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

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

type fuzzy quantifiers determined by fuzzy measures defined on residuated lattices part ii permutation and isomorphism invariances abstract we study the properties of permutation invariance and isomorphism invariance of fuzzy quantifiers of type 1 1 defined using fuzzy measures and integrals these properties hold if fuzzy quantifiers are invariant with respect to permutations bijective mappings on the universe of discourse permutation invariance and with respect to bijections between different universes of discourse isomorphism invariance in other words fuzzy quantifiers with these properties are sensitive only to the cardinality of subsets of the universe of discourse and not to the individual nature of the elements of these subsets we characterize these properties by means of the corresponding properties of functionals used in the definition of the fuzzy quantifiers

weighted sobolev spaces and exterior problems for the helmholtz equation weighted sobolev spaces are used to settle questions of existence and uniqueness of solutions to exterior problems for the helmholtz equation furthermore it is shown that this approach can cater for inhomogeneous terms in the problem that are only required to vanish asymptotically at infinity in contrast to the rellich sommerfeld radiation condition which in a hilbert space setting requires that all radiating solutions of the helmholtz equation should satisfy a condition of the form r i k u l 2 r x r n it is shown here that radiating solutions satisfy a condition of the form 1 r 1 2 ln e r 1 2 u l 2 0 1 2 and moreover such solutions satisfy the classical sommerfeld condition u o r 1 2 n 1 r furthermore the approach avoids many of the difficulties usually associated with applications of the poincare inequality and the sobolev embedding theorems

fuzzy choquet integration of homogeneous possibility and probability distributions the fuzzy integral fi is an extremely flexible and powerful tool for data and information aggregation the fi is parametrized by the fuzzy measure fm a normal and monotone capacity based on the selection of fm the fi produces different aggregation operators in recent years a number of fi extensions have been put forth relative to different types of uncertain information e g real interval and set valued under various constraints herein we study the applicability and behavior of different extensions of the fuzzy choquet integral for fusing homogeneous possibility and probability distributions this analysis is of great utility in terms of understanding what extensions and under what conditions it is possible to aggregate and maintain homogeneity within uncertain information we show that two extensions gfi and ndfi can aggregate both probability and possibility distributions while these extensions do not always maintain homogeneity they do under certain conditions last while we specifically focus on the aggregation of homogeneous uncertain information the propositions put forth also shed light into heterogeneous information aggregation via the gfi and the ndfi

on fuzzy measure and fuzzy integral on fuzzy set abstract on a fuzzy algebra of fuzzy sets we discuss some structural characteristics of a fuzzy measure the pseudo autocontinuity and some other concepts for a fuzzy measure are introduced and the relations between these cre discussed for the case of fuzzy sets riesz s theorems and lebesgue s theorems for a sequence of measurable functions are proved the concept of the fuzzy integral on a fuzzy set is given and some properties of fuzzy integrals on fuzzy sets are studied finally we prove some convergence theorems of a sequence of fuzzy integrals on fuzzy sets

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

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

diffraction by a defect in an open waveguide a mathematical analysis based on a modal radiation condition we consider the scattering of a time harmonic acoustic wave by a defect in a twodimensional open waveguide the scattered wave satisfies the helmholtz equation in a perturbed layered half plane we introduce a modal radiation condition based on a generalized fourier transform which diagonalizes the transverse contribution of the helmholtz operator the uniqueness of the solution is proved by an original technique which combines a property of the energy flux with an argument of analyticity with respect to the generalized fourier variable the existence is then deduced classically from fredholm s alternative by reformulating the scattering problem as a lippmann schwinger equation by means of the green s function for the layered half plane 2009 society for industrial and applied mathematics

an inequality related to minkowski type for sugeno integrals an inequality related to minkowski type for the sugeno integral on abstract spaces is studied in a rather general form some previous results on chebyshev type inequality obtained by the authors are generalized several examples are given to illustrate the validity of this inequality the conditions such that this inequality becomes an equality are also discussed finally conclusions and some problems for further investigations are included

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

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

the limiting absorption and amplitude principles for the diffraction problem with two unbounded media we consider the classical diffraction problem for the wave propagation in the case where the propagation speed is piecewise constant and the surface separating two media is unbounded the validity of the limiting absorption and amplitude principles is proved

