preservation of a c spectrum for random decaying perturbations of square summable high order variation we consider random self adjoint jacobi matrices of the form j u n an u n 1 bn u n an 1 u n 1 on l2 n where an 0 and bn r are sequences of random variables on a probability space dp such that there exists q n such that for any l n 2l al al q and 2l 1 bl bl q are independent random variables of zero mean satisfying n 1 n2 dp let jp be the deterministic periodic of period q jacobi matrix whose coefficients are the mean values of the corresponding entries in j we prove that for a e the a c spectrum of the operator j equals to and fills the spectrum of jp if moreover n 1 n4 dp then for a e the spectrum of j is purely absolutely continuous on the interior of the bands that make up the spectrum of jp

lyapunov exponents one dimensional anderson localization and products of random matrices the concept of lyapunov exponent has long occupied a central place in the theory of anderson localisation its interest in this particular context is that it provides a reasonable measure of the localisation length the lyapunov exponent also features prominently in the theory of products of random matrices pioneered by furstenberg after a brief historical survey we describe some recent work that exploits the close connections between these topics we review the known solvable cases of disordered quantum mechanics involving random point scatterers and discuss a new solvable case finally we point out some limitations of the lyapunov exponent as a means of studying localisation properties

the mean velocity of a brownian motion in a random levy potential a brownian motion in a random levy potential v is the informal solution of the stochastic differential equation dx t db t 1 2v x t dt where b is a brownian motion independent of v we generalize some results of kawazu tanaka who considered for v a brownian motion with drift by proving that x t t converges almost surely to a constant the mean velocity which we compute in terms of the levy exponent of v defined by e e mv t e t m

on surface radiation conditions for an ellipse we compare several on surface radiation boundary conditions in two dimensions for solving the helmholtz equation exterior to an ellipse we also introduce a new boundary condition for an ellipse based on a modal expansion in mathieu functions we compare the osrc to a finite difference method

an improved surface radiation condition for high frequency acoustic scattering problems a new artificial boundary condition for two dimensional acoustic scattering problems is constructed in this paper it is designed from the principal symbol of the dirichlet to neumann map so that both the propagating and evanescent fields are properly modeled in the high frequency regime the new artificial boundary condition can be used as an on surface radiation condition for efficient approximation of the wave field the accuracy and efficiency of our method are demonstrated in numerical examples

three dimensional finite element calculations in acoustic scattering using arbitrarily shaped convex artificial boundaries we report on a generalization of the bayliss gunzburger turkel non reflecting boundary conditions to arbitrarily shaped convex artificial boundaries for elongated scatterers such as submarines we show that this generalization can improve significantly the computational efficiency of finite element methods applied to the solution of three dimensional acoustic scattering problems copyright 2001 john wiley sons ltd

singular continuous spectrum is generic in a variety of contexts we prove that singular continuous spectrum is generic in the sense that for certain natural complete metric spaces of operators those with singular spectrum are a dense g delta

a quasi optimal non overlapping domain decomposition algorithm for the helmholtz equation this paper presents a new non overlapping domain decomposition method for the helmholtz equation whose effective convergence is quasi optimal these improved properties result from a combination of an appropriate choice of transmission conditions and a suitable approximation of the dirichlet to neumann operator a convergence theorem of the algorithm is established and numerical results validating the new approach are presented in both two and three dimensions

fluctuations of random matrix products and 1d dirac equation with random mass we study the fluctuations of certain random matrix products pi n m n cdots m 2m 1 of mathrm sl 2 mathbb r describing localisation properties of the one dimensional dirac equation with random mass in the continuum limit i e when matrices m n s are close to the identity matrix we obtain convenient integral representations for the variance gamma 2 lim n rightarrow infty mathrm var ln pi n n the case studied exhibits a saturation of the variance at low energy varepsilon along with a vanishing lyapunov exponent gamma 1 lim n rightarrow infty ln pi n n leading to the behaviour gamma 2 gamma 1 sim ln 1 varepsilon rightarrow infty as varepsilon rightarrow 0 our continuum description sheds new light on the kappus wegner band center anomaly

on the complexity of the rank syndrome decoding problem in this paper we propose two new generic attacks on the rank syndrome decoding rsd problem let c be a random n k rank code over gf q m and let y x e be a received word such that x in c and rank e r the first attack the support attack is combinatorial and permits to recover an error e of rank weight r in min o n k 3 m 3 q r lfloor km n rfloor o n k 3 m 3 q r 1 lfloor k 1 m n rfloor operations on gf q this new attack improves the exponent for the best generic attack for the rsd problem in the case n m by introducing the ratio m n in the exponential coefficient of the previously best known attacks the second attack the annulator polynomial attack is an algebraic attack based on the theory of q polynomials introduced by ore we propose a new algebraic setting for the rsd problem that permits to consider equations and unknowns in the extension field gf q m rather than in gf q as it is usually the case we consider two approaches to solve the problem in this new setting the linearization technique shows that if n ge k 1 r 1 1 the rsd problem can be solved in polynomial time more generally we prove that if lceil r 1 k 1 n 1 r rceil le k the rsd problem can be solved with an average complexity of o r 3 k 3 q r lceil r 1 k 1 n 1 r rceil operations in the base field gf q we also consider solving with grobner bases for which we discuss theoretical complexity we also consider hybrid solving with grobner bases on practical parameters as an example of application we use our new attacks on all recent cryptosystems parameters which repair the gpt cryptosystem we break all examples of published proposed parameters and some parameters are broken in less than 1 s in certain cases

localization for off diagonal disorder and for continuous schrodinger operators we extend the proof of localization by delyon levy and souillard to accommodate the anderson model with off diagonal disorder and the continuous schrodinger equation with a random potential

