several examples which include sharp spectral multiplier theorems for a class of scattering operators on r3 and new spectral multiplier theorems for the laguerre and hermite expansions

ground state of n coupled nonlinear schrodinger equations in mathbb r n n leq 3 we establish some general theorems for the existence and nonexistence of ground state solutions of steady state n coupled nonlinear schrodinger equations the sign of coupling constants ij s is crucial for the existence of ground state solutions when all ij s are positive and the matrix is positively definite there exists a ground state solution which is radially symmetric however if all ij s are negative or one of ij s is negative and the matrix is positively definite there is no ground state solution furthermore we find a bound state solution which is non radially symmetric when n 3

boundedness of spectral multipliers of generalized laplacians on compact manifolds with boundary consider a second order strongly elliptic negative semidefinite differential operator l may be a system on a compact riemannian manifold overline m with smooth boundary where the domain of l is defined by a coercive boundary condition classically known results and also recent work in duong et al j funct anal 196 443 485 2002 and duong and mcintosh rev math iberoam 15 233 265 1999 establish sufficient conditions for l infty text bmo l continuity of varphi sqrt l where varphi in s 0 1 mathbb r and l is a suitable elliptic operator using a variant of the cheeger gromov taylor functional calculus due to mauceri et al math res lett 16 861 879 2009 and short time upper bounds on the integral kernel of e tl due to greiner arch ration mech anal 41 168 218 1971 we prove that a variant of such sufficient conditions holds for our operator l

analytic hadamard states calderon projectors and wick rotation near analytic cauchy surfaces we consider the klein gordon equation on analytic spacetimes with an analytic cauchy surface in this setting we prove the existence of pure analytic hadamard states the proof is based on considering an elliptic operator obtained by wick rotating the klein gordon operator in a neighborhood of a cauchy hypersurface the cauchy data of hadamard two point functions are constructed as calderon projectors suitably generalized if the hypersurface is non compact for the elliptic operator

noncommutative riemann integration and novikov shubin invariants for open manifolds given a c algebra a with a semicontinuous semifinite trace acting on the hilbert space h we define the family ar of bounded riemann measurable elements w r t as a suitable closure a la dedekind of a in analogy with one of the classical characterizations of riemann measurable functions and show that ar is a c algebra and extends to a semicontinuous semifinite trace on ar then unbounded riemann measurable operators are defined as the closed operators on h which are affiliated to a and can be approximated in measure by operators in ar in analogy with unbounded riemann integration unbounded riemann measurable operators form a a e bimodule on ar denoted by ar and such a bimodule contains the functional calculi of selfadjoint elements of ar under unbounded riemann measurable functions besides extends to a bimodule trace on ar type ii1 singular traces for c algebras can be defined on the bimodule of unbounded riemann measurable operators noncommutative riemann integration and singular traces for c algebras are then used to define novikov shubin numbers for amenable open manifolds to show their invariance under quasi isometries and to prove that they are noncommutative asymptotic dimensions

hadamard states for the klein gordon equation on lorentzian manifolds of bounded geometry we consider the klein gordon equation on a class of lorentzian manifolds with cauchy surface of bounded geometry which is shown to include examples such as exterior kerr kerr de sitter spacetime and the maximal globally hyperbolic extension of the kerr outer region in this setup we give an approximate diagonalization and a microlocal decomposition of the cauchy evolution using a time dependent version of the pseudodifferential calculus on riemannian manifolds of bounded geometry we apply this result to construct all pure regular hadamard states and associated feynman inverses where regular refers to the state s two point function having cauchy data given by pseudodifferential operators this allows us to conclude that there is a one parameter family of elliptic pseudodifferential operators that encodes both the choice of pure regular hadamard state and the underlying spacetime metric

microlocal analysis and interacting quantum field theories renormalization on physical backgrounds we present a perturbative construction of interacting quantum field theories on smooth globally hyperbolic curved space times we develop a purely local version of the stuckelberg bogoliubov epstein glaser method of renormalization by using techniques from microlocal analysis relying on recent results of radzikowski kohler and the authors about a formulation of a local spectrum condition in terms of wave front sets of correlation functions of quantum fields on curved space times we construct time ordered operator valued products of wick polynomials of free fields they serve as building blocks for a local perturbative definition of interacting fields renormalization in this framework amounts to extensions of expectation values of time ordered products to all points of space time the extensions are classified according to a microlocal generalization of steinmann scaling degree corresponding to the degree of divergence in other renormalization schemes as a result we prove that the usual perturbative classification of interacting quantum field theories holds also on curved space times finite renormalizations are deferred to a subsequent paper as byproducts we describe a perturbative construction of local algebras of observables present a new definition of wick polynomials as operator valued distributions on a natural domain and we find a general method for the extension of distributions which were defined on the complement of some surface