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

l fuzzy quantifiers of type determined by fuzzy measures the aim of this paper is first to introduce two new types of fuzzy integrals namely fuzzy integral and fuzzy integral the first integral is based on a fuzzy measure of l fuzzy sets and the second one on a complementary fuzzy measure of l fuzzy sets where l is a complete residuated lattice some of their properties and a relation to the fuzzy sugeno integral are investigated second using these integrals two classes of monadic l fuzzy quantifiers of type are defined these l fuzzy quantifiers can be used for modeling the semantics of natural language quantifiers like all some many none at most half etc several semantic properties of these l fuzzy quantifiers are studied

monadic l fuzzy quantifiers of the type the aim of this paper is to introduce monadic l fuzzy quantifiers of the type and investigate some of their semantic properties that are well known in generalized quantifier theory relationships between the properties of generalized quantifiers are also studied for monadic l fuzzy quantifiers finally a property to be extensional with respect to a given l permutation equivalence is proposed and some properties are shown

fuzzy measures and integrals defined on algebras of fuzzy subsets over complete residuated lattices this paper presents basic notions about fuzzy measures over algebras of fuzzy subsets of a fuzzy set it also presents basic ideas on fuzzy integrals defined using these fuzzy measures definitions of new types of fuzzy measures and integrals are motivated by our research on generalized quantifiers several useful properties of fuzzy measures and fuzzy integrals are stated and proved definitions presented in this paper and its results will be employed in subsequent papers on generalized quantifiers defined using this type of fuzzy integral

transformation theorems for fuzzy integrals on fuzzy sets abstract in this paper sugeno s fuzzy integrals studied in 5 8 9 are generalized some equivalent definitions of fuzzy integrals on fuzzy sets are given and several transformation theorems of such type of fuzzy integrals are proved these results reveal the relations between the fuzzy integrals on fuzzy sets and on classical sets

a partition based relaxation for steiner trees the steiner tree problem is a classical np hard optimization problem with a wide range of practical applications in an instance of this problem we are given an undirected graph g v e a set of terminals r subseteq v and non negative costs c e for all edges e in e any tree that contains all terminals is called a steiner tree the goal is to find a minimum cost steiner tree the vertices v backslash r are called steiner vertices the best approximation algorithm known for the steiner tree problem is a greedy algorithm due to robins and zelikovsky siam j discrete math 19 1 122 134 2005 it achieves a performance guarantee of 1 frac ln 3 2 approx 1 55 the best known linear programming lp based algorithm on the other hand is due to goemans and bertsimas math program 60 145 166 1993 and achieves an approximation ratio of 2 2 r in this paper we establish a link between greedy and lp based approaches by showing that robins and zelikovsky s algorithm can be viewed as an iterated primal dual algorithm with respect to a novel lp relaxation the lp used in the first iteration is stronger than the well known bidirected cut relaxation an instance is b quasi bipartite if each connected component of g backslash r has at most b vertices we show that robins and zelikovsky s algorithm has an approximation ratio better than 1 frac ln 3 2 for such instances and we prove that the integrality gap of our lp is between frac 8 7 and frac 2b 1 b 1

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

absence of singular continuous spectrum for certain self adjoint operators we give a sufficient condition for a self adjoint operator to have the following properties in a neighborhood of a pointe of its spectrum a its point spectrum is finite b its singular continuous spectrum is empty c its resolvent satisfies a class of a priori estimates

pseudo riemann stieltjes integral an extension of the classical riemann stieltjes integral to the field of pseudo analysis is being investigated through this paper core of the construction presented here consists of generalized pseudo operations given by monotone generating function

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

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

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 general approximation technique for constrained forest problems we present a general approximation technique for a large class of graph problems our technique mostly applies to problems of covering at minimum cost the vertices of a graph with trees cycles or paths satisfying certain requirements in particular many basic combinatorial optimization problems fit in this framework including the shortest path minimum cost spanning tree minimum weight perfect matching traveling salesman and steiner tree problems our technique produces approximation algorithms that run in o n 2 log n time and come within a factor of 2 of optimal for most of these problems for instance we obtain a 2 approximation algorithm for the minimum weight perfect matching problem under the triangle inequality our running time of o n 2 log n time compares favorably with the best strongly polynomial exact algorithms running in o n 3 time for dense graphs a similar result is obtained for the 2 matching problem and its variants we also derive the first approximation algorithms for many np complete problems including the non fixed point to point connection problem the exact path partitioning problem and complex location design problems moreover for the prize collecting traveling salesman or steiner tree problems we obtain 2 approximation algorithms therefore improving the previously best known performance guarantees of 2 5 and 3 respectively math programming 59 1993 pp 413 420