efficient encryption from random quasi cyclic codes we propose a framework for constructing efficient code based encryption schemes that do not hide any structure in their public matrix the framework is in the spirit of the schemes first proposed by alekhnovich in 2003 and based on the difficulty of decoding random linear codes from random errors of low weight we depart somewhat from alekhnovich s approach and propose an encryption scheme based on the difficulty of decoding random quasi cyclic codes we propose two new cryptosystems instantiated within our framework the hamming quasi cyclic cryptosystem hqc based on the hamming metric and the rank quasi cyclic cryptosystem rqc based on the rank metric we give a security proof which reduces the indistinguishability under chosen plaintext attack security of our systems to a decision version of the well known problem of decoding random families of quasi cyclic codes for the hamming and rank metrics the respective mathsf qcsd and mathsf rqcsd problems we also provide an analysis of the decryption failure probability of our scheme in the hamming metric case for the rank metric there is no decryption failure our schemes benefit from a very fast decryption algorithm together with small key sizes of only a few thousand bits the cryptosystems are very efficient for low encryption rates and are very well suited to key exchange and authentication asymptotically for lambda the security parameter the public key sizes are respectively in mathcal o lambda 2 for hqc and in mathcal o left lambda frac 4 3 right for rqc practical parameter compares well to the systems based on ring learning parity with noise or the recent moderate density parity check codes system

a fast method for solving acoustic scattering problems in frequency bands abstract we present a methodology for computing efficiently scattered fields in a frequency band the main feature of this methodology is the construction of a series of discrete problems that differ only by their right hand sides rather than by both their left and right hand sides its key steps are a the reformulation of the acoustic scattering problem in a bounded domain using any preferred absorbing boundary condition b the characterization of the repeated derivatives with respect to the frequency of the scattered field as solutions of scattering type problems with different source terms and boundary conditions and c the reconstruction of a scattered field by either the pade approximants or wynn s algorithm we report on several multifrequency acoustic scattering examples that illustrate the accuracy and computational efficiency of the proposed solution methodology

non conforming galerkin finite element methods for local absorbing boundary conditions of higher order a new non conforming finite element discretization methodology for second order elliptic partial differential equations involving higher order local absorbing boundary conditions in 2d and 3d is proposed the novelty of the approach lies in the application of c 0 continuous finite element spaces which is the standard discretization of second order operators to the discretization of boundary differential operators of order four and higher for each of these boundary operators additional terms appear on the boundary nodes in 2d and on the boundary edges in 3d similarly to interior penalty discontinuous galerkin methods which leads to a stable and consistent formulation in this way no auxiliary variables on the boundary have to be introduced and trial and test functions of higher smoothness along the boundary are not required as a consequence the method leads to lower computational costs for discretizations with higher order elements and is easily integrated in high order finite element libraries a priori h convergence error estimates show that the method does not reduce the order of convergence compared to usual dirichlet neumann or robin boundary conditions if the polynomial degree on the boundary is increased simultaneously a series of numerical experiments illustrates the utility of the method and validates the theoretical convergence results

weak commutation relations and eigenvalue statistics for products of rectangular random matrices we study the joint probability density of the eigenvalues of a product of rectangular real complex or quaternion random matrices in a unified way the random matrices are distributed according to arbitrary probability densities whose only restriction is the invariance under left and right multiplication by orthogonal unitary or unitary symplectic matrices respectively we show that a product of rectangular matrices is statistically equivalent to a product of square matrices hereby we prove a weak commutation relation of the random matrices at finite matrix sizes which previously has been discussed for infinite matrix size moreover we derive the joint probability densities of the eigenvalues to illustrate our results we apply them to a product of random matrices drawn from ginibre ensembles and jacobi ensembles as well as a mixed version thereof for these weights we show that the product of complex random matrices yields a determinantal point process while the real and quaternion matrix ensembles correspond to pfaffian point processes our results are visualized by numerical simulations furthermore we present an application to a transport on a closed disordered chain coupled to a particle bath

high order surface radiation conditions for time harmonic waves in exterior domains abstract we formulate a new family of high order on surface radiation conditions to approximate the outgoing solution to the helmholtz equation in exterior domains motivated by the pseudo differential expansion of the dirichlet to neumann operator developed by antoine et al 1999 we design a systematic procedure to apply pseudo differential symbols of arbitrarily high order numerical results are presented to illustrate the performance of the proposed method for solving both the dirichlet and the neumann boundary value problems possible improvements and extensions are also discussed

lyapunov functions identities and the cauchy problem for the hele shaw equation this article is devoted to the study of the hele shaw equation we introduce an approach inspired by the water wave theory starting from a reduction to the boundary introducing the dirichlet to neumann operator and exploiting various cancellations we exhibit parabolic evolution equations for the horizontal and vertical traces of the velocity on the free surface this allows to quasi linearize the equations in a natural way by combining these exact identities with convexity inequalities we prove the existence of hidden lyapunov functions of different natures we also deduce from these identities and previous works on the water wave problem a simple proof of the well posedness of the cauchy problem the analysis contains two side results of independent interest firstly we give a principle to derive estimates for the modulus of continuity of a pde under general assumptions on the flow secondly we prove and give applications of a convexity inequality for the dirichlet to neumann operator

two parameter spectral averaging and localization for non monotonic random schrodinger operators we prove exponential localization at all energies for two types of one dimensional random schr6dinger operators the poisson model and the random displacement model as opposed to anderson type models these operators are not monotonic in the random parameters therefore the classical one parameter version of spectral averaging as used in localization proofs for anderson models breaks down we use the new method of two parameter spectral averaging and apply it to the poisson as well as the displacement case in addition we apply results from inverse spectral theory which show that two parameter spectral averaging works for sufficiently many energies all but a discrete set to conclude localization at all energies

on the dimension of subfield subcodes a new lower bound for the dimension of a subfield subcode is given this bound improves the well known lower bound of p delsarte 1975 and coincides in many cases with the true dimension of the code alternant codes and geometric goppa codes are discussed as special cases

products of random matrices and generalised quantum point scatterers to every product of 2 2 matrices there corresponds a one dimensional schrodinger equation whose potential consists of generalised point scatterers products of random matrices are obtained by making these interactions and their positions random we exhibit a simple one dimensional quantum model corresponding to the most general product of matrices in sl 2 we use this correspondence to find new examples of products of random matrices for which the invariant measure can be expressed in simple analytical terms

