role of the admittance operator in rough surface scattering the linear operator that relates the normal gradient of a wave field to its values on an irregular surface is the basis for a new formulation of the rough surface scattering problem this formulation has led to a class of approximations that surpass traditional methods in accuracy and efficiency the connection between this approach and the standard formulation by means of a boundary integral equation is the subject of this paper while equivalent in all respects the admittance operator formulation provides helpful new perspectives on the physics of surface scattering and lends itself naturally to more elaborate problems for example at a rough boundary between dissimilar media the operators compose themselves into generalized reflection and transmission operators while at a random surface the operator terms lead to analytic forms for the scattered field moments

electromagnetic scattering by an inhomogeneous conducting or dielectric layer on a perfectly conducting plate we consider a two dimensional problem of scattering of a time harmonic electromagnetic plane wave by an inhomogeneous conducting or dielectric layer on a perfectly conducting plate the magnetic permeability is assumed to be a fixed positive constant in the media the material properties of the media are characterized completely by an index of refraction which is a bounded measurable function in the layer and a positive constant above the layer corresponding to a homogeneous dielectric medium in this paper we only examine the tm transverse magnetic polarization case a radiation condition is introduced and equivalence with a second kind lippmann schwinger type integral equation is shown with additional assumptions on the index of refraction in the layer uniqueness of solution is proved existence of solution is then established by employing a form of fredholm alternative using a general result on the solvability of integral equations on unbounded domains published earlier by chandler wilde and zhang an approximate analytic solution for the case of a thin inhomogeneous layer is obtained from the integral equation formulation and is used to show that if the index of refraction is appropriately chosen the scattered field can grow with distance from the plate

scattering by stripe grating when bismuth is substituted at certain lattice points of rare earth garnet crystals a material is formed which exhibits an almost ideal magnetic stripe domain structure the structure consists of a one dimensional periodic array of regions of nearly constants magnetization so that the magnitudes of the magnetization vectors in two adjacent regions are the same but their directions are opposite this condition produces the alternating sign magnetic permeability tensors shown in section 1 7 3 when a beam of light is passed through such a film it splits into many diffracted beams all sharing a plane defined by the periodicity direction i e the y component of all the wave vectors is the same see figure 1 1 the angles of the diffracted beam directions relative to the incident beam direction are determined by the periodicity length

efficient calculation of the green function for acoustic propagation above a homogeneous impedance plane this paper is concerned with the problem of propagation from a monofrequency coherent line source above a plane of homogeneous surface impedance the solution of this problem occurs in the kernel of certain boundary integral equation formulations of acoustic propagation above an impedance boundary and the discussion of the paper is motivated by this application the paper starts by deriving representations as laplace type integrals of the solution and its first partial derivatives the evaluation of these integral representations by gauss laguerre quadrature is discussed and theoretical bounds on the truncation error are obtained specific approximations are proposed which are shown to be accurate except in the very near field for all angles of incidence and a wide range of values of surface impedance the paper finishes with derivations of partial results and analogous laplace type integral representations for the case of a point source

quasilinear parabolic systems of several components in this paper we investigate the blowup criteria of the quasilinear parabolic system u imath t c imath u imath alpha imath delta u imath prod jmath 1 n u jmath p imath jmath imath 1 2 cdots n with homogeneous dirichlet boundary conditions on a bounded domain r n where c i 0 i 0 p i 0 1 i n are constants denote by i the identity matrix and p p i which is assumed to be irreducible that i p is a singular m matrix is shown to be the critical case in which 1 plays a fundamental role where 1 is the first dirichlet eigenvalue of the laplacian on as a result we give a general answer to the question of galaktionov and levine on the porous medium systems

a posteriori error analysis for the conical diffraction problem abstract maxwell s equations for conical diffraction can be reduced to a system of two helmholtz equations in r 2 coupled via quasi periodic transmission conditions on a set of piecewise smooth interfaces a finite element formulation of the conical diffraction problem is presented in a bounded domain by introducing the nonlocal boundary operators an a posteriori error estimate is established when the truncation of the nonlocal boundary operators takes place as our work in wang et al 2015 a duality argument is applied to overcome the difficulty caused by the fact that the truncated pseudo differential mapping does not converge to the original pseudo differential mapping in its operator norm the a posteriori error estimate consists of two parts finite element discretization error and the truncation error of the nonlocal boundary operators in particular the truncation error is exponentially decaying with respect to the truncation parameter

a new spectral method for numerical solution of the unbounded rough surface scattering problem a new spectral method is developed to solve the unbounded rough surface scattering problem an unbounded rough surface is referred to as a non local perturbation of an infinite plane surface such that the whole rough surface lies within a finite distance of the original plane the method uses a transformed field expansion to reduce the boundary value problem with a complex scattering surface into a successive sequence of transmission problems of a planar surface hermite orthonormal basis functions are adopted to further simplify these problems to fully decoupled one dimensional two point boundary value problems which are solved efficiently by the legendre galerkin method numerical results indicate that the method is efficient accurate and well suited for solving the scattering problem by unbounded rough surfaces

strongly self absorbing c algebras say that a separable unital c algebra v c is strongly self absorbing if there exists an isomorphism v such that lx ai e approximately unitarily equivalent homomorphisms we study this class of algebras which includes the cuntz algebras 2 ooo the uhf algebras of infinite type the jiang su algebra z and tensor products of 00 with uhf algebras of infinite type given a strongly self absorbing c algebra v we characterise when a separable c algebra absorbs v tensorially i e is p stable and prove closure properties for the class of separable testable c algebras finally we compute the possible if groups and prove a number of classification results which suggest that the examples listed above are the only strongly self absorbing c algebras

an adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures we develop a finite element adaptive strategy with error control for the wave scattering by periodic structures the unbounded computational domain is truncated to a bounded one by an extension of the perfectly matched layer pml technique which attenuates both the outgoing and evanescent waves in the pml region pml parameters such as the thickness of the layer and the medium property are determined through sharp a posteriori error estimates numerical experiments are included to illustrate the competitive behavior of the proposed adaptive method

finite gratings at high frequencies in this paper we construct geometric optics solutions to an acoustic scattering problem by a finite periodic dielectric grating placed on the interface between semi infinite dielectric media we rigorously justify that the first order approximation is the solution of the infinite periodic grating the method is extended to maxwell s equations

three dimensional quasi periodic shifted green function throughout the spectrum including wood anomalies this work part ii in a series presents an efficient method for evaluation of wave scattering by doubly periodic diffraction gratings at or near what are commonly called wood anomaly frequencies at these frequencies there is a grazing rayleigh wave and the quasi periodic green function ceases to exist we present a modification of the green function by adding two types of terms to its lattice sum the first type are transversely shifted green functions with coefficients that annihilate the growth in the original lattice sum and yield algebraic convergence the second type are quasi periodic plane wave solutions of the helmholtz equation which reinstate certain necessary grazing modes without leading to blow up at wood anomalies using the new quasi periodic green function we establish for the first time that the dirichlet problem of scattering by a smooth doubly periodic scattering surface at a wood frequency is uniquely solvable we also present an efficient high order numerical method based on this new green function for scattering by doubly periodic surfaces at and around wood frequencies we believe this is the first solver able to handle wood frequencies for doubly periodic scattering problems in three dimensions we demonstrate the method by applying it to acoustic scattering

acoustic scattering by mildly rough unbounded surfaces in three dimensions for a nonlocally perturbed half space we consider the scattering of time harmonic acoustic waves a second kind boundary integral equation formulation is proposed for the sound soft case based on a standard ansatz as a combined single and double layer potential but replacing the usual fundamental solution of the helmholtz equation with an appropriate half space green s function due to the unboundedness of the surface the integral operators are noncompact in contrast to the two dimensional case the integral operators are also strongly singular due to the slow decay at infinity of the fundamental solution of the three dimensional helmholtz equation in the case when the surface is sufficiently smooth lyapunov we show that the integral operators are nevertheless bounded as operators on l 2 gamma and on l 2 gamma cap bc gamma and that the operators depend continuously in norm on the wave number and on gamma we further show that for emph mild roughness i e a surface gamma which d

on the solvability of second kind integral equations on the real line abstract this paper considers general second kind integral equations of the form s r k s t t dt s in operator form k k where the functions k and are assumed known with y the space of bounded continuous functions on r and k such that the mapping s k s from r to l1 r is bounded and continuous the function y is the solution to be determined conditions on a set w bc r l1 r are obtained such that a generalised fredholm alternative holds if w satisfies these conditions and i k k is injective for all k w then i k k is also surjective for all k w and moreover the inverse operators i k k 1 on y are uniformly bounded for k w the approximation of the kernel in the integral equation by a sequence kn converging in a weak sense to k is also considered and results on stability and convergence are obtained these general theorems are used to establish results for two special classes of kernels k s t s t z t and k s t s t s t t where l1 r z l r and bc r 0 r kernels of both classes arise in problems of time harmonic wave scattering by unbounded surfaces the general integral equation results are here applied to prove the existence of a solution for a boundary integral equation formulation of scattering by an infinite rough surface and to consider the stability and convergence of approximation of the rough surface problem by a sequence of diffraction grating problems of increasingly large period

efficient calculation of two dimensional periodic and waveguide acoustic green s functions new representations and efficient calculation methods are derived for the problem of propagation from an infinite regularly spaced array of coherent line sources above a homogeneous impedance plane and for the green s function for sound propagation in the canyon formed by two infinitely high parallel rigid or sound soft walls and an impedance ground surface the infinite sum of source contributions is replaced by a finite sum and the remainder is expressed as a laplace type integral a pole subtraction technique is used to remove poles in the integrand which lie near the path of integration obtaining a smooth integrand more suitable for numerical integration and a specific numerical integration method is proposed numerical experiments show highly accurate results across the frequency spectrum for a range of ground surface types it is expected that the methods proposed will prove useful in boundary element modeling of noise propagation in canyon streets and in ducts and for problems of scattering by periodic surfaces

on analyticity of linear waves scattered by a layered medium abstract the scattering of linear waves by periodic structures is a crucial phenomena in many branches of applied physics and engineering in this paper we establish rigorous analytic results necessary for the proper numerical analysis of a class of high order perturbation of surfaces methods for simulating such waves more specifically we prove a theorem on existence and uniqueness of solutions to a system of partial differential equations which model the interaction of linear waves with a multiply layered periodic structure in three dimensions this result provides hypotheses under which a rigorous numerical analysis could be conducted for recent generalizations to the methods of operator expansions field expansions and transformed field expansions

decomposition rank and z stability we show that separable simple nonelementary unital c algebras with finite decomposition rank absorb the jiang su algebra mathcal z tensorially this has a number of consequences for elliott s program to classify nuclear c algebras by their k theory data in particular it completes the classification of c algebras associated to uniquely ergodic smooth minimal dynamical systems by their ordered k groups

numerical analysis of graded mesh methods for a class of second kind integral equations on the real line abstract in this paper we are concerned with the numerical analysis of the collocation method based on graded meshes of second kind integral equations on the real line of the form s s r s t z t t dt s r where l 1 r z l r and bc r the space of bounded continuous complex valued functions on r are assumed known and the function bc r is to be determined we introduce some new graded meshes for the collocation method of the integral equation which are different from those used previously for the wiener hopf integral equation in the case when the solution decays exponentially at infinity and establish optimal local and global l norm error estimates under the condition that the solution decays only polynomially at infinity