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

pseudo additive measures and integrals abstract we suggest pseudo additive measures based on a pseudo addition and discuss integrals with respect to pseudo additive measures a pseudo additive measure is a special type of fuzzy measures to define an integral a multiplication corresponding to a pseudo addition is introduced the resulting integral is an extension of the lebesgue integral in this context radon nikodym like theorems are shown

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

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

coupling of finite element and boundary integral methods for electromagnetic scattering in a two layered medium consider a time harmonic electromagnetic plane wave incident on an inhomogeneity embedded in a two layered medium in this paper a method of coupling of finite element and boundary integral equation methods is presented for the solutions of electromagnetic scattering in both transverse electric and magnetic polarization cases the well posedness of the continuous and discrete problems as well as optimal error estimates for the coupled variational approximations are obtained numerical results are included to illustrate the accuracy with the optimal convergence property of the proposed method and to show the wave features in a two layered medium

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

pseudo optimal measures based on a pseudo addition on 0 pseudo optimal measures related to non negative set functions vanishing at the empty set non negative games are introduced for a fixed pseudo addition an equivalence on non negative set functions is studied a special stress is given to pure pseudo optimal measures admitting no other equivalent set function

a jensen type inequality for fuzzy integrals in this paper we show a jensen type inequality for the sugeno integral we also discuss some conditions assuring the satisfaction of opposite inequality reverse jensen inequality

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

stable and efficient evaluation of periodized green s functions for the helmholtz equation at high frequencies a difficulty that arises in the context of infinite d periodic rough surface scattering relates to the effective numerical evaluation of the corresponding quasi periodic green function g q p due to its relevance in a variety of applications this problem has generated significant interest over the last 40 years and a variety of numerical methods have been devised for this purpose none of these methods to evaluate g q p however were designed for high frequency calculations as a result in this regime these methods become prohibitively expensive and or unstable here we present a novel scheme that can be shown to outperform every alternative numerical evaluation procedure and is especially effective for high frequency calculations our new algorithm is based on the use of some exact integrals that arise on judicious manipulation of the integral representation of g q p and which reduce the overall problem to that of evaluation of a sequence of simpler integrals that can be effectively handled by standard quadrature formulas we include a variety of numerical results that confirm that indeed our algorithm compares favorably with alternative methods

type 1 1 fuzzy quantifiers determined by fuzzy measures on residuated lattices part iii extension conservativity and extensionality we study the properties of extension conservativity and extensionality of fuzzy quantifiers of type 1 1 defined using fuzzy measures and integrals the property of extension states that truth values of quantifier applications are invariant with respect to possible extensions of the universe conservativity expresses the property that quantifiers are sensitive in their second argument only to objects that lie in the intersection of their arguments extensionality represents a form of the smoothness of quantifiers we characterize these properties by the corresponding properties of functionals used in the definition of fuzzy quantifiers

radiation condition bounds on manifolds with ends abstract we study spectral theory for the schrodinger operator on manifolds possessing an escape function a particular class of examples are manifolds with euclidean and or hyperbolic ends certain exterior domains for possibly unbounded obstacles are included we prove rellich s theorem the limiting absorption principle radiation condition bounds and the sommerfeld uniqueness result striving to extending and refining previously known spectral results on manifolds the proofs are given by an extensive use of commutator arguments these arguments have a classical spirit essentially not involving energy cutoffs or microlocal analysis and require presumably minimum regularity and decay properties of perturbations this paper has interest of its own right but it also serves as a basis for the stationary scattering theory developed fully in the sequel 19

a universal integral as common frame for choquet and sugeno integral the choquet and the sugeno integral provide a useful tool in many problems in engineering and social choice where the aggregation of data is required however their applicability is restricted because of the special operations used in the construction of these integrals therefore we provide a concept of integrals generalizing both the choquet and the sugeno case for functions with values in the nonnegative real numbers universal integrals are introduced and investigated which can be defined on arbitrary measurable spaces and for arbitrary monotone measures for a fixed pseudo multiplication on the nonnegative real numbers the smallest and the greatest universal integrals are given finally another construction method for obtaining universal integrals is introduced and the restriction to the unit interval i e to fuzzy integrals is considered

combinatorial optimization of matrix vector multiplication in finite element assembly it has been shown that combinatorial optimization of matrix vector multiplication can lead to faster evaluation of finite element stiffness matrices based on a graph model characterizing relationships between rows an efficient set of operations can be generated to perform matrix vector multiplication for this problem we improve the graph model by extending the set of binary row relationships and solve this combinatorial optimization problem optimally for the binary row relationships implemented yielding significantly improved results over previously published graph models we also extend the representation by using hypergraphs to model more complicated row relationships expressing a three row relationship with a three vertex hyperedge for example our initial greedy algorithm for this hypergraph model has yielded significantly better results than the graph model for many matrices