fluctuations of the product of random matrices and generalized lyapunov exponent i present a general framework allowing to carry out explicit calculation of the moment generating function of random matrix products varpi n m nm n 1 ldots m 1 where m i s are i i d following tutubalin theor probab appl 10 1 15 27 1965 the calculation of the generating function is reduced to finding the largest eigenvalue of a certain transfer operator associated with a family of representations of the group the formalism is illustrated by considering products of random matrices from the group mathrm sl 2 mathbb r where explicit calculations are possible for concreteness i study in detail transfer matrix products for the one dimensional schrodinger equation where the random potential is a levy noise derivative of a levy process in this case i obtain a general formula for the variance of ln varpi n and for the variance of ln psi x where psi x is the wavefunction in terms of a single integral involving the fourier transform of the invariant density of the matrix product finally i discuss the continuum limit of random matrix products matrices close to the identity in particular i investigate a simple case where the spectral problem providing the generalized lyapunov exponent can be solved exactly

absorbing boundary conditions on arbitrary boundaries for the scalar and vector wave equations the solution of open region scattering problems involving inhomogeneous arbitrarily shaped objects may be performed through the use of partial differential equation techniques which require enclosing the scatterer by an outer boundary on which an absorbing boundary condition abc is applied in order to minimize the size of the domain to be meshed and consequently the number of unknowns if may be advisable to implement abc s devised for outer boundaries of arbitrary shapes such abc s are obtained for the 3d scalar and vector wave equations they incorporate most of existing boundary conditions when used in conjunction with a finite element technique the numerical results derived by using a simplified form of these abc s compare favourably to those obtained by using a rigorous hybrid finite element integral equation formulation these boundary conditions have been obtained in the frequency domain framework they may however be used in time domain calculations

on metrics for error correction in network coding the problem of error correction in both coherent and noncoherent network coding is considered under an adversarial model for coherent network coding where knowledge of the network topology and network code is assumed at the source and destination nodes the error correction capability of an outer code is succinctly described by the rank metric as a consequence it is shown that universal network error correcting codes achieving the singleton bound can be easily constructed and efficiently decoded for noncoherent network coding where knowledge of the network topology and network code is not assumed the error correction capability of a subspace code is given exactly by a new metric called the injection metric which is closely related to but different than the subspace metric of koumltter and kschischang in particular in the case of a non constant dimension code the decoder associated with the injection metric is shown to correct more errors then a minimum subspace distance decoder all of these results are based on a general approach to adversarial error correction which could be useful for other adversarial channels beyond network coding

on the similarities between generalized rank and hamming weights and their applications to network coding rank weights and generalized rank weights have been proved to characterize error and erasure correction and information leakage in linear network coding in the same way as hamming weights and generalized hamming weights describe classical error and erasure correction and information leakage in wire tap channels of type ii and code based secret sharing although many similarities between both the cases have been established and proved in the literature many other known results in the hamming case such as bounds or characterizations of weight preserving maps have not been translated to the rank case yet or in some cases have been proved after developing a different machinery the aim of this paper is to further relate both weights and generalized weights show that the results and proofs in both cases are usually essentially the same and see the significance of these similarities in network coding some of the new results in the rank case also have new consequences in the hamming case

absence of diffusion in the anderson tight binding model for large disorder or low energy we prove that the green s function of the anderson tight binding hamiltonian decays exponentially fast at long distances on v with probability 1 we must assume that either the disorder is large or the energy is sufficiently low our proof is based on perturbation theory about an infinite sequence of block hamiltonians and is related to kam methods

singular continuous spectrum under rank one perturbations and localization for random hamiltonians we consider a selfadjoint operator a and a selfadjoint rank one projection p onto a vector which is cyclic for a in terms of the spectral measure d a we give necessary and sufficient conditions for a p to have empty singular continuous spectrum or to have only point spectrum for a e we apply these results to questions of localization in the one and multi dimensional anderson models

fine structure of the zeros of orthogonal polynomials iv a priori bounds and clock behavior we prove locally uniform spacing for the zeros of orthogonal polynomials on the real line under weak conditions jacobi parameters approach the free ones and are of bounded variation we prove that for ergodic discrete schrodinger operators poisson behavior implies a positive lyapunov exponent both results depend on a priori bounds on eigenvalue spacings for which we provide several proofs

analytical and numerical investigation of the performance of the bgt2 condition for low frequency acoustic scattering problems a mathematical and numerical analysis is performed to assess the performance of the second order bayliss gunzburger turkel bgt2 condition when applied to solving low frequency acoustic scattering problems in the case of elongated scatterers this investigation suggests that bgt2 retains an acceptable level of accuracy for relatively low wavenumber a damping effect is incorporated to the bgt2 condition in order to extend the range of satisfactory performance this damping procedure consists in adding only a constant imaginary part to the wavenumber the numerical results indicate that the modified version of bgt2 extends the range of satisfactory performance by improving the level of accuracy by up to two orders of magnitude guidelines on the appropriate choice of the damping coefficient are provided

absorbing boundary conditions for relativistic quantum mechanics equations this paper is devoted to the derivation of absorbing boundary conditions for the klein gordon and dirac equations modeling quantum and relativistic particles subject to classical electromagnetic fields microlocal analysis is the main ingredient in the derivation of these boundary conditions which are obtained in the form of pseudo differential equations basic numerical schemes are derived and analyzed to illustrate the accuracy of the derived boundary conditions

phase reduction models for improving the accuracy of the finite element solution of time harmonic scattering problems i general approach and low order models this paper introduces a new formulation of high frequency time harmonic scattering problems in view of a numerical finite element solution it is well known that pollution error causes inaccuracies in the finite element solution of short wave problems to partially avoid this precision problem the strategy proposed here consists in firstly numerically computing at a low cost an approximate phase of the exact solution through asymptotic propagative models secondly using this approximate phase a slowly varying unknown envelope is introduced and is computed using coarser mesh grids the global procedure is called phase reduction in this first paper the general theoretical procedure is developed and low order propagative models are numerically investigated in detail improved solutions based on higher order models are discussed showing the potential of the method for further developments

