reflection of an acoustic line source by an impedance surface with uniform flow abstract an exact analytic solution is derived for the 2d acoustic pressure field generated by a time harmonic line mass source located above an impedance surface with uniform grazing flow closed form asymptotic solutions in the far field are also provided the analysis is valid for both locally reacting and nonlocally reacting impedances as is demonstrated by analyzing a nonlocally reacting effective impedance representing the presence of a thin boundary layer over the surface the analytic solution may be written in a form suggesting a generalization of the method of images to account for the impedance surface the line source is found to excite surface waves on the impedance surface some of which may be leaky waves which contradict the assumption of decay away from the surface predicted in previous analyses of surface waves with flow the surface waves may be treated either correctly as unstable waves or artificially as stable waves enabling comparison with previous numerical or mathematical studies which make either of these assumptions the computer code for evaluating the analytic solution and far field asymptotics is provided in the supplementary material it is hoped this work will provide a useful benchmark solution for validating 2d numerical acoustic codes

relatively weakly open subsets of the unit ball in functions spaces for an infinite hausdorff compact set k and for any banach space x we show that every nonempty weak open subset relative to the unit ball of the space of x valued functions that are continuous when x is equipped with the weak respectively norm weak topology has diameter 2 as consequence we improve known results about nonexistence of denting points in these spaces also we characterize when every nonempty weak open subset relative to the unit ball has diameter 2 for the spaces of bochner integrable and essentially bounded measurable x valued functions

on the extremal structure of the unit balls of banach spaces of weakly continuous functions and their duals a sufficient and then a necessary condition are given for a function to be an extreme point of the unit ball of the banach space c k x w of continuous functions under the supremum norm from a compact hausdorff topological space k into a banach space x equipped with its weak topology w strongly extreme points of the unit ball of c k x w are characterized as the norm one functions that are uniformly strongly extreme point valued on a dense subset of k it is shown that a variety of stronger types of extreme points e g denting points never exist in the unit ball of c k x w lastly some naturally arising and previously known extreme points of the unit ball of c k x w are shown to actually be strongly exposed points

ground effect over hard rough surfaces laboratory and outdoor measurements are reported of the relative sound pressure level spectrum over hard surfaces containing either random or periodically spaced arrays of 2 d roughnesses the resulting data have been compared with predictions obtained analytically and with numerical predictions of a boundary element code effective impedances of the rough surfaces have been calculated from the boss theory developed by twersky a classical asymptotic approximation for propagation near grazing incidence from a point source over an impedance boundary has been modified heuristically to allow for diffraction grating effects the resulting predictions are found to be in tolerable agreement with the data except for close and random packing the boundary element code is found to give superior results for larger roughnesses but computational restrictions on element size reduce its usefulness for roughnesses with small width or height

octahedral norms in spaces of operators abstract we study octahedral norms in the space of bounded linear operators between banach spaces in fact we prove that l x y has octahedral norm whenever x and y have octahedral norm as a consequence the space of operators l l 1 x has octahedral norm if and only if x has octahedral norm these results also allow us to get the stability of strong diameter 2 property for projective tensor products of banach spaces which is an improvement of the known results about the size of nonempty relatively weakly open subsets in the unit ball of the projective tensor product of banach spaces

sound propagation above an impedance boundary an asymptotic solution of the scalar wave field due to a point source above a locally reacting plane surface is obtained by a modified saddle point method numerical calculation indicates that the relatively simple solution is more accurate than the thomasson asymptotic solution and the chien soroka solution furthermore it is shown that another solution simplified still more is also in good agreement with the exact solution

displacement rank and quasitriangular decomposition for r toeplitz matrices abstract an r toeplitz r t matrix can be regarded as a block toeplitz matrix of order mr from which an l shaped border of k r rows and columns is removed the concept of displacement rank is extended to r t matrices and it is shown that both an r t matrix and its inverse can be expressed as the sum of at most two products of r t matrices this result reduces to a standard lu decomposition in the case when an r t matrix becomes an arbitrary matrix an explicit expression for the inverse is given which contains three triangular matrices and one quasitriangular matrix furthermore the inverse of an r t matrix is completely determined by the elements in its first and last r rows and columns the results presented generalize known properties of toeplitz and block toeplitz matrices

a three critical points theorem revisited abstract in this paper the following result is proved let x be a reflexive real banach space i r an interval x r a sequentially weakly lower semicontinuous c 1 functional bounded on each bounded subset of x whose derivative admits a continuous inverse on x j x r a c 1 functional with compact derivative assume that lim x x j x for all i and that there exists r such that sup i inf x x x j x inf x x sup i x j x then there exist a non empty open set a i and a positive real number r with the following property for every a and every c 1 functional x r with compact derivative there exists 0 such that for each 0 the equation x j x x 0 has at least three solutions in x whose norms are less than r