effective limiting absorption principles and applications the limiting absorption principle asserts that if h is a suitable schrodinger operator and f lives in a suitable weighted l2 space namely h 0 1 2 sigma for some sigma 0 then the functions r lambda i varepsilon f h lambda i varepsilon 1 f converge in another weighted l2 space h 0 1 2 sigma to the unique solution u of the helmholtz equation h lambda u f which obeys the sommerfeld outgoing radiation condition in this paper we investigate more quantitative or effective versions of this principle for the schrodinger operator on asymptotically conic manifolds with short range potentials and in particular consider estimates of the form r lambda i varepsilon f h 0 1 2 sigma leq c lambda h f h 0 1 2 sigma we are particularly interested in the exact nature of the dependence of the constants c lambda h on both lambda and h it turns out that the answer to this question is quite subtle with distinctions being made between low energies lambda ll 1 medium energies lambda sim 1 and large energies lambda gg 1 and there is also a non trivial distinction between qualitative estimates on a single operator h possibly obeying some spectral condition such as non resonance or a geometric condition such as non trapping and quantitative estimates which hold uniformly for all operators h in a certain class using elementary methods integration by parts and ode techniques we give some sharp answers to these questions as applications of these estimates we present a global in time local smoothing estimate and pointwise decay estimates for the associated time dependent schrodinger equation as well as an integrated local energy decay estimate and pointwise decay estimates for solutions of the corresponding wave equation under some additional assumptions on the operator h

exponential bounds and absence of positive eigenvalues for n body schrodinger operators for a large class ofn body potentialsv we prove that if is an eigenfunction of v with eigenvaluee then sup 2 e 0 exp x l2 is either a threshold or consequences of this result are the absence of positive eigenvalues and optimal l2 exponential lower bounds

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

scattering for schrodinger operators in a class of domains with non compact boundaries abstract schrodinger operators in a class of domains with asymptotic cones are considered a generalized fourier transform representing the absolutely continuous part of the schrodinger operator as multiplication by 2 in the asymptotic cone is constructed wave operators relating the free laplacian to schrodinger operators are computed using the generalized fourier transform the wave operators relating schrodinger operators acting in domains with the same asymptotic cone are computed and shown to be complete

radiation condition and limiting amplitude principle for acoustic propagators with two unbounded media consider the diffraction problem for perturbed acoustic propagators with perturbations decreasing slowly at infinity the propagation speed is discontinuous at the interface of two unbounded media and the interface may be an arbitrary and smooth surface locally a sommerfeld radiation condition is introduced for the acoustic propagator and is then used to establish the limiting absorption principle and the resolvent estimate at low frequencies for such an operator furthermore we prove the existence of a unique solution to the diffraction problem and the validity of the limiting amplitude principles for the acoustic propagator

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

extensions of fuzzy relational compositions based on generalized quantifiers abstract the aim of this paper is to extend the fuzzy relational compositions motivated by their applications to medical diagnosis problem by bandler and kohout these compositions employ two quantifiers the universal quantifier and the existential quantifier as there exists a big gap between these two quantifiers that may be appropriately filled in by intermediate generalized quantifiers we take this as a motivation for our study in particular we introduce the concept of fuzzy relational compositions based on generalized quantifiers furthermore we show that the properties that are typically valid for standard fuzzy relational compositions are also valid for the compositions based on generalized quantifiers yet sometimes in a weaken form apart from fuzzy relational compositions the use of generalized quantifiers is also applied to images and preimages of fuzzy sets under fuzzy relations

residuated variants of sugeno integrals sugeno integrals and their particular cases such as weighted minimum and maximum have been used in multiple criteria aggregation when the evaluation scale is qualitative this paper proposes two new variants of weighted minimum and maximum where the criteria weights play the role of tolerance thresholds these variants require the use of a residuated structure equipped with an involutive negation we propose residuated counterparts of sugeno integrals where the weights bear on subsets of criteria and we study their properties showing they are analogous to sugeno integrals to a large extent finally we propose dual aggregation operations we call desintegrals where an item is evaluated in terms of its defects rather than in terms of its positive features desintegrals are maximal when no defects at all are present while integrals are maximal when all merits are sufficiently present qualitative integrals and desintegrals suggest a possible approach to bipolar evaluation processes where items are judged both in terms of merits and defects that are not independent of one another

mathematical analysis of the junction of two acoustic open waveguides the present paper concerns the scattering of a time harmonic acoustic wave by the junction of two open uniform waveguides where the junction is limited to a bounded region we consider a two dimensional problem for which wave propagation is described by the scalar helmholtz equation the main difficulty in the modeling of the scattering problem lies in the choice of conditions which characterize the outgoing behavior of a scattered wave we use here modal radiation conditions which extend the classical conditions used for closed waveguides they are based on the generalized fourier transforms which diagonalize the transverse contributions of the helmholtz operator on both sides of the junction we prove the existence and uniqueness of the solution which seems to be the first result in this context the originality lies in the proof of uniqueness which combines a natural property related to energy fluxes with an argument of analyticity with respect to the generalized fourier variable