imbedded singular continuous spectrum for schrodinger operators 1 1 hv v x and some self adjoint boundary condition at zero the operator 1 1 describes a charged particle such as an electron in the electric field v x when v x is decaying quickly one expects the spectral and dynamical properties of hy to remain close to those of the free operator ho recall that with the operator 1 1 one can associate in a canonical way a spectral measure p see for example 6 2 which contains much information about the quantum system the classical result in this direction going back to the beginning of the century is that if v g l1 the spectral measure on the positive semi axis remains purely absolutely continuous in one dimension an absolutely continuous spectrum corresponds to the ballistic rate of propagation of the particle a natural question to ask is what are the critical rates of decay at which some changes in the spectral and dynamical properties of hy may happen as far as v x has any decay at all the essential spectrum coincides with 0 oo but the quality of the spectrum and dynamics may change in 1928 wigner and von neumann 27 showed that there exist potentials v x satisfying v x jsuch that hy has positive imbedded eigenvalue e 1 this is a purely quantum resonance phenomenon since v can be chosen arbitrarily small and the bound state originates from long range correlations in the potential rather than the usual confining effect naboko 17 and simon 25 provided constructions which show that much more drastic changes are possible if the potential decays arbitrarily slower than coulomb namely for any positive function h x which grows at infinity there exist potentials v such that v x yjgand hy has a dense set of positive eigenvalues due to some recent results 3 21 it is known that the absolutely continuous spectrum on the positive semi axis is preserved for v o c l x l 2 e and in fact for v g l2 8 more precisely for such potentials the absolutely continuous part of the spectral measure iac gives positive weight to any subset of 0 oo of positive lebesgue measure therefore the naboko

one dimensional schrodinger operators with random decaying potentials we investigate the spectrum of the following random schrodinger operators h omega frac d 2 dt 2 a t f x t omega wheref xt is a markovian potential studied by the russian school 8 we completely describe the transition of the spectrum from pure point type to absolutely continuous type as the decreasing order ofa t grows this is an extension to a continuous case of the result due to delyon simon souillard 6 who deal with the lattice case

wide angle one way wave equations a one way wave equation also known as a paraxial or parabolic wave equation is a differential equation that permits wave propagation in certain directions only such equations are used regularly in underwater acoustics in geophysics and as energy absorbing numerical boundary conditions the design of a one way wave equation is connected with the approximation of 1 s2 1 2 on 1 1 by a rational function which has usually been carried out by pade approximation this article presents coefficients for l2 l and other alternative classes of approximants that have better wide angle behavior for theoretical results establishing the well posedness of these wide angle equations see the work of trefethen and halpern well posedness of one way wave equations and absorbing boundary conditions math comput 47 421 435 1986

anderson localization for one and quasi one dimensional systems we prove almost sure exponential localization of all the eigenfunctions and nondegeneracy of the spectrum for random discrete schrodinger operators on one and quasi one dimensional lattices this paper provides a much simpler proof of these results than previous approaches and extends to a much wider class of systems we remark in particular that the singular continuous spectrum observed in some quasiperiodic systems disappears under arbitrarily small local perturbations of the potential our results allow us to prove that e g for strong disorder the smallest positive lyapunov exponent of some products of random matrices does not vanish as the size of the matrices increases to infinity

bayliss turkel like radiation conditions on surfaces of arbitrary shape this paper addresses the extension of the bayliss turkel second order radiation condition to an arbitrarily shaped surface the derivation is based mainly on the pseudo differential calculus as well as on the introduction of a criterion providing a precise handling of the approximation process involved in the derivation of the radiation condition the radiation condition then ranges among the most accurate of those of order two as a by product of the derivation almost all known radiation conditions of order less than or equal to two are recovered and their respective accuracies are compared

reducible rank codes and their applications to cryptography we present a new family of so called reducible rank codes which are a generalization of rank product codes this family includes maximal rank distance mrd codes for lengths n n in the field f sub n we give methods for encoding and decoding reducible rank codes a public key cryptosystem based on these codes and on the idea of a column scrambler is proposed the column scrambler mixes columns of a generator parity check matrix of a code it makes the system more resistant to structural attacks such as gibson s attacks possible attacks on the system are thoroughly studied the system is found to be secure against known attacks for public keys of about 16 kbits and greater

a high order absorbing boundary condition for 2d time harmonic elastodynamic scattering problems abstract in this paper we are concerned with the construction of a new high order absorbing boundary condition abc for 2d elastic scattering problems it is defined by an approximate local dirichlet to neumann dtn map first we explain the derivation of this approximation next a detailed analytical study in terms of hankel functions in the circular case is addressed the new abc is compared with the standard low order lysmer kuhlemeyer abc finally its accuracy and efficiency are investigated for various numerical examples particularly at high frequencies

corner treatments for high order local absorbing boundary conditions in high frequency acoustic scattering this paper deals with the design and validation of accurate local absorbing boundary conditions set on convex polygonal computational domains for the finite element solution of high frequency acoustic scattering problems while high order absorbing boundary conditions habcs are accurate for smooth fictitious boundaries the precision of the solution drops in the presence of corners if no specific treatment is applied we present and analyze two strategies to preserve the accuracy of pade type habcs at corners first by using compatibility relations derived for right angle corners and second by regularizing the boundary at the corner we show that the former strategy is well adapted to right corners and efficient for nearly right corners while the later is better for very obtuse corners numerical results are proposed to analyze and compare the approaches for two and three dimensional problems

boundary conditions for the numerical solution of elliptic equations in exterior regions elliptic equations in exterior regions frequently require a boundary condition at infinity to ensure the weill posedness of the problem examples of practical applications include the helmholtz equation and laplace s equation computational procedures based on a direct discretization of the elliptic problem require the replacement of the condition at infinity by a boundary condition on a finite artificial surface direct imposition of the condition at infinity along the finite boundary results in large errors a sequence of boundary conditions is developed which provides increasingly accurate approximations to the problem in the infinite domain estimates of the error due to the finite boundary are obtained for several cases computations are presented which demonstrate the increased accuracy that can be obtained by the use of the higher order boundary conditions the examples are based on a finite element formulation but finite difference methods can also be used

a generalized thouless formula as a criterion for anderson localization in two and three dimensional systems abstract a generalized thouless formula is proposed as an approximate criterion for anderson localization in two and three dimensional systems the criterion is exact for ordered systems and in the mean field limit of infinite dimensionality for random systems it is expected that possible corrections are small ample numerical evidence is provided through an exact diagonalization of several large size hamiltonians the predictions of the thouless formula whose evaluation only needs the eigenvalues are critically checked against the inverse participation ratio ipr of the corresponding eigenvectors included are the lloyd model for sufficiently low disorder mobility edges cannot be excluded and surprising new results on the binary alloy problem the criterion predicts four as the upper and two as the lower critical dimension

