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

indefinite sequences with a finite order singularity in the complex moment problem in this paper we consider indefinite sequences having singularity at the point 1 1 i p of order at most 2x we show that they are related to both the classical moment problem and the l6vy khinchin formula for negative definite sequences we establish an integral representation for these sequences and characterize the exponentially bounded ones by means of finitely many positive definiteness conditions even in the case x 1 our approach provides new l6vy khinchin type integral representations let no be the set of all non negative integers and fix the positive integer p we denote by l z i the algebra of polynomials in z and where z zl zp e r p this algebra acts on the sequences ct m n m n e ng n following the correspondence

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

on ensembles of low density parity check codes asymptotic distance distributions we derive expressions for the average distance distributions in several ensembles of regular low density parity check codes ldpc among these ensembles are the standard one defined by matrices having given column and row sums ensembles defined by matrices with given column sums or given row sums and an ensemble defined by bipartite graphs

a recursive approach to low complexity codes a method is described for constructing long error correcting codes from one or more shorter error correcting codes referred to as subcodes and a bipartite graph a graph is shown which specifies carefully chosen subsets of the digits of the new codes that must be codewords in one of the shorter subcodes lower bounds to the rate and the minimum distance of the new code are derived in terms of the parameters of the graph and the subeodes both the encoders and decoders proposed are shown to take advantage of the code s explicit decomposition into subcodes to decompose and simplify the associated computational processes bounds on the performance of two specific decoding algorithms are established and the asymptotic growth of the complexity of decoding for two types of codes and decoders is analyzed the proposed decoders are able to make effective use of probabilistic information supplied by the channel receiver e g reliability information without greatly increasing the number of computations required it is shown that choosing a transmission order for the digits that is appropriate for the graph and the subcodes can give the code excellent burst error correction abilities the construction principles

harmonic analysis on conelike bodies and holomorphic functions on tube domains in this paper we establish bochner weil type theorems and integral formulas of levy khinchin type in the setting of locally compact commutative semigroups with involution these results are used to prove some new holomorphic extension results for a conelike involution semigroup s in a finite dimensional real space v we establish integral characterizations for continuous positive definite and definitizable functions on s in particular we obtain holomorphic and cauchy riemann extensions of these functions by means of an integral representation we prove that ifz0is a point in the complexified vector space w v iv and if a function is defined onz0 s then either positive definiteness or definitizability offextends holomorphy from a domain containingz0to a tube domain in w containing the sum of 0 and the convex cone spanned by s we also obtain the holomorphic extension as a generalized fourier laplace transform of some measure or a levy khinchin type integral representation

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

weight distribution of low density parity check codes we derive the average weight distribution function and its asymptotic growth rate for low density parity check ldpc code ensembles we show that the growth rate of the minimum distance of ldpc codes depends only on the degree distribution pair it turns out that capacity achieving sequences of standard unstructured ldpc codes under iterative decoding over the binary erasure channel bec known to date have sublinearly growing minimum distance in the block length

narrow escape problem with a mixed trap and the effect of orientation we consider the mean first passage time mfpt of a two dimensional diffusing particle to a small trap with a distribution of absorbing and reflecting sections high order asymptotic formulas for the mfpt and the fundamental eigenvalue of the laplacian are derived which extend previously obtained results and show how the orientation of the trap affects the mean time to capture we obtain a simple geometric condition which gives the optimal trap alignment in terms of the gradient of the regular part of a regular part of a green s function and a certain alignment vector we find that subdividing the absorbing portions of the trap reduces the mean first passage time of the diffusing particle in the scenario where the trap undergoes prescribed motion in the domain the mfpt is seen to be particularly sensitive to the orientation of the trap

distance distributions in ensembles of irregular low density parity check codes we derive asymptotic expressions for the average distance distributions in ensembles of irregular low density parity check ldpc codes the ensembles are defined by matrices with given profiles of column and row sums

asymptotic enumeration methods for analyzing ldpc codes we show how asymptotic estimates of powers of polynomials with nonnegative coefficients can be used in the analysis of low density parity check ldpc codes in particular we show how these estimates can be used to derive the asymptotic distance spectrum of both regular and irregular ldpc code ensembles we then consider the binary erasure channel bec using these estimates we derive lower bounds on the error exponent under iterative decoding of ldpc codes used over the bec both regular and irregular code structures are considered these bounds are compared to the corresponding bounds when optimal maximum likelihood ml decoding is applied

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

eigenvalue asymptotics of the neumann laplacian of regions and manifolds with cusps abstract we study the eigenvalue asymptotics of a neumann laplacian n in unbounded regions of r2 with cusps at infinity a typical example is x y r 2 x 1 y x 2 and prove that n e n n e h v e 2 vol where hv is the canonical one dimensional schrodinger operator associated to the problem we establish a similar formula for manifolds with cusps and derive the eigenvalue asymptotics of a dirichlet laplacian d for a class of cusp type regions of infinite volume

weyl asymptotic formula for the laplacian on domains with rough boundaries we study asymptotic distribution of eigenvalues of the laplacian on a bounded domain in n our main results include an explicit remainder estimate in the weyl formula for the dirichlet laplacian on an arbitrary bounded domain sufficient conditions for the validity of the weyl formula for the neumann laplacian on a domain with continuous boundary in terms of smoothness of the boundary and a remainder estimate in this formula in particular we show that the weyl formula holds true for the neumann laplacian on a lip domain whenever d 1 d prove that the remainder in this formula is o d 1 and give an example where the remainder estimate o d 1 is order sharp we use a new version of the variational technique which does not require the extension theorem

