topological hochschild homology of l and ko we calculate the integral homotopy groups of thh ell at any prime and of thh ko at p 2 where ell is the adams summand of the connective complex p local k theory spectrum and ko is the connective real k theory spectrum

on inverse spectral theory for self adjoint extensions mixed types of spectra abstract lethbe a symmetric operator in a separable hilbert space h suppose thathhas some gapj we shall investigate the question about what spectral properties the self adjoint extensions ofhcan have inside the gapjand provide methods on how to construct self adjoint extensions ofhwith prescribed spectral properties insidej under some weak assumptions about the operatorhwhich are satisfied e g provided the deficiency indices ofhare infinite and the operator h 1is compact for one regular point ofh we shall show that for every auxiliary self adjoint operatorm in the hilbert space h and every open subsetj0of the gapjofhthere exists a self adjoint extensionhofhsuch that insidejthe self adjoint extensionhofhhas the same absolutely continuous and the same point spectrum as the given operatorm and the singular continuous spectrum ofhinjequals the closure ofj0inj moreover we shall present a method of how to construct such a self adjoint extensionh via our methods it is possible to construct new kinds of self adjoint realizations of the laplacian on a bounded domain in r d d 1 with spectral properties very different from the spectral properties of the self adjoint realizations known before mathematics subject classification 1991 47a10 47a60 47b25 47e05 47f05

limits of quantum graph operators with shrinking edges abstract we address the question of convergence of schrodinger operators on metric graphs with general self adjoint vertex conditions as lengths of some of graph s edges shrink to zero we determine the limiting operator and study convergence in a suitable norm resolvent sense it is noteworthy that as edge lengths tend to zero standard sobolev type estimates break down making convergence fail for some graphs we use a combination of functional analytic bounds on the edges of the graph and lagrangian geometry considerations for the vertex conditions to establish a sufficient condition for convergence this condition encodes an intricate balance between the topology of the graph and its vertex data in particular it does not depend on the potential on the differences in the rates of convergence of the shrinking edges or on the lengths of the unaffected edges

the essential spectrum of neumann laplacians on some bounded singular domains in the present paper we consider neumann laplacians on singular domains of the type rooms and passages or combs and we show that in typical situations the essential spectrum can be determined from the geometric data moreover given an arbitrary closed subset s of the non negative reals we construct domains s such that the essential spectrum of the neumann laplacian on is just this set s

an example of unitary equivalence between self adjoint extensions and their parameters abstract the spectral problem for self adjoint extensions is studied using the machinery of boundary triplets for a class of symmetric operators having weyl functions of a special type we calculate explicitly the spectral projections in the form of operator valued integrals this allows one to give a constructive proof of the fact that in certain intervals the resulting self adjoint extensions are unitarily equivalent to the parameterizing boundary operator acting in a smaller space and one is able to provide an explicit form for the associated unitary transform applications to differential operators on metric graphs and to direct sums are discussed

the schrodinger propagator for scattering metrics let g be a scattering metric on a compact manifold x with boundary i e a smooth metric giving the interior x the structure of a complete riemannian manifold with asymptotically conic ends an example is any compactly supported perturbation of the standard metric on rn consider the operator h 1 2 v where is the positive laplacian with respect to g and v is a smooth real valued function on x vanishing to second order at iox assuming that g is nontrapping we construct a global parametrix u z w t for the kernel of the schri odinger propagator u t e ith where z w io x and t 0 the parametrix is such that the difference between u and u is smooth and rapidly decreasing both as t iae 0 and as z iae iox uniformly for w on compact subsets of x let r x 1 where x is a boundary defining function for x be an asymptotic radial variable and let w t be the kernel e ir2 2tu t using the parametrix we show that w t belongs to a class of i legendre distributionsi on x i x i r 0 previously considered by hassell vasy when the metric is trapping then the parametrix construction goes through microlocally in the nontrapping part of the phase space we apply this result to determine the singularities of u t f for any tempered distribution f and for any fixed t 0 in terms of the oscillation of f near iox if the metric is nontrapping then we precisely determine the wavefront set of u t f and hence also precisely determine its singular support more generally we are able to determine the wavefront set of u t f for t 0 resp t 0 on the non backward trapped resp non forward trapped subset of the phase space this generalizes results of craig kappeler strauss and wunsch