on surface radiation condition for multiple scattering of waves the formulation of the on surface radiation condition osrc is extended to handle wave scattering problems in the presence of multiple obstacles the new multiple osrc simultaneously accounts for the outgoing behavior of the wave fields as well as the multiple wave reflections between the obstacles like boundary integral equations bie this method leads to a reduction in dimensionality from volume to surface of the discretization region however as opposed to bie the proposed technique leads to boundary integral equations with smooth kernels hence these fredholm integral equations can be handled accurately and robustly with standard numerical approaches without the need to remove singularities moreover under weak scattering conditions this approach renders a convergent iterative method which bypasses the need to solve single scattering problems at each iteration inherited from the original osrc the proposed multiple osrc is generally a crude approximate method if accuracy is not satisfactory this approach may serve as a good initial guess or as an inexpensive pre conditioner for krylov iterative solutions of bie

convergence of krylov subspace solvers with schwarz preconditioner for the exterior maxwell problem the consideration of an integral representation as an exact boundary condition for the finite element resolution of wave propagation problems in exterior domain induces algorithmic difficulties in this paper we are interested in the resolution of an exterior maxwell problem in 3d as a first step we focus on the justification of an algorithm described in literature using an interpretation as a schwarz method the study of the convergence indicates that it depends significantly on the thickness of the domain of computation this analysis suggests the use of the finite element term of schwarz method as a preconditioner for use of krylov iterative solvers an analytical study of the case of a spherical perfect conductor indicates the efficiency of such approach the consideration of the preconditioner suggested by the schwarz method leads to a superlinear convergence of the gmres predicted by the analytical study and verified numerically

franco american meeting on the mathematics of random and almost periodic potentials we present the abstracts of papers given at a conference held at the ecole polytechnique in june 1983 on the mathematics of random and almost periodic potentials

an approximate boundary condition in acoustics an approximate formula connecting the field and its normal derivative on the surface of a body is derived from comparison with exact results for a sphere and geometrical acoustics for a general obstacle the formula appears to be effective in the middle range of frequencies although it is easy to apply and its structure is simple

local absorbing boundary conditions for elliptical shaped boundaries we compare several local absorbing boundary conditions for solving the helmholtz equation by a finite difference or finite element method exterior to a general scatterer these boundary conditions are imposed on an artificial elliptical or prolate spheroid outer surface in order to compare the computational solution with an analytical solution we consider as an example scattering about an ellipse we solve the helmholtz equation with both finite differences and finite elements we also introduce a new boundary condition for an ellipse based on a modal expansion

on the hardness of the decoding and the minimum distance problems for rank codes we give a randomized reduction for the rank syndrome decoding problem and rank minimum distance problem for rank codes over extension fields our results are based on embedding linear codes in the hamming space into linear codes over an extension field equipped with the rank metric we prove that if any of the previous problems for the rank metric is in zpp rp cap corp then we would have np zpp we also give complexity results for the respective rank metric approximation problems

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

on the statistics of superlocalized states in self affine disordered potentials we investigate the statistics of eigenstates in a weak self affine disordered potential in one dimension whose gaussian fluctuations grow with distance with a positive hurst exponent h typical eigenstates are superlocalized on samples much larger than a well defined crossover length which diverges in the weak disorder regime we present a parallel analytical investigation of the statistics of these superlocalized states in the discrete and the continuum formalisms for the discrete tight binding model the effective localization length decays logarithmically with the sample size and the logarithm of the transmission is marginally self averaging for the continuum schrodinger equation the superlocalization phenomenon has more drastic effects the effective localization length decays as a power of the sample length and the logarithm of the transmission is fully non self averaging

localization in one dimensional random media a scattering theoretic approach we use scattering theoretic methods to prove exponential localization for random displacement models in one dimension the operators we consider model both quantum and classical wave propagation our main tools are the reflection and transmission coefficients for compactly supported single site perturbations we show that randomly displaced non reflectionless single sites lead to localization

exact solutions for spectra and green s functions in random one dimensional systems for chains of harmonic oscillators with random masses a set of equations is derived which determine the spatial fourier components of the average one particle green s function these equations are valid for complex values of the frequency a relation between the spectral density and functions introduced by schmidt is discussed exact solutions for this green s function and the less complicated characteristics function the analytic continuation into the complex frequency plane of the accumulated spectral density and the inverse localization length of the eigenfunctions are derived for exponential distributions of the masses for some cases the characteristic function is calculated numerically for gamma distributions the equations are cast in the form of ordinary higher order differential equations these have been solved numerically for determining the characteristic function for arbitrary mass distributions a cumulant expansion and a peculiar symmetry of the green s function are discussed the method is also applied to chains where the spring constants and or the masses have random values also for these systems exact solutions are discussed for exponential distributions e g of both masses and spring constants the characteristic function is expressed in bessel functions the relation with certain random relaxation models is shown finally x y hamiltonians with random exchange constants and or magnetic fields or equivalently tight binding electron models with diagonal and or off diagonal disorder are considered here the green s function does not depend on the wave number if the distribution of exchange constants is symmetric around the origin new solutions for the characteristic function and green s function are derived for a number of cases including exponentially distributed magnetic fields and power law distributed exchange constants

unbounded trace orbits of thue morse hamiltonian it is well known that an energy is in the spectrum of fibonacci hamiltonian if and only if the corresponding trace orbit is bounded however it is not known whether the same result holds for the thue morse hamiltonian in this paper we give a negative answer to this question more precisely we construct two subsets sigma ii and sigma iii of the spectrum of the thue morse hamiltonian both of which are dense and uncountable such that each energy in sigma ii cup sigma iii corresponds to an unbounded trace orbit exact estimates on the norm of the transfer matrices are also obtained for these energies for e in sigma ii cup sigma iii the norms of the transfer matrices behave like begin aligned e c 1 gamma sqrt n le vert t n e vert le e c 2 gamma sqrt n end aligned however two types of energies are quite different in the sense that each energy in sigma ii is associated with a two sided pseudo localized state while each energy in sigma iii is associated with a one sided pseudo localized state the difference is also reflected by the local dimensions of the spectral measure the local dimension is 0 for energies in sigma ii and is larger than 1 for energies in sigma iii as a comparison we mention another known countable dense subset sigma i each energy in sigma i corresponds to an eventually constant trace map and the associated eigenvector is an extended state in summary the thue morse hamiltonian exhibits mixed spectral nature