unique determination of periodic polyhedral structures by scattered electromagnetic fields this work is concerned with the unique determination of a periodic diffraction grating profile in three dimensions by some scattered electromagnetic fields measured above the grating in general it is well known that global uniqueness may not be true when the measurement is only taken for one incident field our goal is to completely characterize the global uniqueness properties when the periodic structure is of polyhedral type corresponding to each incident plane wave we are able to classify all unidentifiable structures into three classes and show that any periodic polyhedral structure can be uniquely determined by one incident field if and only if it belongs to none of the three classes consequently the minimum number of incident waves required for the unique determination of a periodic polyhedral structure can be easily read

a well posed integral equation formulation for three dimensional rough surface scattering we consider the problem of scattering of time harmonic acoustic waves by an unbounded sound soft rough surface recently a brakhage werner type integral equation formulation of this problem has been proposed based on an ansatz as a combined single and double layer potential but replacing the usual fundamental solution of the helmholtz equation with an appropriate half space green s function moreover it has been shown in the three dimensional case that thi si ntegral equation is uniquely solvable in the space l 2 g when the scattering surface g does not differ too much from a plane in this paper we show that this integral equation is uniquely solvable with no restriction on the surface elevation or slope moreover we construct explicit bounds on the inverse of the associated boundary integral operator as a function of the wave number the parameter coupling the single and double layer potentials and the maximum surface slope these bounds show that the norm of the inverse operator is bounded uniformly in the wave number k f orko0 if the coupling parameter h is chosen proportional to the wave number in the case when g is a plane we show that the choice hzk 2 is nearly optimal in terms of minimizing the condition number

the floquet bloch transform and scattering from locally perturbed periodic surfaces abstract we use the floquet bloch transform to reduce variational formulations of surface scattering problems for the helmholtz equation from periodic and locally perturbed periodic surfaces to equivalent variational problems formulated on bounded domains to this end we establish various mapping properties of that transform between suitable weighted sobolev spaces on periodic strip like domains and coupled families of quasiperiodic sobolev spaces our analysis shows in particular that the decay of solutions to surface scattering problems from locally perturbed periodic surfaces is precisely characterized by the smoothness of its bloch transform in the quasiperiodicity

infinite non simple c algebras absorbing the cuntz algebra o abstract the first named author has given a classification of all separable nuclear c algebras a that absorb the cuntz algebra o we say that a absorbs o if a is isomorphic to a o motivated by this classification we investigate here if one can give an intrinsic characterization of c algebras that absorb o this investigation leads us to three different notions of pure infiniteness of a c algebra all given in terms of local algebraic conditions on the c algebra the strongest of the three properties strongly purely infinite is shown to be equivalent to absorbing o for separable nuclear c algebras that either are stable or have an approximate unit consisting of projections in a previous paper 2000 amer j math 122 637 666 we studied an intermediate and perhaps more natural condition purely infinite that extends a well known property for simple c algebras the weakest condition of the three weakly purely infinite is shown to be equivalent to the absence of quasitraces in an ultrapower of the c algebra the three conditions may be equivalent for all c algebras and we prove this to be the case for c algebras that are either simple of real rank zero or approximately divisible

global existence and finite time blow up for a degenerate reaction diffusion system abstract this paper investigates the blow up and global existence of solutions of the degenerate reaction diffusion system u t u m u v p v t v n u q v x t 0 t with homogeneous dirichlet boundary data where r n is a bounded domain with smooth boundary m n 1 0 and p q 0 it is proved that if m n and pq m n every nonnegative solution is global whereas if m or n or pq m n there exist both global and blow up nonnegative solutions when m n and pq m n we show that there exists 1 which depends on the parameters p q m n such that all positive solutions are global if 1 while if 1 1 all positive solutions blow up in finite time where 1 is the first dirichlet eigenvalue for the laplacian on

scattering by infinite one dimensional rough surfaces we consider the dirichlet boundary value problem for the helmholtz equation in a non locally perturbed half plane this problem models time harmonic electromagnetic scattering by a one dimensional infinite rough perfectly conducting surface the same problem arises in acoustic scattering by a sound soft surface chandler wilde and zhang have suggested a radiation condition for this problem a generalization of the rayleigh expansion condition for diffraction gratings and uniqueness of solution has been established recently an integral equation formulation of the problem has also been proposed and in the special case when the whole boundary is both lyapunov and a small perturbation of a flat boundary the unique solvability of this integral equation has been shown by chandler wilde and ross by operator perturbation arguments in this paper we study the general case with no limit on surface amplitudes or slopes and show that the same integral equation has exactly one solution in the space of bounded and continuous functions for all wavenumbers as an important corollary we prove that for a variety of incident fields including the incident plane wave the dirichlet boundary value problem for the scattered field has a unique solution

a uniqueness result for scattering by infinite rough surfaces consider the dirichlet boundary value problem for the helmholtz equation in a non locally perturbed half plane with an unbounded piecewise lyapunov boundary this problem models time harmonic electromagnetic scattering in transverse magnetic polarization by one dimensional rough perfectly conducting surfaces a radiation condition is introduced for the problem which is a generalization of the usual one used in the study of diffraction by gratings when the solution is quasi periodic and allows a variety of incident fields including an incident plane wave to be included in the results obtained we show in this paper that the boundary value problem for the scattered field has at most one solution for the case when the whole boundary is lyapunov and is a small perturbation of a flat boundary we also prove existence of solution and show a limiting absorption principle

