boundary layer techniques for solving the helmholtz equation in the presence of small inhomogeneities we consider solutions to the helmholtz equation in two and three dimensions based on layer potential techniques we provide for such solutions a rigorous systematic derivation of complete asymptotic expansions of perturbations resulting from the presence of diametrically small inhomogeneities with constitutive parameters different from those of the background medium it is expected that our results will find important applications for developing effective algorithms for reconstructing small dielectric inhomogeneities from boundary measurements

classification of quaternary hermitian self dual codes of length 20 a classification of quaternary hermitian self dual codes of length 20 is given using this classification a classification of extremal quaternary hermitian self dual codes of length 22 is also given

music type electromagnetic imaging of a collection of small three dimensional inclusions in this paper we consider the localization of a collection of small three dimensional bounded homogeneous inclusions via time harmonic electromagnetic means typically using arrays of electric or magnetic dipole transmitters and receivers with given polarization s at some distance from the collection possibly also lying in the far field the inclusions somewhat apart or closely spaced are buried within a homogeneous medium and are of arbitrary contrast of permittivity conductivity and permeability vis ag vis this embedding medium the problem is formulated as an inverse scattering problem for the full maxwell equations and it involves a robust asymptotic modeling of the multistatic response matrix no specific application is studied at this stage but characterization of obstacles in subsoils nondestructive evaluation of man made structures and medical imaging are primary fields of application envisaged the proposed approach uses a music multiple signal classification type algorithm and it yields fast numbering accurate localization and estimates of the electromagnetic and geometric parameters polarization tensors of the inclusions the mathematical machinery is detailed first some specific attention being given to triaxial ellipsoidal inclusions and degenerate spherical shapes for the latter known results are retrieved then the viability of this algorithm which would be easily extended to planarly layered environments by introduction of their green s functions is documented by a variety of numerical results from synthetic noiseless and severely noisy field data

an algebraic formula for the accelerated computation of the low frequency scattering coefficients an explicit expression for the rapid computation of the low frequency scattering coefficients is presented although the demonstration of the methodology is restricted to the simple case of an acoustically soft spherical scatterer the introduced main concept can be applied to a wide class of scattering problems the procedure based on algebraic calculations only provides an analytic expression of the corresponding scattering coefficients in a straightforward fashion the proposed algorithm is readily implemented furnishing an efficient way for obtaining any order approximation of the total field in the exterior of the scatterer

generalized polarization tensors for shape description with each domain an infinite number of tensors called the generalized polarization tensors gpts is associated the gpts contain significant information on the shape of the domain in the recent paper ammari et al in math comput 81 367 386 2012 a recursive optimal control scheme to recover fine shape details of a given domain using gpts is proposed in this paper we show that the gpts can be used for shape description we also show that high frequency oscillations of the boundary of a domain are only contained in its high order gpts indeed we provide an original stability and resolution analysis for the reconstruction of small shape changes from the gpts by developing a level set version of the recursive optimization scheme we make the change of topology possible and show that the gpts can capture the topology of the domain we also propose an indicator of topology which could be used in some particular cases to test whether we have the correct number of connected components in the reconstructed image we provide analytical and numerical evidence that gpts can capture topology and high frequency shape oscillations the results of this paper clearly show that the concept of gpts is a very promising new tool for shape description

the generalized polarization tensors for resolved imaging part ii shape and electromagnetic parameters reconstruction of an electromagnetic inclusion from multistatic measurements this paper deals with the problem of reconstructing the electromagnetic parameters and the shape of a target from multi static response matrix measurements at a single frequency the target is of characteristic size less than the operating wavelength using long wavelength asymptotic expansions of the measurements of high order we show how the electromagnetic parameters and the equivalent ellipse of the target can be reconstructed the asymptotic expansions of the measurements are written in terms of the new concept of frequency dependent polarization tensors moreover we extend the optimization approach proposed in part i 9 to image geometric details of an electromagnetic target that are finer than the equivalent ellipse the equivalent ellipse still provides a good initial guess for the optimization procedure however compared to the conductivity case in 9 the cost functional measures the discrepancy between the computed and measured high order frequency dependent polarization tensors rather than between the generalized polarization tensors the main reason for such a modification of the cost functional is the fact that the measured frequency dependent polarization tensors can be easily obtained from multistatic measurements by solving a linear system while the derivation of the generalized polarization tensors from measurements requires more delicate treatment the proposed methods are numerically implemented to demonstrate their validity and efficiency mathematics subject classification msc2000 35r30 35b30