numerical accuracy of a pade type non reflecting boundary condition for the finite element solution of acoustic scattering problems at high frequency the present text deals with the numerical solution of two dimensional high frequency acoustic scattering problems using a new high order and asymptotic pade type artificial boundary condition the pade type condition is easy to implement in a galerkin least squares iterative finite element solver for arbitrarily convex shaped boundaries the method accuracy is investigated for different model problems and for the scattering problem by a submarine shaped scatterer as a result relatively small computational domains optimized according to the shape of the scatterer can be considered while yielding accurate computations for high frequencies copyright 2005 john wiley sons ltd

breaking supersymmetry in a one dimensional random hamiltonian the one dimensional supersymmetric random hamiltonian where x is a gaussian white noise of zero mean and variance g presents particular spectral and localization properties at low energy a dyson singularity in the integrated density of states idos n e 1 ln2e and a delocalization transition related to the behavior of the lyapunov exponent inverse localization length vanishing like e 1 ln e as e 0 we study how this picture is affected by breaking supersymmetry with a scalar random potential h hsusy v x where v x is a gaussian white noise of variance in the limit g3 a fraction of states n 0 g ln2 g3 migrate to the negative spectrum and the lyapunov exponent reaches a finite value 0 g ln g3 at e 0 the exponential lifshits tail of the idos for e is studied in detail and is shown to involve a competition between the two noises x and v x whatever the larger is this analysis relies on analytic results for n e and e obtained by two different methods a stochastic method and the replica method the problem of extreme value statistics of eigenvalues is also considered distribution of the nth excited state energy the results are analyzed in the context of classical diffusion in a random force field in the presence of random annihilation creation local rates

schrodinger operators with an electric field and random or deterministic potentials we prove that the schrodinger operatorh d2 dx2 v x f x has purely absolutely continuous spectrum for arbitrary constant external fieldf for a large class of potentials this result applies to many periodic almost periodic and random potentials and in particular to random wells of independent depth for which we prove that whenf 0 the spectrum is almost surely pure point with exponentially decaying eigenfunctions

universal secure network coding via rank metric codes the problem of securing a network coding communication system against an eavesdropper is considered the network implements linear network coding to deliver n packets from source to each receiver and the adversary can eavesdrop on arbitrarily chosen links the objective is to provide reliable communication to all receivers while guaranteeing that the source information remains information theoretically secure from the adversary a coding scheme is proposed that can achieve the maximum possible rate of n packets the scheme which is based on rank metric codes has the distinctive property of being universal it can be applied on top of any communication network without requiring knowledge of or any modifications on the underlying linear network code the only requirement of the scheme is that the packet length be at least n which is shown to be strictly necessary for universal communication at the maximum rate a further scenario is considered where the adversary is allowed not only to eavesdrop but also to inject up to t erroneous packets into the network and the network may suffer from a rank deficiency of at most in this case the proposed scheme can be extended to achieve the rate of n 2t packets this rate is shown to be optimal under the assumption of zero error communication

individual energy level distributions for one dimensional diagonal and off diagonal disorder we study the distribution of the n th energy level for two different one dimensional random potentials this distribution is shown to be related to the distribution of the distance between two consecutive nodes of the wave function we first consider the case of a white noise potential and study the distributions of energy level both in the positive and the negative part of the spectrum it is demonstrated that in the limit of a large system l the distribution of the n th energy level is given by a scaling law which is shown to be related to the extreme value statistics of a set of independent variables in the second part we consider the case of a supersymmetric random hamiltonian potential v x x 2 x we study first the case of x being a white noise with zero mean it is in particular shown that the ground state energy which behaves on average like exp l 1 3 in agreement with previous work is not a self averaging quantity in the limit l as is seen in the case of diagonal disorder then we consider the case when x has a non zero mean value

supersymmetric quantum mechanics with levy disorder in one dimension we consider the schrodinger equation with a random potential of the form open image in new window where w is a levy noise we focus on the problem of computing the so called complex lyapunov exponent open image in new window where n is the integrated density of states of the system and is the lyapunov exponent in the case where the levy process is non decreasing we show that the calculation of reduces to a stieltjes moment problem we ascertain the low energy behaviour of the density of states in some generality and relate it to the distributional properties of the levy process we review the known solvable cases where can be expressed in terms of special functions and discover a new one

artificial boundary conditions for one dimensional cubic nonlinear schrodinger equations this paper addresses the construction of nonlinear integro differential artificial boundary conditions for one dimensional nonlinear cubic schrodinger equations several ways of designing such conditions are provided and a theoretical classification of their accuracy is given semidiscrete time schemes based on the method developed by duran and sanz serna ima j numer anal 20 2000 pp 235 261 are derived for these unusual boundary conditions stability results are stated and several numerical tests are performed to analyze the capacity of the proposed approach

analytic preconditioners for the electric field integral equation since the advent of the fast multipole method large scale electromagnetic scattering problems based on the electric field integral equation efie formulation are generally solved by a krylov iterative solver a well known fact is that the dense complex non hermitian linear system associated to the efie becomes ill conditioned especially in the high frequency regime as a consequence this slows down the convergence rate of krylov subspace iterative solvers in this work a new analytic preconditioner based on the combination of a finite element method with a local absorbing boundary condition is proposed to improve the convergence of the iterative solver for an open boundary some numerical tests precise the behaviour of the new preconditioner moreover comparisons are performed with the analytic preconditioner based on the calderon s relations for integral equations for several kinds of scatterers copyright 2004 john wiley sons ltd

on the essential spectrum of schrodinger operators with spherically symmetric potentials on determine le spectre essentiel des operateurs de schrodinger avec un potentiel a symetrie spherique v x q x