analytic continuation of the resolvent of the laplacian on symmetric spaces of noncompact type let m g be a globally symmetric space of noncompact type of arbitrary rank and its laplacian we introduce a new method to analyze and the resolvent 1 this has origins in quantum n body scattering but is independent of the classical theory of spherical functions and is analytically much more robust we expect that suitably modified it will generalize to locally symmetric spaces of arbitrary rank as an illustration of this method we prove the existence of a meromorphic continuation of the resolvent across the continuous spectrum to a riemann surface multiply covering the plane we also show how this continuation may be deduced using the theory of spherical functions in summary this paper establishes a long suspected connection between the analysis on symmetric spaces and n body scattering

decay for the wave and schrodinger evolutions on manifolds with conical ends part ii let c n be a compact imbedded riemannian manifold of dimension d 1 and define the d 1 dimensional riemannian manifold m x r x w x with r 0 and smooth and the natural metric ds 2 1 r x 2 dx 2 r 2 x ds 2 we require that m has conical ends r x x o x 1 as x the hamiltonian flow on such manifolds always exhibits trapping dispersive estimates for the schrodinger evolution e it m and the wave evolution e it m are obtained for data of the form f x y n u x where y n are eigenfunctions of with eigenvalues u 2 n in this paper we discuss all cases d n 1 if n 0 there is the following accelerated local decay estimate with 0 1 e it m y n f l m c n m t d 1 2 w 1 f l 1 m where w x x and similarly for the wave evolution our method combines two main ingredients a a detailed scattering analysis of schrodinger operators of the form 2 v 2 1 4 2 u on the line where u is real valued and smooth with u l 0 3 l for all l 0 as and v 0 in particular we introduce the notion of a zero energy resonance for this class and derive an asymptotic expansion of the wronskian between the outgoing jost solutions as the energy tends to zero in particular the division into part i and part ii can be explained by the former being resonant at zero energy where the present paper deals with the nonresonant case b estimation of oscillatory integrals by non stationary phase

the geometrical approach to multidimensional inverse scattering we prove that in multidimensional short range potential scattering the high velocity limit of the scattering operator of an n body system determines uniquely the potential for a given long range potential the short range potential of the n body system is uniquely determined by the high velocity limit of the modified dollard scattering operator moreover we prove that any one of the dollard scattering operators determines uniquely the total potential we obtain as well a reconstruction formula with an error term our simple proof uses a geometrical time dependent method

the localization sequence for the algebraic k theory of topological k theory we verify a conjecture of rognes by establishing a localization cofiber sequence of spectra k mathbb z to k ku to k ku to sigma k mathbb z for the algebraic k theory of topological k theory we deduce the existence of this sequence as a consequence of a devissage theorem identifying the k theory of the waldhausen category of finitely generated finite stage postnikov towers of modules over a connective a infty ring spectrum r with the quillen k theory of the abelian category of finitely generated pi 0 r modules

the semiclassical resolvent and the propagator for non trapping scattering metrics abstract consider a compact manifold with boundary m with a scattering metric g or equivalently an asymptotically conic manifold m g euclidean r n with a compactly supported metric perturbation is an example of such a space let be the positive laplacian on m g and v a smooth potential on m which decays to second order at infinity in this paper we construct the kernel of the operator h 2 v 0 i 0 2 1 at a non trapping energy 0 0 uniformly for h 0 h 0 h 0 0 small within a class of legendre distributions on manifolds with codimension three corners using this we construct the kernel of the propagator e i t 2 v t 0 t 0 as a quadratic legendre distribution we also determine the global semiclassical structure of the spectral projector poisson operator and scattering matrix