global existence and nonexistence for a class of degenerate parabolic systems this paper investigates the finite time blow up and global existence of the positive solution of the nonlinear degenerate parabolic system ut f1 u u a1v vt f2 v v a2u x t 0 with homogeneous dirichlet boundary data a necessary and sufficient condition for global existence of the solution is derived

an adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures an edge element adaptive strategy with error control is developed for wave scattering by biperiodic structures the unbounded computational domain is truncated to a bounded one by a perfectly matched layer pml technique the pml parameters such as the thickness of the layer and the medium properties are determined through sharp a posteriori error estimates numerical experiments are presented to illustrate the competitive behavior of the proposed adaptive method

diffraction at a curved grating tm and te cases homogenization a hierarchy of effective boundary conditions is derived for a helmholtz equation with dirichlet or neumann boundary conditions these generalized impedance boundary conditions approximate the effect of thin dielectric layers with perfectly conducting inclusions covering an object well posedness of the problems with these new boundary conditions is proved and optimal error estimates are given

non periodic acoustic and electromagnetic scattering from periodic structures in 3d abstract scattering of non periodic waves from unbounded structures is difficult to treat as one typically formulates the problem in an unbounded domain covering the unbounded periodic structure the floquet bloch transform reduces the latter problem to a family of decoupled periodic scattering problems this reduction is in particular interesting from the point of view of numerical computations we analyze a corresponding fully discrete numerical solution algorithm for three dimensional scattering problems in acoustics and electromagnetics proving in particular convergence rates under suitable assumptions on the geometry and the material coefficients a crucial part of our analysis relies on the continuous dependence of the family of solutions to the quasiperiodic scattering problems on the quasiperiodicity the latter part is actually more difficult to establish than for corresponding two dimensional problems we further provide a numerical example in 3d acoustics that illustrates feasibility of the proposed algorithm

equivariant kirchberg phillips type absorption for amenable group actions we show an equivariant kirchberg phillips type absorption theorem for pointwise outer actions of discrete amenable groups on kirchberg algebras with respect to natural model actions on the cuntz algebras mathcal o infty and mathcal o 2 this generalizes results known for finite groups and poly mathbb z groups the model actions are shown to be determined up to strong cocycle conjugacy by natural abstract properties which are verified for some examples of actions arising from tensorial shifts we also show the following homotopy rigidity result which may be understood as a precursor to a general kirchberg phillips type classification theory if two outer actions of an amenable group on a unital kirchberg algebra are equivariantly homotopy equivalent then they are conjugate this marks the first c dynamical classification result up to cocycle conjugacy that is applicable to actions of all amenable groups

a numerical approach to rough surface scattering by the parabolic equation method the parabolic equation method is an effective approach when the acoustic wave field is incident at low grazing angles onto a rough surface the method consists of an integral equation and an integral the first of which yields the surface field derivative the main part of this paper is concerned with an approximation to this equation valid when wavenumber times surface height is up to order unity the approximation has several advantages first it allows a decomposition of the equation into deterministic and stochastic components the stochastic part depends only locally upon the surface in certain regimes and this can give rise to a great reduction in computational expense some basic statistical moments of the stochastic component are also considered these are nonstationary but for the incident field a simple stationary transformation is found which can therefore be compared with monte carlo simulations using far fewer realizations these results are demonstrated computationally the final part of the paper describes the numerical implementation of the full parabolic equation method both of the integrals contain singularities and these are treated semianalytically

near field imaging of infinite rough surfaces this paper is concerned with an inverse rough surface scattering problem in near field optical imaging which is to reconstruct the scattering surface with a resolution beyond the diffraction limit the surface is assumed to be a small and smooth deformation of a plane surface based on a transformed field expansion the boundary value problem with complex scattering surface is converted into a successive sequence of two point boundary value problems in the frequency domain where an analytic solution for the direct scattering problem is derived from the method of integrated solution by neglecting the high order terms in the power series expansion the nonlinear inverse problem is linearized and an explicit inversion formula is obtained a spectral cut off regularization is adopted to suppress the exponential growth of the noise in the evanescent wave components which carry high spatial frequency of the scattering surface and contribute to the superresolution in the near field regime the method works f

acoustic scattering by an inhomogeneous layer on a rigid plate the problem of scattering of time harmonic acoustic waves by an inhomogeneous fluid layer on a rigid plate in r 2 is considered the density is assumed to be unity in the media within the layer the sound speed is assumed to be an arbitrary bounded measurable function the problem is modelled by the reduced wave equation with variable wavenumber in the layer and a neumann condition on the plate to formulate the problem and prove uniqueness of solution a radiation condition appropriate for scattering by infinite rough surfaces is introduced a generalization of the rayleigh expansion condition for diffraction gratings with the help of the radiation condition the problem is reformulated as a system of two second kind integral equations over the layer and the plate under additional assumptions on the wavenumber in the layer uniqueness of solution is proved and the nonexistence of guided wave solutions of the homogeneous problem established general results on the solvability of systems of integral equations on unbounded domains are used to establish existence and continuous dependence in a weighted norm of the solution on the given data

the validity of the perturbation approximation for rough surface scattering using a gaussian roughness spectrum the validity of the perturbation approximation for rough surface scattering is examined 1 by comparison with exact results obtained by solving an integral equation and 2 through comparison of low order perturbation predictions with higher order predictions the pressure release boundary condition is assumed and the field quantity calculated is the bistatic scattering cross section a gaussian roughness spectrum is used and the surfaces have height variations in only one direction it is found in general that the condition kh 1 k is the acoustic wavenumber h is the rms surface height is insufficient to guarantee the accuracy of first order or higher order perturbation theory when the surface correlation length l becomes too large or too small with h held fixed higher order perturbation terms can make larger contributions to the scattering cross section than lower order terms an explanation for this result is given the regions of validity for low order perturbation theory are also given th