the lyapunov exponent of products of random 2 2 matrices close to the identity we study products of arbitrary random real 2 2 matrices that are close to the identity matrix using the iwasawa decomposition of sl 2 we identify a continuum regime where the mean values and the covariances of the three iwasawa parameters are simultaneously small in this regime the lyapunov exponent of the product is shown to assume a scaling form in the general case the corresponding scaling function is expressed in terms of gauss hypergeometric function a number of particular cases are also considered where the scaling function of the lyapunov exponent involves other special functions airy bessel whittaker elliptic the general solution thus obtained allows us among other things to recover in a unified framework many results known previously from exactly solvable models of one dimensional disordered systems

nested linear lattice codes for structured multiterminal binning network information theory promises high gains over simple point to point communication techniques at the cost of higher complexity however lack of structured coding schemes limited the practical application of these concepts so far one of the basic elements of a network code is the binning scheme wyner 1974 1978 and other researchers proposed various forms of coset codes for efficient binning yet these schemes were applicable only for lossless source or noiseless channel network coding to extend the algebraic binning approach to lossy source or noisy channel network coding previous work proposed the idea of nested codes or more specifically nested parity check codes for the binary case and nested lattices in the continuous case these ideas connect network information theory with the rich areas of linear codes and lattice codes and have strong potential for practical applications we review these developments and explore their tight relation to concepts such as combined shaping and precoding coding for memories with defects and digital watermarking we also propose a few novel applications adhering to a unified approach

h and p adaptivity driven by recovery and residual based error estimators for pht splines applied to time harmonic acoustics abstract in this work we demonstrate the application of pht splines for time harmonic acoustic problems modeled by the helmholtz equation solutions of the helmholtz equation have two features global oscillations associated with the wave number and local gradients caused by geometrical irregularities we show that after a sufficient number of degrees of freedom is used to approximate global oscillations adaptive refinement can capture local features of the solution we compare residual based and recovery based error estimators and investigate the performance of p refinement the simulations are done in the context of recently introduced geometry independent field approximation gift where pht splines are only used to approximate the solution while the computational domain is parameterized with nurbs this approach builds on the natural adaptation ability of pht splines and avoids the re parameterization of the nurbs geometry during the solution refinement process

semianalytical solution of the random product problem of matrices and discrete time random evolution we consider the product of a large number of matrices chosen randomly with some correlation as a or b we introduce an exact functional equation whose solution gives the limiting distribution and the lyapunov exponent for the product of large number of matrices thus we suggest new semianalytical method to solve the random product problem of correlated matrices the method can be easily generalized to high order matrices as we deal with a larger systems of functional equations we have provided a known numerical algorithm for the solution of the matrix product problem we have also applied our method for a solution of discrete time random evolution our functional equation method provides exact transition point

construction of local boundary conditions for an eigenvalue problem using micro local analysis application to optical waveguide problems we present a general procedure based on the pseudo differential calculus for deriving artificial boundary conditions for an eigenvalue problem that characterizes the propagation of guided modes in optical waveguides this new approach allows the construction of local conditions that a are independent of the frequency regime b preserve the sparsity pattern of the finite element discretization and c are applicable to arbitrarily shaped convex artificial boundaries the last feature has the potential for reducing the size of the computational domain numerical results are presented to highlight the potential of conditions of order 1 2 and 1 for improving significantly the computational efficiency of finite element methods for the solution of optical waveguide problems

relative generalized rank weight of linear codes and its applications to network coding by extending the notion of minimum rank distance this paper introduces two new relative code parameters of a linear code mathcal c 1 of length n over a field extension mathbb f q m and its subcode mathcal c 2 subsetneqq mathcal c 1 one is called the relative dimension intersection profile rdip and the other is called the relative generalized rank weight rgrw we clarify their basic properties and the relation between the rgrw and the minimum rank distance as applications of the rdip and the rgrw the security performance and the error correction capability of secure network coding guaranteed independently of the underlying network code are analyzed and clarified we propose a construction of secure network coding scheme and analyze its security performance and error correction capability as an example of applications of the rdip and the rgrw silva and kschischang showed the existence of a secure network coding in which no part of the secret message is revealed to the adversary even if any dim mathcal c 1 1 links are wiretapped which is guaranteed over any underlying network code however the explicit construction of such a scheme remained an open problem our new construction is just one instance of secure network coding that solves this open problem

formulation and accuracy of on surface radiation conditions for acoustic multiple scattering problems the aim of this paper is 1 to address an extension of the on surface radiation condition osrc method to multiple scattering by a cluster of convex obstacles and 2 to analyze the numerical accuracy of the formulation for some model problems first the acoustic scattering problem is considered and materials about integral equations and osrcs are recalled then an osrc integral equation formulation is developed for the dirichlet multiple scattering problems finally an extensive numerical study of the method is proposed in the special case of the scattering by circular cylinders even if this situation is not completely general it provides an assessment of the accuracy of the approximation with respect to various parameters including thanks to the choice of the osrc operators

a fictitious domain decomposition method for the solution of partially axisymmetric acoustic scattering problems part 2 neumann boundary conditions we present a fictitious domain decomposition method for the fast solution of acoustic scattering problems characterized by a partially axisymmetric sound hard scatterer we apply this method to the solution of a mock up submarine problem and highlight its computational advantages and intrinsic parallelism a key component of our method is an original idea for addressing a neumann boundary condition in the general framework of a fictitious domain method this idea is applicable to many other linear partial differential equations besides the helmholtz equation copyright 2003 john wiley sons ltd

l 2 exponential lower bounds to solutions of the schrodinger equation we study decay properties of solutions of the schrodinger equation v e typical of our results is one which shows that ifv o x 1 2 at infinity or ifv is a homogeneousn body potential for example atomic or molecular then ife 0 and alpha sqrt e e alpha left x right psi notin l 2 left mathbb r n right we also construct examples to show that previous essential spectrum dependent upper bounds can be far from optimal if is not the ground state

constructive proof of localization in the anderson tight binding model we prove that for large disorder or near the band tails the spectrum of the anderson tight binding hamiltonian with diagonal disorder consists exclusively of discrete eigenvalues the corresponding eigenfunctions are exponentially well localized these results hold in arbitrary dimension and with probability one in one dimension we recover the result that all states are localized for arbitrary energies and arbitrarily small disorder our techniques extend to other physical systems which exhibit localization phenomena such as infinite systems of coupled harmonic oscillators or random schrodinger operators in the continuum