the radial curvature of an end that makes eigenvalues vanish in the essential spectrum i the concern of this paper is to clarify a relationship between the curvatures at infinity and the spectral structure of the laplacian in particular this paper discusses the question of whether there is an eigenvalue of the laplacian embedded in the essential spectrum or not the borderline behavior of the radial curvatures for this problem will be determined we will assume that the radial curvature k rad of an end converges to a constant 1 at infinity with the decay order k rad 1 o r 1 and prove the absence of eigenvalues embedded in the essential spectrum furthermore in order to show that this decay order k rad 1 o r 1 is sharp we will construct a manifold with the radial curvature decay k rad 1 o r 1 and with an eigenvalue frac n 1 2 4 1 embedded in the essential spectrum frac n 1 2 4 infty of the laplacian

periodic manifolds with spectral gaps abstract we investigate spectral properties of the laplace operator on a class of non compact riemannian manifolds for a given number n we construct periodic manifolds such that the essential spectrum of the corresponding laplacian has at least n open gaps we use two different methods first we construct a periodic manifold starting from an infinite number of copies of a compact manifold connected by small cylinders in the second construction we begin with a periodic manifold which will be conformally deformed in both constructions a decoupling of the different period cells is responsible for the gaps

localization sequences for logarithmic topological hochschild homology we study the logarithmic topological hochschild homology of ring spectra with logarithmic structures and establish localization sequences for this theory our results apply for example to connective covers of periodic ring spectra like real and complex topological k theory mathematics subject classification 14f10 19d55 55p43

scattering theory for the laplacian on manifolds with bounded curvature abstract in this paper we study the behaviour of the continuous spectrum of the laplacian on a complete riemannian manifold of bounded curvature under perturbations of the metric the perturbations that we consider are such that its covariant derivatives up to some order decay with some rate in the geodesic distance from a fixed point especially we impose no conditions on the injectivity radius one of the main results are conditions on the rate of decay depending on geometric properties of the underlying manifold that guarantee the existence and completeness of the wave operators

scattering asymptotics for riemann surfaces in this article we prove the optimal polynomial lower bound for the number of resonances of a surface with hyperbolic ends we also give weyl asymptotics for the relative scattering phase of such a surface the proofs are based on trace formulae analogous to those of the euclidean odd dimensional scattering the main technical ingredient is a new proof of the poisson formula theorem 5 7 which is applicable in the euclidean case as well our lower bound seems to be the first example of an optimal polynomial lower bound for the number of resonances holding for a general class of higher dimensional elliptic operators with no symmetries the previous general lower bounds or asymptotics were either nonoptimal 25 58 9 one dimensional or radial 65 67 and 54 41 1 or they required some degeneracy of the

model selection and the principle of minimum description length this article reviews the principle of minimum description length mdl for problems of model selection by viewing statistical modeling as a means of generating descriptions of observed data the mdl framework discriminates between competing models based on the complexity of each description this approach began with kolmogorov s theory of algorithmic complexity matured in the literature on information theory and has recently received renewed attention within the statistics community here we review both the practical and the theoretical aspects of mdl as a tool for model selection emphasizing the rich connections between information theory and statistics at the boundary between these two disciplines we find many interesting interpretations of popular frequentist and bayesian procedures as we show mdl provides an objective umbrella under which rather disparate approaches to statistical modeling can coexist and be compared we illustrate the mdl principle by considering problems in regression nonpar

local smoothing for scattering manifolds with hyperbolic trapped sets we prove a resolvent estimate for the laplace beltrami operator on a scattering manifold with a hyperbolic trapped set and as a corollary deduce local smoothing we use a result of nonnenmacher zworski to provide an estimate near the trapped region a result of burq and cardoso vodev to provide an estimate near infinity and the microlocal calculus on scattering manifolds to combine the two