there is no 24 12 10 self dual quaternary code an attempt at directly constructing a 24 12 10 extremal quaternary code is reported the attempt at directly constructing the generator matrix of the code is detailed it was found that no such code exists

classification of extremal self dual quaternary codes of lengths 30 and 32 we classify extremal hermitian self dual quaternary codes of lengths 30 and 32 with an automorphism of odd prime order we prove that there exists exactly one extremal hermitian self dual 30 15 12 quaternary code with a nontrivial automorphism of odd prime order up to equivalence the order of its automorphism group is 36 540 in fact this code is equivalent to the extended quadratic residue code we also prove that there exists no extremal hermitian self dual 32 16 12 quaternary code with a nontrivial automorphism of odd prime order

the topological asymptotic for the helmholtz equation with dirichlet condition on the boundary of an arbitrarily shaped hole the aim of the topological sensitivity analysis is to obtain an asymptotic expansion of a design functional with respect to the creation of a small hole in the domain in this paper such an expansion is obtained for the helmholtz equation in two and three space dimensions with a dirichlet condition on the boundary of an arbitrarily shaped hole in this case the main difficulty is related to the nonhomogeneous symbol of the helmholtz operator in the numerical part of this work we will show that the topological sensitivity method is very promising for solving shape inverse problems in electromagnetic applications

regularity properties of solutions to transmission problems we show that the gradients of solutions to certain elliptic and parabolic transmission problems with internal lipschitz boundary and constant co efficients at each side of the internal boundary are square integrable along the internal boundary

reconstruction of a small inclusion in a two dimensional open waveguide we consider wave propagation in a perturbed open waveguide we provide a new asymptotic expansion for the scattered wave when the inclusion is of small diameter we design a music multiple signal classification type of algorithm for locating the inclusion and illustrate its viability in numerical examples

identification of small inhomogeneities of extreme conductivity by boundary measurements a theorem on continuous dependence we consider an electrostatic problem for a conductor consisting of finitely many small inhomogeneities of extreme conductivity embedded in a spatially varying reference medium firstly we establish an asymptotic formula for the voltage potential in terms of the reference voltage potential the location of the inhomogeneities and their geometry secondly we use this representation formula to prove a lipschitz continuous dependence estimate for the corresponding inverse problem this estimate bounds the difference in the location and in certain geometric properties of two sets of inhomogeneities by the difference in the boundary voltage potentials corresponding to a fixed current distribution

enhancement of near cloaking part ii the helmholtz equation the aim of this paper is to extend the method of ammari et al commun math phys 2012 to scattering problems we construct very effective near cloaking structures for the scattering problem at a fixed frequency these new structures are before using the transformation optics layered structures and are designed so that their first scattering coefficients vanish inside the cloaking region any target has near zero scattering cross section for a band of frequencies we analytically show that our new construction significantly enhances the cloaking effect for the helmholtz equation

layer potential techniques in spectral analysis part i complete asymptotic expansions for eigenvalues of the laplacian in domains with small inclusions we provide a rigorous derivation of new complete asymptotic expansions for eigenvalues of the laplacian in domains with small inclusions the inclusions somewhat apart from or nearly touching the boundary are of arbitrary shape and arbitrary conductivity contrast vis a vis the background domain with the limiting perfectly conducting inclusion by integral equations we reduce this problem to the study of the characteristic values of integral operators in the complex plane powerful techniques from the theory of meromorphic operator valued functions and careful asymptotic analysis of integral kernels are combined for deriving complete asymptotic expansions for eigenvalues our asymptotic formulae in this paper may be expected to lead to efficient algorithms not only for solving shape optimization problems for laplacian eigenvalues but also for determining specific internal features of an object based on scattering data measurements

characterization of quaternary extremal codes of lengths 18 and 20 we prove that up to equivalence there is a unique extremal quaternary hermitian self dual code of length 18 and that there are two inequivalent extremal quaternary hermitian self dual codes of length 20