a fast two grid and finite section method for a class of integral equations on the real line with application to an acoustic scattering problem in the half plane we consider the numerical treatment of second kind integral equations on the real line of the form phi s psi s int infty infty k s t z t phi t dt s in mathbb r abbreviated k z in which l 1 z l and bc the space of bounded continuous functions on are assumed known and bc is to be determined we first derive sharp error estimates for the finite section approximation reducing the range of integration to a a via bounds on i k z 1 as an operator on spaces of weighted continuous functions numerical solution by a simple discrete collocation method on a uniform grid on is then analysed in the case whenz is compactly supported this leads to a coefficient matrix which allows a rapid matrix vector multiply via the fft to utilise this possibility we propose a modified two grid iteration a feature of which is that the coarse grid matrix is approximated by a banded matrix and analyse convergence and computational cost in cases wherez is not compactly supported a combined finite section and two grid algorithm can be applied and we extend the analysis to this case as an application we consider acoustic scattering in the half plane with a robin or impedance boundary condition which we formulate as a boundary integral equation of the class studied our final result is that ifz related to the boundary impedance in the application takes values in an appropriate compact subsetq of the complex plane then the difference between s and its finite section approximation computed numerically using the iterative scheme proposed is c 1 khlog 1 kh 1 1 2 ka 1 2 in the interval a a 1 forkh sufficiently small wherek is the wavenumber andh the grid spacing moreover this numerical approximation can be computed in c 2 n logn operations wheren 2a h is the number of degrees of freedom the values of the constantsc 1 andc 2 depend only on the setq and not on the wavenumberk or the support ofz

monte carlo simulations of large scale problems of random rough surface scattering and applications to grazing incidence with the bmia canonical grid method scattering of a te incident wave from a perfectly conducting one dimensional random rough surface is studied with the banded matrix iterative approach canonical grid bmia cag method the bmia cag is an improvement over the previous bmia the key idea of bmia cag is that outside the near field interaction the rest of the interactions can be translated to a canonical grid by taylor series expansion the use of a flat surface as a canonical grid for a rough surface facilitates the use of the fast fourier transform for nonnear field interaction the method can be used for monte carlo simulations of random rough surface problems with a large surface length including all the coherent wave interactions within the entire surface we illustrate results up to a surface length of 2500 wavelengths with 25000 surface unknowns the method is also applied to study scattering from random rough surfaces at near grazing incidence the numerical examples illustrate the importance of using a large surface length for some backscattering problems

an adaptive finite element method for the diffraction grating problem with transparent boundary condition the diffraction grating problem is modeled by a boundary value problem governed by a helmholtz equation with transparent boundary conditions an a posteriori error estimate is derived when the truncation of the nonlocal boundary operators takes place to overcome the difficulty caused by the fact that the truncated dirichlet to neumann dtn mapping does not converge to the original dtn mapping in its operator norm a duality argument without assuming more regularity than the weak solution is applied the a posteriori error estimate consists of two parts the finite element discretization error and the truncation error of boundary operators which decays exponentially with respect to the truncation parameter based on the a posteriori error control a finite element adaptive strategy is established for the diffraction grating problem such that the truncation parameter is determined through the truncation error and the mesh elements for local refinements are marked through the finite element discretization

some degenerate and quasilinear parabolic systems not in divergence form abstract this paper deals with positive solutions of degenerate and quasilinear parabolic systems not in divergence form u t u p u av v t v q v bu with null dirichlet boundary conditions and positive initial conditions where p q a and b are all positive constants the local existence and uniqueness of classical solution are proved moreover it will be proved that all solutions exist globally if and only if ab 1 2 where 1 is the first eigenvalue of in with homogeneous dirichlet boundary condition

second harmonic generation in nonlinear optical films second harmonic generation an important phenomenon in nonlinear optics is modeled in this work the model is derived from a nonlinear system of maxwell s equations which overcomes the known shortcomings of some commonly used models in the literature existence and uniqueness of solutions are established by a combination of a variational approach and the contraction mapping principle some numerical results are also presented

on the solvability of a class of second kind integral equations on unbounded domains we consider integral equations of the form x x k x y z y y dy in operator form kz where is some subset ofrn n 1 the functionsk z and are assumed known withz l and y the space of bounded continuous functions on the function yis to be determined the class of domains and kernelskconsidered includes the case rnandk x y x y with l1 rn in which case ifzis the characteristic function of some setg the integral equation is one of wiener hopf type the main theorems proved using arguments derived from collectively compact operator theory are conditions on a setw l which ensure that ifi kzis injective for allz wtheni kzis also surjective and moreover the inverse operators i kz 1onyare bounded uniformly inz these general theorems are used to recover classical results on wiener hopf integral operators of21and19 and generalisations of these results and are applied to analyse the lippmann schwinger integral equation

solvability and spectral properties of integral equations on the real line i weighted spaces of continuous functions abstract we consider in this paper the solvability of linear integral equations on the real line in operator form k where c and k is an integral operator we impose conditions on the kernel k of k which ensure that k is bounded as an operator on x bc r let xa denote the weighted space x a x s o s a as s our first result is that if additionally k s t s t with l 1 r and s o s b as s for some b 1 then the spectrum of k is the same on xa as on x for 0 k k k w which ensure that if 0 and kk has only the trivial solution in x for all k w then for 0 a b k has exactly one solution xa for every k w and xa these conditions ensure further that k 1 x a x a is bounded uniformly in k w for 0 a b as a particular application we consider the case when the kernel takes the form k s t s t z t with l 1 r z l r and s o s b as s for some b 1 as an example where kernels of this latter form occur we discuss a boundary integral equation formulation of an impedance boundary value problem for the helmholtz equation in a half plane