distribution of resonances for asymptotically euclidean manifolds in this paper we discuss meromorphic continuation of the resolvent and bounds on the number of resonances for scattering manifolds a class of manifolds generalizing euclidian n space subject to the basic assumption of analyticity near infinity we show that resolvent of the laplacian has a meromorphic continuation to a conic neighborhood of the continuous spectrum this involves a geometric interpretation of the complex scaling method in terms of deformations in the grauert tube of the manifold we then show that the number of resonances poles of the meromorphic continuation of the resolvent in a conic neighborhood of mathbb r of absolute value less than r 2 is mathcal o r n under the stronger assumption of global analyticity and hyperbolicity of the geodesic flow we prove a finer weyl type upper bound for the counting function for resonances in small neighborhoods of the real axis this estimate has an exponent which involves the dimension of the trapped set of the geodesic flow

on the homology spectral sequence for topological hochschild homology marcel b6kstedt has computed the homotopy type of the topological hochschild homology of z p using his definition of topological hochschild homology for a functor with smash product here we show that easy conceptual proofs of his main technical result of are possible in the context of the homotopy theory of s algebras as introduced by elmendorf kriz mandell and may we give algebraic arguments based on naturality properties of the topological hochschild homology spectral sequence in the process we demonstrate the utility of the unstable lower notation for the dyer lashof algebra

on the lowest eigenvalue of the laplacian with neumann boundary condition at a small obstacle we study the lowest eigenvalue 1 e of the laplacian in a bounded domain rd d 2 from which a small compact set ke be has been deleted imposing dirichlet boundary conditions along and neumann boundary conditions on ke we are mainly interested in results that require minimal regularity of ke expressed in terms of a poincare condition for the domains e 1ke we then show that 1 e converges to 1 the first dirichlet eigenvalue of as e 0 assuming some more regularity we also obtain asymptotic bounds on 1 e 1 for e small where we employ an idea of burenkov and davies j differential equations 186 2002 485 508

algebraic k theory of topological k theory we are interested in the arithmetic of ring spectra to make sense of this we must work with structured ring spectra such as s algebras ekmm symmetric ring spectra hss or rings ly we will refer to these as salgebras the commutative objects are then commutative s algebras the category of rings is embedded in the category of s algebras by the eilenberg maclane functor r hr we may therefore view an s algebra as a generalization of a ring in the algebraic sense the added flexibility of s algebras provides room for new examples and constructions which may eventually also shed light upon the category of rings itself in algebraic number theory the arithmetic of the ring of integers in a number field is largely captured by its picard group its unit group and its brauer group these are

pure point spectrum under 1 parameter perturbations and instability of anderson localization we consider a selfadjoint operator a and a selfadjoint rank one projection p onto a vector which is cyclic fora we study the set of all eigenvalues of the operatora t a tp t that belong to its essential spectrum which does not depend on the parametert we prove that this set is empty for a dense set of values oft then we apply this result or its idea to questions of anderson localization for 1 dimensional schrodinger operators discrete and continuous

on open scattering channels for manifolds with ends abstract in the framework of time dependent geometric scattering theory we study the existence and completeness of the wave operators for perturbations of the riemannian metric for the laplacian on a complete manifold of dimension n the smallness condition for the perturbation is expressed intrinsically and coordinate free in purely geometric terms using the harmonic radius therefore the size of the perturbation can be controlled in terms of local bounds on the injectivity radius and the ricci curvature as an application of these ideas we obtain a stability result for the scattering matrix with respect to perturbations of the riemannian metric this stability result implies that a scattering channel which interacts with other channels preserves this property under small perturbations

on the absolutely continuous spectrum of self adjoint extensions abstract let be any bounded domain in r d d 1 and j a gap of the minimal laplacian on we show that within j each kind of absolutely ontinuous spectrum can be generated by a self adjoint realization of the laplacian on and in addition give results on mixed types of spectra i e absolutely continuous singular continuous and point spectrum thus for bounded domains in r d with smooth boundary we give self adjoint realizations of the laplacian on with spectral properties very different from the properties of the self adjoint realizations studied before both in order to have very simple and clear concepts and in order to enlarge the possible range of applications we shall work within the much more general framework of self adjoint extensions of so called significantly deficient operators