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

when trees collide an approximation algorithm for thegeneralized steiner problem on networks we give the first approximation algorithm for the generalized network steiner problem a problem in network design an instance consists of a network with link costs and for each pair i j of nodes an edge connectivity requirement r ij the goal is to find a minimum cost network using the available links and satisfying the requirements our algorithm outputs a solution whose cost is within 2 lceil log 2 r 1 rceil of optimal where r is the highest requirement value in the course of proving the performance guarantee we prove a combinatorial min max approximate equality relating minimum cost networks to maximum packings of certain kinds of cuts as a consequence of the proof of this theorem we obtain an approximation algorithm for optimally packing these cuts we show that this algorithm has application to estimating the reliability of a probabilistic network

generalized fuzzy integrals part 3 convergent theorems abstract in this part we continue the work of preceding papers to consider all kinds of convergent theorems of g fuzzy integrals

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

randomized parallel algorithms for matroid union and intersection with applications to arboresences and edge disjoint spanning trees the strong link between matroids and matching is used to extend the ideas that resulted in the design of random nc rnc algorithms for matching to obtain rnc algorithms for the matroid union intersection and matching problems and for linearly representable matroids as a consequence rnc algorithms for the well known problems of finding an arboresence and a maximum cardinality set of edge disjoint spanning trees in a graph are obtained the key tools used are linear algebra and randomization

on the probabilistic hausdorff distance and a class of probabilistic decomposable measures in this paper some useful properties associated with the probabilistic hausdorff distance are further derived especially we provide a direct proof for an existing important result afterwards the t norm based probabilistic decomposable measure is presented in which the value of measure is characterized by a probability distribution function meantime several examples are constructed to illustrate different notions and then further properties are examined moreover for a given menger pm space a probabilistic decomposable measure can be induced by means of the resulting probabilistic hausdorff distance we prove that this type of measure is s email protected probabilistic subdecomposable measure for the strongest t norm furthermore we also prove that the class of all measurable sets forms an algebra finally an outer probabilistic measure is induced by a class of probabilistic decomposable measures and the t norm based on this kind of measure a menger probabilistic pseudometric space can be obtained for a non strict continuous archimedean t norm

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

boundary integral equation methods for the scattering problem by an unbounded sound soft rough surface with tapered wave incidence in this paper we consider the scattering problem of tapered acoustic wave by an unbounded sound soft surface the scattering problem is modeled as a boundary value problem governed by the helmholtz equation with dirichlet boundary condition although the tapered wave is often introduced to realize asymptotic truncation for unbounded rough surface the standard helmholtz integral equations which derive for the scattering of plane waves by an arbitrary bounded obstacle are often used to generate benchmark numerical solutions different from the scattering of plane waves by an arbitrary bounded obstacle we use the angular spectrum representation radiation condition to replace the sommerfeld radiation condition and derive a boundary integral equation for studying the scattering problem then we study the integral equation by the truncation method whereby the integral equation posed on an unbounded region is approximated by an integral equation on a bounded region some properties of the integral equation in an energy space with weights are proved then the collocation method is used to solve the integral equation on a bounded region and its convergence is also obtained a rigorous derivation of bie is presented for studying the scattering problem some properties of the integral equation in an energy space with weights are proved the convergence of the numerical method is also obtained

type fuzzy quantifiers determined by fuzzy measures on residuated lattices part i basic definitions and examples abstract we study fuzzy quantifiers of type 1 1 defined using fuzzy measures and integrals residuated lattices are considered as structures of truth values we present basic notions on fuzzy measures over algebras of fuzzy subsets of a given fuzzy set and the definition and necessary properties of so called fuzzy integrals residuated lattice operations are established to derive operations between fuzzy sets describing degrees in which formulas expressing relations between fuzzy sets are true finally a general definition of type 1 1 fuzzy quantifiers determined by fuzzy measures and integrals is introduced and several examples of important natural language quantifiers are modeled using this approach

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