linear interactive encoding and decoding for lossless source coding with decoder only side information linear interactive encoding and decoding ied for near lossless source coding with decoder only side information is considered where the interactive encoder uses linear codes described by parity check matrices over a finite field x for encoding it is first demonstrated how to convert any classical universal lossless code cn with block length n and with side information available to both the encoder and decoder into a universal random linear ied scheme based on gallager s parity check ensemble it is then shown that there is no performance loss by restricting ied to linear ied and that the universal random linear ied scheme based on gallager s parity check ensemble achieves essentially the same rate performance as does cn for each and every individual sequence pair xn yn while the word decoding error probability goes to 0 as n define the density of a linear ied scheme as the percentage of nonzero entries in its parity check matrix to reduce the encoding complexity of linear ied low density linear ied is further investigated in terms of the trade off among its rate decoding error probability and density

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

new nonasymptotic channel coding theorems for structured codes new nonasymptotic random coding theorems with error probability epsilon and finite block length n based on gallager parity check ensemble and general parity check ensembles are derived in this paper the resulting nonasymptotic achievability bounds when combined with nonasymptotic equipartition properties developed in this paper can be easily computed analytically these nonasymptotic achievability bounds are shown to be asymptotically tight up to the second order of the coding rate as n goes to infinity with either constant or subexponentially decreasing epsilon in the case of gallager parity check ensemble and to imply that low density parity check ldpc codes be capacity achieving in the case of ldpc ensembles numerically they are also compared favorably for finite n and epsilon of practical interest with existing nonasymptotic achievability bounds in the literature

log sobolev inequalities and regions with exterior exponential cusps abstract we begin by studying certain semigroup estimates which are more singular than those implied by a sobolev embedding theorem but which are equivalent to certain logarithmic sobolev inequalities we then give a method for proving that such log sobolev inequalities hold for euclidean regions which satisfy a particular hardy type inequality our main application is to show that domains which have exterior exponential cusps and hence have no sobolev embedding theorem satisfy such heat kernel bounds provided the cusps are not too sharp finally we consider a rotationally invariant domain with an exponentially sharp cusp and prove that ultracontractivity breaks down when the cusp becomes too sharp

interactive encoding and decoding based on binary ldpc codes with syndrome accumulation interactive encoding and decoding based on binary low density parity check codes with syndrome accumulation sa ldpc ied is proposed and investigated assume that the source alphabet is gf 2 and the side information alphabet is finite it is first demonstrated how to convert any classical universal lossless code cn with block length n and side information available to both the encoder and decoder into a universal sa ldpc ied scheme it is then shown that with the word error probability approaching 0 subexponentially with n the compression rate including both the forward and backward rates of the resulting sa ldpc ied scheme is upper bounded by a functional of that of cn which in turn approaches the compression rate of cn for each and every individual sequence pair xn yn and the conditional entropy rate h x y for any stationary ergodic source and side information x y as the average variable node degree l of the underlying ldpc code increases without bound when applied to the class of binary source and side information x y correlated through a binary symmetrical channel with crossover probability unknown to both the encoder and decoder the resulting sa ldpc ied scheme can be further simplified yielding even improved rate performance versus the bit error probability when l is not large simulation results coupled with linear time belief propagation decoding on binary source side information pairs confirm the theoretic analysis and further show that the sa ldpc ied scheme consistently outperforms the slepian wolf coding scheme based on the same underlying ldpc code as a by product probability bounds involving ldpc established in the course are also interesting on their own and expected to have implications on the performance of ldpc for channel coding as well

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

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

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

sequences with finitely many negative squares this work shows how the more general theory of sequences with finitely many negative squares can be reduced to the classical theory of moments via linear recursion specifically if f n is a real sequence with negative squares then the main theorem asserts the existence of a minimal definitizing polynomial i e a nonnegative polynomial q x i 02 aixi of degree 2 so that g n i 02 aif n i is a moment sequence i e of the form g n xndv with v 0 the question of uniqueness of a minimal definitizing polynomial is discussed a levy khinchin type integral formula for f is then explicitly obtained by solving a difference equation of the form i 02 aif n i xn dv as a non trivial application of the main theorem it is shown that many sequences of number theoretic importance such as the sequence of primes have infinitely many positive and infinitely many negative squares some of our results are contained in the work of krein and langer but our proofs are different and they are to a great extent independent of the theory of spaces

fourier tauberian theorems and applications abstract the main aim of the paper is to present a general version of the fourier tauberian theorem for monotone functions this result together with berezin s inequality allows us to obtain a refined version of the li yau estimate for the counting function of the dirichlet laplacian which implies the li yau estimate itself and at the same time the asymptotic results obtained by the variational method

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

seminormal composition operators on l2 spaces induced by matrices the laplace density case abstract bounded composition operators with matrix symbols on l 2 where is a positive borel measure with a laplace type density function on the d dimensional euclidean space are studied hyponormality and subnormality of adjoints of such operators are characterized explicitly in terms of their symbols