inverse spectral theory for symmetric operators with several gaps scalar type weyl functions let s be the orthogonal sum of infinitely many pairwise unitarily equivalent symmetric operators with non zero deficiency indices let j be an open subset of r if there exists a self adjoint extension s0 of s such that j is contained in the resolvent set of s0 and the associated weyl function of the pair s s0 is monotone with respect to j then for any self adjoint operator r there exists a self adjoint extension s such that the spectral parts s j and rj are unitarily equivalent it is shown that for any extension s of s the absolutely continuous spectrum of s0 is contained in that one of s moreover for a wide class of extensions the absolutely continuous parts of s and s are even unitarily equivalent

scattering theory for riemannian laplacians we introduce a notion of scattering theory for the laplace beltrami operator on non compact connected and complete riemannian manifolds a principal condition is given by a certain positive lower bound of the second fundamental form of angular submanifolds at infinity another condition is certain bounds of derivatives up to order one of the trace of this quantity these conditions are shown to be optimal for existence and completeness of a wave operator our theory does not involve prescribed asymptotic behavior of the metric at infinity like asymptotic euclidean or hyperbolic metrics studied previously in the literature a consequence of the theory is spectral theory for the laplace beltrami operator including identification of the continuous spectrum and absence of singular continuous spectrum

sharp remainder estimates in the weyl formula for the neumann laplacian on a class of planar regions we obtain estimates for the counting function of the neumann laplacian on a planar domain bounded by the graph of a lower semicontinuous l1 function these estimates imply necessary and sufficient conditions for the validity of the classical one term weyl formula for the counting function and under certain restrictions give an order sharp remainder estimate in this formula

on the algebraic k theory of the complex k theory spectrum let p 5 be a prime let ku be the connective complex k theory spectrum and let k ku be the algebraic k theory spectrum of ku in this paper we study the p primary homotopy type of the spectrum k ku by computing its mod p v1 homotopy groups we show that up to a finite summand these groups form a finitely generated free module over the polynomial algebra mathbb f p b where b is a class of degree 2p 2 defined as a higher bott element

operators with singular continuous spectrum ii rank one operators for an operator a with cyclic vector we studya p wherep is the rank one projection onto multiples of if spec a anda has no a c spectrum we prove thata p has purely singular continuous spectrum on for a denseg of s

generalized resolvents and the boundary value problems for hermitian operators with gaps a hermitian operator a with gaps j j 1 j m is studied the self adjoint extensions which put exactly kj eigenvalues into each gap j j in particular for kj 0 1 j m the extensions preserving the gaps are described in terms of boundary conditions the generalized resolvents of the extensions with the indicated properties are described also a solvability criterion and description of all the solutions of the hamburger moment problem with supports in r j 1m j j are obtained in terms of the nevanlinna matrix

scattering theory for high order operators in domains with infinite boundary abstract a general scattering theory is studied for high order elliptic operators in domains with infinite boundary

poincare inequalities in punctured domains the classic poincare inequality bounds the l q norm of a function f in a bounded domain r n in terms of some l p norm of its gradient in we generalize this in two ways in the first generalization we remove a set from and concentrate our attention on this new domain might not even be connected and hence no poincare inequality can generally hold for it or if it does hold it might have a very bad constant this is so even if the volume of is arbitrarily small a poincare inequality does hold h owever if one makes the additional assumption that f has a finite l p gradient norm on the whole of not just on the important point is that the poincare inequality thus obtained bounds the l q norm of f in terms of the l p gradient norm on not plus an additional term that goes to zero as the volume of g oe st ozero this error term depends on only through its volume apart from this additive error term the constant in the inequality remains that of the nice domain in the second generalization we are given a vector field a and replace by ia x geometrically a connection on a u 1 bundle unlike the a 0 case the infimum of ia fp over all f with a given fq is in general not zero this permits an improvement of the inequality by the addition of a term whose sharp value we derive we describe some open problems that arise from these generalizations

spectral analysis of second order elliptic operators on noncompact manifolds on demontre l estimation de mourre et les bornes qui y sont reliees pour une classe d operateurs elliptiques qui inclut des laplaciens pour des quotients cofinis et coinfinis d espaces hyperboliques