quasi locality and property a abstract let x be a metric space with bounded geometry p 0 1 and let e be a banach space the main result of this paper is that either if x has yu s property a and p 1 or without any condition on x when p 0 1 then quasi local operators on l p x e belong to the appropriate variant of the roe algebra of x this generalises the existing results of this type by lange and rabinovich engel tikuisis and the first author and li wang and the second author as consequences we obtain that uniform l p roe algebras of spaces with property a are closed under taking inverses and another condition characterising property a akin to the operator norm localisation for quasi local operators

the cumulant theory of cyclostationary time series i foundation the problem of characterizing the sine wave components in the output of a polynomial nonlinear system with a cyclostationary random time series input is investigated the concept of a pure nth order sine wave is introduced and it is shown that pure nth order sine wave strengths in the output time series are given by scaled fourier coefficients of the polyperiodic temporal cumulant of the input time series the higher order moments and cumulants of narrowband spectral components of time series are defined and then idealized to the case of infinitesimal bandwidth such spectral moments and cumulants are shown to be characterized by the fourier transforms of the temporal moments and cumulants of the time series it is established that the temporal and spectral cumulants have certain mathematical and practical advantages over their moment counterparts to put the contributions of the paper in perspective a uniquely comprehensive historical survey that traces the development of the ideas of temporal and spectral cumulants from their inception is provided

exotic cohomology and index theory the main result of this paper is a version of the atiyah singer index theorem for dirac type operators on noncompact complete riemannian manifolds the statement of the theorem involves a novel cohomology theory for such manifolds this theory called exotic cohomology depends on the structure at infinity of a space more precisely it depends on the way that large bounded sets fit together for each cohomology class in this theory we define a higher index of a dirac type operator enjoying the stability and vanishing properties of the usual atiyah singer index these higher indices are analogous to the novikov higher signatures our main theorem will compute these higher indices in terms of standard topological invariants applications of this index theorem include a different approach to some of the results of gromov and lawson 10 on topological obstructions to positive scalar curvature the concept of index that we will use involves the a theory functors k0 and kx for operator algebras 3 14 suppose that b is an ideal in a unital algebra c and let t e c be invertible modulo b in the classical atiyah singer index theorem one takes c to be the bounded operators on the l space of some compact manifold b the compact operators and t an elliptic pseudodifferential operator of order zero then t has an index in the theory group kq b in the classical case this is just z and one recovers the usual fredholm index now let m be a complete riemannian manifold possibly noncompact in 18 i introduced an algebra m which is defined as follows sf m consists of all bounded operators a on l m that have a kernel representation

the microlocal spectrum condition initial value formulations and background independence we analyze implications of the microlocal spectrum hadamard condition for states in a linear quantum field theory on a globally hyperbolic spacetime m in the context of a distributional initial value formulation more specifically we work in 3 1 split m and give a bound independent of the spacetime metric on the wave front sets of the initial data for a quasi free hadamard state in a quantum field theory defined by a normally hyperbolic differential operator p acting in a vector bundle e m this aims at a possible way to apply the concept of hadamard states within approaches to quantum field theory gravity relying on a hamiltonian formulation potentially without a classical background metric g

c0 coarse geometry and scalar curvature in this paper we introduce an alternative form of coarse geometry on proper metric spaces which is more delicate at infinity than the standard metric coarse structure there is an assembly map from the k homology of a space to the k theory of the c algebra associated to the new coarse structure which factors through the coarse k homology of the space with the new coarse structure a dirac type operator on a complete riemannian manifold m gives rise to a class in k homology and its image under assembly gives a higher index in the k theory group the main result of this paper is a vanishing theorem for the index of the dirac operator on an open spin manifold for which the scalar curvature x tends to infinity as x tends to infinity this is derived from a spectral vanishing theorem for any dirac type operator with discrete spectrum and finite dimensional eigenspaces