electrical impedance tomography by elastic deformation this paper presents a new algorithm for conductivity imaging our idea is to extract more information about the conductivity distribution from data that have been enriched by coupling impedance electrical measurements to localized elastic perturbations using asymptotics of the fields in the presence of small volume inclusions we relate the pointwise values of the energy density to the measured data through a nonlinear pde our algorithm is based on this pde and takes full advantage of the enriched data we give numerical examples that illustrate the performance and the accuracy of our approach

approximation by multipoles of the multiple acoustic scattering by small obstacles in three dimensions and application to the foldy theory of isotropic scattering the asymptotic analysis carried out in this paper for the problem of a multiple scattering in three dimensions of a time harmonic wave by obstacles whose size is small as compared with the wavelength establishes that the effect of the small bodies can be approximated at any order of accuracy by the field radiated by point sources among other issues this asymptotic expansion of the wave furnishes a mathematical justification with optimal error estimates of foldy s method that consists in approximating each small obstacle by a point isotropic scatterer finally it is shown how this theory can be further improved by adequately locating the center of phase of the point scatterers and the taking into account of self interactions in this way it is established that the usual foldy model may lead to an approximation whose asymptotic behavior is the same than that obtained when the multiple scattering effects are completely neglected

an algebraic calculation method for the acoustic low frequency expansion abstract in the present work we propose an analytical method for the algebraic calculation of the low frequency expansion of the solution of a scalar scattering problem upon a smooth starshaped scatterer the method is based on the far field expansion theorem introduced by atkinson and wilcox in the middle of the last century applied in the low frequency realm in this view the need for solving elliptical boundary value problems to obtain the low frequency approximations is replaced by an analytical procedure easily encoded in an algorithm including only algebraic operators as a demonstration example the method is applied to the acoustic scattering problem with plane wave excitation upon three different non penetrable spherical scatterers the first low frequency approximations are deduced yielding the validity of the proposed method by both recovering already known results and accurately deriving higher orders of approximation

restricted weak type endpoint estimates for k spherical maximal functions in this paper we use the approximation formula for the fourier transform of the solution set of lattice points on k spheres and methods of bourgain and ionescu to refine the ell p mathbb z d boundedness results for discrete k spherical maximal functions to a restricted weak type result at the endpoint we introduce a density parameter which may be viewed as a discrete version of minkowski dimension used in related works on the continuous analgoue in order to exploit recent progress of wooley and bourgain demeter guth on the vinogradov mean value conjectures via a novel approximation formula for a single average and obtain improved bounds for lacunary discrete k spherical maximal functions when k ge 3

new self dual codes over gf 4 with the highest known minimum weights the purpose of this correspondence is to construct new hermitian self dual codes over gf 4 of lengths 22 24 26 32 and 34 which have the highest known minimum weights in particular for length 22 we construct eight new extremal self dual 22 11 8 codes over gf 4 which do not have a nontrivial automorphism of odd order the existence of such codes has been left open since 1991 by huffman

on the classification of hermitian self dual additive codes over rm gf 9 additive codes over gf 9 that are self dual with respect to the hermitian trace inner product have a natural application in quantum information theory where they correspond to ternary quantum error correcting codes however these codes have so far received far less interest from coding theorists than self dual additive codes over gf 4 which correspond to binary quantum codes self dual additive codes over gf 9 have been classified up to length 8 and in this paper we extend the complete classification to codes of lengths 9 and 10 the classification is obtained by using a new algorithm that combines two graph representations of self dual additive codes the search space is first reduced by the fact that every code can be mapped to a weighted graph and a different graph is then introduced that transforms the problem of code equivalence into a problem of graph isomorphism by an extension technique we are able to classify all optimal codes of lengths 11 and 12 there are 56 005 876 11 311 5 codes and 6493 12 312 6 codes we also find the smallest codes with trivial automorphism group

identification of small inhomogeneities asymptotic factorization we consider the boundary value problem of calculating the electrostatic potential for a homogeneous conductor containing finitely many small insulating inclusions we give a new proof of the asymptotic expansion of the electrostatic potential in terms of the background potential the location of the inhomogeneities and their geometry as the size of the inhomogeneities tends to zero such asymptotic expansions have already been used to design direct i e noniterative reconstruction algorithms for the determination of the location of the small inclusions from electrostatic measurements on the boundary e g music type methods our derivation of the asymptotic formulas is based on integral equation methods it demonstrates the strong relation between factorization methods and music type methods for the solution of this inverse problem