on the completeness of wave operators under loss of local compactness abstract we present a general criterion for the existence and completeness of the wave operators in obstacle or potential scattering which does not require local compactness near the boundary of the obstacle the scope of our basic theorem is illustrated by new classes of examples namely exterior rooms and passages domains where completeness holds and exterior jelly roll domains where it fails

interactions along brownian paths in via the posilicano method schrodinger operators with singular potentials supported by brownian paths in the configuration space are constructed the essential absolutely continuous and singular continuous spectra are determined almost surely with respect to wiener measure it is shown that the set of positive eigenvalues is discrete and that the wave operators exist and are asymptotically complete almost surely if d 3 then the set of positive eigenvalues is even empty almost surely a trace formula for the number counting multiplicities of negative eigenvalues is derived

on the k theory of local fields in this paper we establish a connection between the quillen k theory of certain local fields and the de rham witt complex of their rings of integers with logarithmic poles at the maximal ideal the fieldsk we consider are complete discrete valuation fields of characteristic zero with perfect residue field k of characteristic p 2 when k contains the pth roots of unity the relationship between the k theory with z p coefficients and the de rham witt complex can be described by a sequence

on schrodinger operators with discrete spectrum on etudie le caractere discret des equations de schrodinger a valeur operateur on considere l operateur differentiel x 2 q avec un potentiel localement integrable q agissant sur c 0 r r 0

absence of embedded eigenvalues for riemannian laplacians in this paper we study absence of embedded eigenvalues for schrodinger operators on non compact connected riemannian manifolds a principal example is given by a manifold with an end possibly more than one in which geodesic coordinates are naturally defined in this case one of our geometric conditions is a positive lower bound of the second fundamental form of angular submanifolds at infinity inside the end another condition is an upper bound of the trace of this quantity while a third one is a bound of the derivatives of part of the trace some oscillatory behaviour of the trace is allowed in addition to geometric bounds we need conditions on the potential a regularity property of the domain of the schrodinger operator and the unique continuation property examples include ends endowed with asymptotic euclidean or hyperbolic metrics

the minimum description length principle in coding and modeling we review the principles of minimum description length and stochastic complexity as used in data compression and statistical modeling stochastic complexity is formulated as the solution to optimum universal coding problems extending shannon s basic source coding theorem the normalized maximized likelihood mixture and predictive codings are each shown to achieve the stochastic complexity to within asymptotically vanishing terms we assess the performance of the minimum description length criterion both from the vantage point of quality of data compression and accuracy of statistical inference context tree modeling density estimation and model selection in gaussian linear regression serve as examples

on the topological hochschild homology of bu i the purpose of this paper and its sequel is to determine the homotopy groups of the spectrum thh here p is an odd prime is the adams summand of p local connective k theory see for example 25 and thh is the topological hochschild homology construction introduced by bokstedt in 3 in the present paper we will determine the mod p homotopy groups of thh and also the integral homotopy groups of thh l where l denotes the periodic adams summand in the sequel we will investigate the integral homotopy groups of thh using our present results as a starting point the thh construction appears to be of basic importance in algebraic k theory because it combines two useful properties it can be used to construct good approximations to the algebraic k theory functor and it is very accessible to calculation we shall review what is known about the first property in a moment the second property was demonstrated by bokstedt s calculation in his paper 4 of the homotopy groups of thh hz p and thh hz here hz p and hz denote the evident eilenberg mac lane spectra it is natural to ask about thh r for other popular ring spectra r and our work is a first step in this direction we pay special attention to the connective case because this is the case which is likely to be relevant in applications see subsection 1 4 below the calculation which we present in this paper is a homotopy theoretic one which uses the adams spectral sequence hereafter abbreviated ass this calculation has several interesting features in particular it is a pleasing example of an ass calculation in which although there are infinitely many differentials it is still possible to get the complete answer here is a summary of the contents of the paper in section 2 we review the facts we need to know about ordinary hochschild homology in section 3 we do the same