a new approach to the analysis of rough surface scattering a novel approach to the problem of scattering by a randomly rough surface is developed based on combining normalization and the method of smoothing previous uses of smoothing have been forced to assume small surface heights in order to guarantee the dominance of the scattered field by its average or specular part the term that causes the average scattered field to be so sensitive to the height is identified and is normalized out of the basic integral equation smoothing is then applied to the normalized integral equation with a subsequent reintroduction of the normalizing factor after smoothing the results are shown to extend beyond existing approximations and they contain none of the divergent integral behavior that has recently been observed with a pure second order iterative approximation an examination of the conditions under which the technique provides very accurate results shows that it is essentially an extension of the so called composite surface scattering model

the mathematics of scattering by unbounded rough inhomogeneous layers in this paper we study via variational methods a boundary value problem for the helmholtz equation modelling scattering of time harmonic waves by a layer of spatially varying refractive index above an unbounded rough surface on which the field vanishes in particular in the 2d case with te polarization the boundary value problem models the scattering of time harmonic electromagnetic waves by an inhomogeneous conducting or dielectric layer above a perfectly conducting unbounded rough surface with the magnetic permeability a fixed positive constant in the medium via analysis of an equivalent variational formulation we show that this problem is well posed in two important cases when the frequency is small enough and when the medium in the layer has some energy absorption in this latter case we also establish exponential decay of the solution with depth in the layer an attractive feature is that all constants in our estimates are bounded by explicit functions of the index of refraction and the geometry of the scatterer

the nuclear dimension of c algebras abstract we introduce the nuclear dimension of a c algebra this is a noncommutative version of topological covering dimension based on a modification of the earlier concept of decomposition rank our notion behaves well with respect to inductive limits tensor products hereditary subalgebras hence ideals quotients and even extensions it can be computed for many examples in particular it is finite for all uct kirchberg algebras in fact all classes of nuclear c algebras which have so far been successfully classified consist of examples with finite nuclear dimension and it turns out that finite nuclear dimension implies many properties relevant for the classification program surprisingly the concept is also linked to coarse geometry since for a discrete metric space of bounded geometry the nuclear dimension of the associated uniform roe algebra is dominated by the asymptotic dimension of the underlying space

maxwell s equations in a periodic structure consider a diffraction of a beam of particles in r3 when the dielectric coefficient is a constant cl above a surface s and a constant e2 below a surface s and the magnetic permeability is constant throughout rii s is assumed to be periodic in the xl direction and of the form x1 f1 s x3 f3 s x2 arbitrary we prove that there exists a unique solution to the time harmonic maxwell equations in r 3 having the form of refracted waves for x3 1 and of transmitted waves for x3 1 if and only if there exists a unique solution to a certain system of two coupled fredholm equations thus in particular for all the e s except for a discrete number there exists a unique solution to the maxwell equations introduction in this paper we consider the maxwell equations for time harmonic solutions in the entire space r3 with piecewise constant dielectric coefficient having jump across a periodic surface the magnetic permeability u is assumed to be constant whereas the dielectric coefficient e is given by e 8 above a surface s x3 f xl and e e2 below the surface s e and 82 are different constants if s is a half space x3 0 then the solution eo ho can be computed explicitly we assume in this paper that s is periodic i e f xl l f xl vx1 e r l 0 we wish to find a solution e h such that e eo and h ho are superpositions of transmitted waves 0 1 in x3 a where a maxlfl in 1 7 we assume that f e c2 and we reduce the solution of the maxwell equations to a fredholm system of four integral equations in 8 we reduce it further to a fredholm system of two integral equations thus for all but a discrete sequence of values of the physical parameters there exists a unique solution to the integral equations yielding a solution of the maxwell equations received by the editors january 25 1989 1980 mathematics subject classification 1985 revision primary 78a10 78a45 secondary 35p25 47a40

on the nuclear dimension of strongly purely infinite c algebras abstract we show that separable nuclear and strongly purely infinite c algebras have finite nuclear dimension in fact the value is at most three this exploits a deep structural result of kirchberg and rordam on strongly purely infinite c algebras that are homotopic to zero in an ideal system preserving way

regularity properties in the classification program for separable amenable c algebras we report on recent progress in the program to classify separable amenable c algebras our emphasis is on the newly apparent role of regularity properties such as finite decomposition rank strict comparison of positive elements and z stability and on the importance of the cuntz semigroup we include a brief history of the program s successes since 1989 a more detailed look at the villadsen type algebras which have so dramatically changed the landscape and a collection of announcements on the structure and properties of the cuntz semigroup

approximate solution of second kind integral equations on infinite cylindrical surfaces the paper considers second kind integral equations of the form x g x fs k x y 0 y ds y abbreviated 0 g ko in which s is an infinite cylindrical surface of arbitrary smooth cross section the truncated equation abbreviated 4 a eag ka4 a obtained by replacing s by sa a closed bounded surface of class c2 the boundary of a section of the interior of s of length 2a is also discussed conditions on k are obtained in particular implying that k commutes with the operation of translation in the direction of the cylinder axis which ensure that i k is invertible that i ka is invertible and i ka is uniformly bounded for all sufficiently large a and that ka converges to k in an appropriate sense as a oo uniform stability and convergence results for a piecewise constant boundary element collocation method for the truncated equations are also obtained a boundary integral equation which models three dimensional acoustic scattering from an infinite rigid cylinder illustrates the application of the above results to prove existence of solution of the integral equation and the corresponding boundary value problem and convergence of a particular collocation method