characterization of partial derivatives with respect to material parameters in a fluid solid interaction problem abstract for a fluid solid interaction problem with lipschitz interface we investigate the partial frechet differentiability of the solutions and the approximate far field pattern with respect to solid material parameters differentiability is shown in standard sobolev framework and the derivatives are characterized as solutions to inhomogeneous fluid solid transmission problems to validate the accuracy of the characterization we compare analytical values with numerical ones given by interior penalty discontinuous galerkin ipdg in a setting with circular obstacles our comparisons also show that ipdg gives results with high precision and incurs almost no effect of discretization error accumulation

exact electronic spectra and inverse localization lengths in one dimensional random systems i random alloy liquid metal and liquid alloy analytic continuations into the complex energy plane of dyson schmidt type of equations for the calculation of the density of states are constructed for a random alloy model a liquid metal and for a liquid alloy in all these models the characteristic function follows from the solution of this equation its imaginary part yields the accumulated density of states and its real part is a measure for the inverse of the localization length of the eigenfunctions the equations have been solved exactly for some distributions of the random variables in the random alloy case the strengths of the delta potentials have an exponential distribution they may also have finite exponentially distributed values with probability 0 p 1 and be infinite with probability q 1 p in the liquid metal the liquid particles are assumed to behave like hard rods this implies an exponential distribution of the distances between the particles the common electronic potential may be arbitrary but is assumed to vanish outside the rods in the one dimensional liquid alloy there is apart from positional randomness of the liquid particles a distribution of the strengths of the electronic delta potentials for cauchy distributions an argument of lloyd is extended to obtain the characteristic function from the one in the model with equal strengths for the case of a liquid of point particles a three parameter class of distributions of the strengths is shown to yield a solution in the form of known functions of the equation mentioned above for several cases numerical calculations of the density of states and the inverse localization length of the eigenfunctions are presented and discussed new results are found exponential decay of the density of states near special energies in the random alloy and liquid metal divergence of the density of states at certain energies with non classical exponent 13 in the random alloy if the average of the potential strengths vanishes exponentially small broadening of the bound state levels for low concentrations of the liquid particles peak in the localization length at the bound state energies which becomes exponentially narrow for low concentrations different exponent in the decay of the inverse localization length at large energies for delta potentials and square well potentials further an expression for the grand potential is given involving a sum over the characteristic function at certain points and divergence of the zero point energy is found for cauchy distributions of the delta potential strengths

the effect of boundaries on the spectrum of a one dimensional random mass dirac hamiltonian the average density of states dos of the one dimensional dirac hamiltonian with a random mass on a finite interval 0 l is derived our method relies on the eigenvalue distributions extreme value statistics problem which are obtained explicitly the well known dyson singularity is recovered above the crossover energy below c we find a log normal suppression of the average dos

localization in general one dimensional random systems we discuss two ways of extending the recent ideas of localization from discrete schrodinger operators jacobi matrices to the continuum case one case allows us to prove localization in the goldshade molchanov pastur model for a larger class of functions than previously the other method studies the model v wherev is a random constant in each hyper cube we extend wegner s result on the lipschitz nature of the ids to this model

on the existence and the uniqueness of the solution of a fluid structure interaction scattering problem the existence and uniqueness of the solution of a fluid structure interaction problem is investigated the proposed analysis distinguishes itself from previous studies by employing a weighted sobolev space framework the dtn operator properties and the fredholm theory the proposed approach allows to extend the range of validity of the standard existence and uniqueness results to the case where the elastic scatterer is assumed to be only lipschitz continuous which is of more practical interest

one dimensional schrodinger operators with random or deterministic potentials new spectral types abstract we prove sufficient conditions involving only potential asymptotic near one of the infinities in order to have purely absolutely continuous components in the spectrum these deterministic results are then applied to random cases and exhibit classes of models for which with probability one one component of the spectrum is purely absolutely continuous and the rest is dense pure point with exponentially decaying eigenfunctions

generalized rank weights of reducible codes optimal cases and related properties reducible codes for the rank metric were introduced for cryptographic purposes they have fast encoding and decoding algorithms include maximum rank distance mrd codes and can correct many rank errors beyond half of their minimum rank distance which makes them suitable for error correction in network coding in this paper we study their security behavior against information leakage on networks when applied as coset coding schemes giving the following main results 1 we give lower and upper bounds on their generalized rank weights grws which measure worst case information leakage to the wire tapper 2 we find new parameters for which these codes are mrd meaning that their first grw is optimal and use the previous bounds to estimate their higher grws 3 we show that all linear over the extension field codes whose grws are all optimal for fixed packet and code sizes but varying length are reducible codes up to rank equivalence and 4 we show that the information leaked to a wire tapper when using reducible codes is often much less than the worst case given by their optimal in some cases grws we conclude with some secondary related properties conditions to be rank equivalent to cartesian products of linear codes and conditions to be rank degenerate duality properties and mrd ranks

a general approach for high order absorbing boundary conditions for the helmholtz equation when solving a scattering problem in an unbounded space one needs to implement the sommerfeld condition as a boundary condition at infinity to ensure no energy penetrates the system in practice solving a scattering problem involves truncating the region and implementing a boundary condition on an artificial outer boundary bayliss gunzburger and turkel bgt suggested an absorbing boundary condition abc as a sequence of operators aimed at annihilating elements from the solution s series representation their method was practical only up to a second order condition later hagstrom and hariharan hh suggested a method which used auxiliary functions and enabled implementation of higher order conditions we compare various absorbing boundary conditions abcs and introduce a new method to construct high order abcs generalizing the hh method we then derive from this general method abcs based on different series representations of the solution to the helmholtz equation in polar elliptical and spherical coordinates some of these abcs are generalizations of previously constructed abcs and some are new these new abcs produce accurate solutions to the helmholtz equation which are much less dependent on the various parameters of the problem such as the value of k or the eccentricity of the ellipse in addition to constructing new abcs our general method sheds light on the connection between various abcs computations are presented to verify the high accuracy of these new abcs