image subband coding using context based classification and adaptive quantization adaptive compression methods have been a key component of many proposed subband or wavelet image coding techniques this paper deals with a particular type of adaptive subband image coding where we focus on the image coder s ability to adjust itself on the fly to the spatially varying statistical nature of image contents this backward adaptation is distinguished from more frequently used forward adaptation in that forward adaptation selects the best operating parameters from a predesigned set and thus uses considerable amount of side information in order for the encoder and the decoder to operate with the same parameters specifically we present backward adaptive quantization using a new context based classification technique which classifies each subband coefficient based on the surrounding quantized coefficients we couple this classification with online parametric adaptation of the quantizer applied to each class a simple uniform threshold quantizer is employed as the baseline quantizer for which adaptation is achieved our subband image coder based on the proposed adaptive classification quantization idea exhibits excellent rate distortion performance in particular at very low rates for popular test images it is comparable or superior to most of the state of the art coders in the literature

large scale behavior of kernels of schrodinger operators in this paper we study the asymptotic behavior of solutions to schrodinger equations on manifolds with nonnegative ricci curvature and euclidean volume growth as a first application we get a sharp asymptotic estimate on the green s function for the laplacian which sharpens results of li and yau next we establish a definite gap in the possible rates of decay of exterior harmonic functions on these manifolds when the manifold also has quadratic curvature decay we get supremum estimates on the c 2 norm of the green s function this has function theoretic implications for these manifolds we then give a generalization to manifolds with nonnegative ricci curvature outside a compact set finally we study schrodinger operators where the potential decays quadratically at infinity in this setting we obtain a geometric version at infinity of a classical theorem of bers

spectral stability of the neumann laplacian we prove the equivalence of hardy and sobolev type inequalities certain uniform bounds on the heat kernel and some spectral regularity properties of the neumann laplacian associated with an arbitrary region of finite measure in euclidean space we also prove that if one perturbs the boundary of the region within a uniform holder category then the eigenvalues of the neumann laplacian change by a small and explicitly estimated amount

coding and compression a happy union of theory and practice the mathematical theory behind coding and compression began a little more than 50 years ago with the publication of claude shannon s 1948 a mathematical theory of communication in the bell systems technical journal this article laid the foundation for what is now known as information theory in a mathematical framework that is probabilistic see e g cover and thomas 1991 verdui 1998 that is shannon modeled the signal or message process by a random process and a communication channel by a random transition matrix that may distort the message in the five decades that followed information theory provided fundamental limits for communication in general and coding and compression in particular these limits predicted by information theory under probabilistic models are now being approached in real products such as computer modems because these limits or fundamental communication quantities such as entropy and channel capacity vary from signal process to signal process or from channel to channel they must be estimated for each communication setup in this sense information theory is intrinsically statistical moreover the algorithmic theory of information has inspired an extension of shannon s ideas that provides a formal measure of information of the kind long sought in statistical inference and modeling this measure has led to the minimum description length mdl principle for modeling in general and model selection in particular barron rissanen and yu 1998 hansen and yu 1998 rissanen 1978 1989 a coding or compression algorithm is used when one surfs the web listens to a cd uses a cellular phone or works on a computer in particular when a music file is downloaded through the internet a losslessly compressed file often having a much smaller size is transmitted instead of the original file lossless compression works because the music signal is statistically redundant and this redundancy can be removed through statistical prediction for digital signals integer prediction can be easily done based on the past signals that are available to both the sender and receiver and so we need to transmit only the residuals from the prediction these residuals can be coded at a much lower rate than the original signal see e g edler huang schuller and yu 2000

on the absolutely continuous spectrum of one dimensional schrodinger operators with square summable potentials for continuous and discrete one dimensional schrodinger operators with square summable potentials the absolutely continuous part of the spectrum is essentially supported by 0 and 2 2 respectively this fact is proved by considering a priori estimates for the transmission coefficient

topological hochschild homology of connective complex k theory let ku be the connective complex k theory spectrum completed at an odd prime p we present a computation of the mod p 1 homotopy algebra of the topological hochschild homology spectrum of ku