quasidiagonality of nuclear c algebras we prove that faithful traces on separable and nuclear c algebras in the uct class are quasidiagonal this has a number of consequences firstly by results of many hands the classification of unital separable simple and nuclear c algebras of finite nuclear dimension which satisfy the uct is now complete secondly our result links the finite to the general version of the toms winter conjecture in the expected way and hence clarifies the relation between decomposition rank and nuclear dimension finally we confirm the rosenberg conjecture discrete amenable groups have quasidiagonal c algebras

a least squares finite element analysis for diffraction problems the diffraction of a time harmonic wave incident upon a grating or periodic structure is treated by a least squares finite element method that incorporates the jump conditions at interfaces into the objective functional two fundamental polarizations are considered coercivity of the quadratic functional is established and optimal discretization error estimates are obtained in both cases the theoretical results indicate that for sufficiently smooth interfaces the error estimates are superior to those of standard finite element methods

a new numerical method for rough surface scattering calculations a new approach to solving the magnetic field integral equation mfie for the current induced on a infinite perfectly conducting rough surface is presented by splitting the propagator matrix into contributions from the left and from the right of the point of observation a second kind integral equation can be formed with a new born term and a new kernel following discretization of this new integral equation the unknown currents can be determined more rapidly and with significantly less storage requirements than conventional lu decomposition where the time saving factor is roughly n 3 where n is the number of current samples on the surface and the usual storage requirements associated with matrix inversion are eliminated while the new born term is usually adequate for scattered field calculations the new discretized integral equation can be iterated to amy desired accuracy with no apparent convergence problems results are presented for one dimensional rough surfaces with rms heights exceeding one wavelength and rms slopes exceeding 40 which illustrate the robustness of the new born term

uniqueness of a scattering problem with l2 boundary values in this paper we consider the scattering problem for a locally perturbed half plane with l 2 boundary values and we obtain a uniqueness result to the solution especially we prove our result for the two dimensional dirichlet boundary value problem in l 2 sense for the helmholtz equation in a locally perturbed half plane

the numerical solution of scattering by infinite rough interfaces based on the integral equation method in this paper we describe a nystrom integration method for the integral operator t which is the normal derivative of the double layer potential arising in problems of two dimensional acoustic scattering by infinite rough interfaces the hypersingular kernel and unbounded integral interval of t are the key difficulties by using a mollifier we separately deal with these two difficulties and propose its nystrom integration method furthermore we establish convergence of the method finally we apply the method to the scattering problem by infinite rough interfaces and carry out some numerical experiments to show the validity

scattering of scalar waves by two dimensional gratings of arbitrary shape application to rough surfaces at near grazing incidence we present the numerical study of scattering of scalar waves from impenetra ble two dimensional periodic surfaces of arbitrary shape nearly all numerical simulations of scattering of waves from rough surfaces in the past have been limited to one dimensional surfaces and moderate angles of incidence by making the surface infinite and bi periodic it becomes possible to simulate numerically scattering from two dimensional surfaces even down to grazing angle only impenetrable surfaces are considered some calculations are presented and are used to compare with the small perturbation or rayleigh rice theory it is found that for near grazing incidence neumann boundary condition the small perturbation theory gives inaccurate values especially near the backscatter direction

artificial boundary conditions for axisymmetric eddy current probe problems we study different strategies for the truncation of computational domains in the simulation of eddy current probes of elongated axisymmetric tubes for axial fictitious boundaries an exact dirichlet to neumann map is proposed and mathematically analyzed via a non selfadjoint spectral problem under general assumptions we show convergence of the solution to an eddy current problem involving a truncated dirichlet to neumann map to the solution on the entire unbounded axisymmetric domain as the truncation parameter tends to infinity under stronger assumptions on the physical parameters of the eddy current problem convergence rates are shown we further validate our theoretical results through numerical experiments for a realistic physical setting inspired by eddy current probes of nuclear reactor core tubes

solution of a boundary value problem for the helmholtz equation via variation of the boundary into the complex domain in this paper we deal with the problem of diffraction of electromagnetic waves by a periodic interface between two materials this corresponds to a two dimensional quasi periodic boundary value problem for the helmholtz equation we prove that solutions behave analytically with respect to variations of the interface the interest of this result is both theoretical the legitimacy of power series expansions in the parameters of the problem has indeed been questioned and perhaps more importantly practical we have found that the solution can be computed on the basis of this observation the simple algorithm that results from such boundary variations is described to establish the property of analyticity of the solution for the grating with respect to the height we present a holomorphic formulation of the problem using surface potentials we show that the densities entering into the potential theoretic formulation are analytic with respect to variations of the boundary or in other words that the integral operator that results from the transmission conditions at the interface is invertible in a space of holomorphic functions of the variables x y this permits us to conclude in particular that the partial derivatives of u with respect to at 0 satisfy certain boundary value problems for the helmholtz equation in regions with plane boundaries which can be solved in a closed form

the time harmonic maxwell equations in a doubly periodic structure abstract consider the diffraction of a beam of particles in r 3 when the dielectric coefficient is a constant e1 above a surface s and a different constant e2 below s and the magnetic permeability is constant throughout r 3 s is assumed to be a doubly periodic surface say z f x y with f x ml1 y nl2 f x y for integers m n the existence and uniqueness of a solution satisfying a radiation condition at infinity is reduced to a system of fredholm equations thus for all but a discrete set of e s there exists a unique solution

monte carlo simulation of electromagnetic scattering from two dimensional random rough surfaces the fast multipole method fast fourier transform fmm fft method is developed to compute the scattering of an electromagnetic wave from a two dimensional 2 d rough surface the resulting algorithm computes a matrix vector multiply in o n log n operations this algorithm is shown to be more efficient than another o n log n algorithm the multilevel fast multipole algorithm mlfma for surfaces of small height for surfaces with larger roughness the mlfma is found to be more efficient using the mlfma monte carlo simulations are carried out to compute the statistical properties of the electromagnetic scattering from 2 d random rough surfaces using a workstation for the rougher surface backscattering enhancement is clearly observable as a pronounced peak in the backscattering direction of the computed bistatic scattering coefficient for the smoother surface the monte carlo results compare well with the results of the approximate kirchhoff theory

inverse problems for scattering by periodic structures suppose the 3 dimensional space is filled with three materials having dielectric constants 1 above s1 x2 f1 x1 x3 arbitrary 2 below s2 x2 f2 x1 x3 arbitrary and o in f2 x1 0 if the measurements for the two pairs are close then tilde s 1 and tilde s 2 are 0 close to s1 and s2 respectively

the domain derivative in rough surface scattering and rigorous estimates for first order perturbation theory we investigate frechet differentiability of the scattered field with respect to variation in the boundary in the case of time harmonic acoustic scattering by an unbounded sound soft one dimensional rough surface we rigorously prove the differentiability of the scattered field and derive a characterization of the frechet derivative as the solution to a dirichlet boundary value problem as an application of these results we give rigorous error estimates for first order perturbation theory justifying small perturbation methods that have a long history in the engineering literature as an application of our rigorous estimates we show that a plane acoustic wave incident on a sound soft rough surface can produce an unbounded scattered field

a new treatment of rough surface scattering this paper treats the problem of the scattering of a narrow beam at a randomly rough surface where the height distribution is gaussian and the correlation of heights is gaussian the field at the surface is assumed to satisfy a dirichlet boundary condition the problem is formulated using the standard integral equation approach in the case when the rms height is very large numerical solution of this equation for a realization of the surface involves the solution of a very large matrix problem particularly if the beam is scattered to other parts of the surface that are at a considerable distance from the initial illumination here we provide a straightforward technique motivated by ray theory that allows only those parts of the surface that are actually illuminated at any stage to be treated thus for the first reflection of the beam one solves an integral equation over the initially illuminated area this solution is used to determine the parts of the surface which give rise to doubly scattered waves and one then solves the integral equation over the newly illuminated area and so on until all reflections have been accounted for this procedure can be carried out with no explicit reference to any ray model working simply with the discrete matrix form of the integral equation in particular for one dimensional surface variation a strategy of successive sweeps from left to right and back again for a beam incident from the left is shown to be successful any standard iterative or direct solution technique can be used for the sequence of sub problems so introduced the general method is also applicable to scattering from surfaces varying in two dimensions

numerical analysis of diffraction by periodic structures tm polarization the diffraction of a time harmonic wave in tm polarization by some periodic inhomogeneous material is studied in this paper the diffraction problem is modeled by a generalized helmholtz equation with transparent boundary conditions existence and uniqueness are proved for the model problem uniform error estimates for the finite element approximation of the solution are established error estimates are also obtained when the truncation of the nonlocal boundary operators takes place

an inverse problem for scattering by a doubly periodic structure consider scattering of electromagnetic waves by a doubly periodic structure s x3 f xl x2 with f xl nial x2 n2a2 f xl x2 for integers nl n2 above the structure the medium is assumed to be homogeneous with a constant dielectric coefficient the medium is a perfect conductor below the structure an inverse problem arises and may be described as follows for a given incident plane wave the tangential electric field is measured away from the structure say at x3 b for some large b to what extent can one determine the location of the periodic structure that separates the dielectric medium from the conductor in this paper results on uniqueness and stability are established for the inverse problem a crucial step in our proof is to obtain a lower bound for the first eigenvalue of the following problem in a convex domain q l au in q v u 0 in q n xu 0 on q

on the derivation of boundary integral equations for scattering by an infinite one dimensional rough surface a crucial ingredient in the formulation of boundary value problems for acoustic scattering of time harmonic waves is the radiation condition this is well understood when the scatterer is a bounded obstacle for plane wave scattering by an infinite rough impenetrable surface s the physics of the problem suggests that all scattered waves must travel away from or along the surface this condition is used together with green s theorem and the free space green s function to derive boundary integral equations over s this requires careful consideration of certain integrals over a large semicircle of radius r it is known that these integrals vanish as r if the scattered field satisfies the sommerfeld radiation condition but that is not the case here reflected plane waves must be present the integral equations obtained are helmholtz integral equations they must be modified for grazing incident waves as such integral equations are often claimed to be exact and are often used to generate benchmark n

finite element approximation of time harmonic waves in periodic structures consider the diffraction of a time harmonic wave incident on a periodic surface of some inhomogeneous material it is shown that the scattering diffraction problem may be modeled by a helmholtz equation with transparent boundary conditions the diffraction problem may be solved by a finite element method in this paper optimal error estimates for the finite element method are established the error estimates are also established when the truncation of the nonlocal transparent boundary operators takes place

a scattering model for nondifferentiable periodic surface roughness ice dendrites that form when seawater freezes produce a surface with periodic roughness this surface may be modeled by joining a concave upward semiellipse to a concave downward semiellipse and replicating this fundamental surface element to form a surface that is everywhere continuous but is not differentiable at the points where the two ellipses are joined i e the derivative is infinite this surface is regarded as realization of the class of periodic rough surfaces that are not differentiable at a finite number of points within a period and includes surfaces that are piecewise and nonpiecewise smooth a model is described to calculate plane wave scattering from one dimensional realizations of these types of surfaces to calculate scattering from these types of surfaces for the physical parameters of interest the integral equation formalism of holford j acoust soc am 70 1116 1128 1980 is used and the surface roughness profile is approximated by truncating its fourier series representatio