a hybrid bie ffp scheme for predicting barrier efficiency outdoors to assess the acoustical performance of noise screens in the presence of an arbitrary sound speed profile a numerical scheme based on a combination of the boundary integral and fast field program methods is developed the green s function required in the boundary integral is evaluated by a fast field formulation the procedure is validated by comparing its predictions with other numerical results for simple cases and with measurements for indoor model experiments simulating downward refracting conditions it is predicted that the performance of a noise screen placed 50 m from the source is reduced considerably by moderate downwind conditions

relatively weakly open sets in closed balls of banach spaces and real jb triples of finite rank we prove that given a real jb triple x there exists a nonempty relatively weakly open subset of the closed unit ball of x with diameter less than 2 if and only if the banach space of x is isomorphic to a hilbert space moreover we give the structure of real jb triples whose banach spaces are isomorphic to hilbert spaces such real jb triples are also characterized in two different purely algebraic ways

big slices versus big relatively weakly open subsets in banach spaces abstract we study the unknown differences between the size of slices and relatively weakly open subsets of the unit ball in banach spaces we show that every banach space containing c 0 isomorphically can be equivalently renormed so that every slice of its unit ball has diameter 2 and still its unit ball contains nonempty relatively weakly open subsets with diameter arbitrarily small which answers an open question and stresses the differences between the size of slices and relatively weakly open subsets of the unit ball of banach spaces

extreme differences between weakly open subsets and convex combinations of slices in banach spaces abstract we show that every banach space containing isomorphic copies of c 0 can be equivalently renormed so that every nonempty relatively weakly open subset of its unit ball has diameter 2 and however its unit ball still contains convex combinations of slices with diameter arbitrarily small which improves in an optimal way the known results about the size of this kind of subsets in banach spaces

symmetric strong diameter two property in tensor products of banach spaces abstract we continue the investigation of the behavior of diameter two properties in tensor products of banach spaces our main result shows that the symmetric strong diameter two property is stable by taking projective tensor products we also prove a result for the symmetric strong diameter two property for the injective tensor product

relatively weakly open sets in closed balls of banach spaces and the centralizer we prove that if the centralizer of a banach space x is infinite dimensional then every nonempty relatively weakly open subset of the closed unit ball of x has diameter equal to 2 this result together with a suitable refinement also proven in the paper contains and improves in some cases previously known facts for c algebras jb triples spaces of vector valued continuous functions and spaces of operators

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

weakly open sets in the unit ball of the projective tensor product of banach spaces abstract a banach space is said to have the diameter two property if every non empty relatively weakly open subset of its unit ball has diameter two we prove that the projective tensor product of two banach spaces whose centralizer is infinite dimensional has the diameter two property the same statement also holds for x y if the centralizer of x is infinite dimensional and the unit sphere of y contains an element of numerical index one we provide examples of classical banach spaces satisfying the assumptions of the results if k is any infinite compact hausdorff topological space then c k y has the diameter two property for any nonzero banach space y we also provide a result on the diameter two property for the injective tensor product

sound propagation above an inhomogeneous impedance plane abstract boundary integral equation bie methods are described for the prediction of sound propagation in particular from a line source over a flat plane of inhomogeneous impedance approximate methods which satisfy reciprocity for the calculation of the wave field over a two impedance plane are proposed these approximations applied to propagation from a line source give results agreeing well with those of the bie method when they are applied to propagation from a point source agreement with experiment is shown

existence of three solutions for p q biharmonic systems abstract in this paper we establish the existence of at least three solutions to a navier boundary problem involving the p q biharmonic systems the technical approach is mainly based on a three critical points theorem of b ricceri

variational formulation using integral equations to solve sound scattering above an absorbing plane the boundary element method is used to model two dimensional acoustic radiation and scattering from a body of arbitrary shape above an infinite plane of flat surface and homogeneous impedance the particularity of the study is the use of an indirect integral representation of the solution given in terms of the jumps of pressure and its normal derivative through the boundaries a variational formulation is associated with the boundary indirect integral equations modeling our problem the major difficulty in the formulation is the infinite feature of the plane which is avoided by introducing an appropriate green s function numerical results of the attenuation of sound by noise barriers are presented they show good agreement with other results in the literature

prediction of the attenuation of road traffic noise with distance abstract a method is described of calculating the sound pressure level received from single representative light and heavy vehicles after propagation over terrain composed of various combinations of hard ground and grass land these results are then used to derive correction contours for the attenuation with distance of the leq level received from streams of each class of vehicle the results apply to conditions where the boundary between the soft and hard areas is parallel to the traffic stream and the soft ground is on the receiver side various positions of the boundary are considered a simple prediction method for leq is developed incorporating the distance corrections similar calculations are carried out for the attenuation with distance of the l10 noise index for a representative stream of vehicles based on these results simple prediction equations for the excess attenuation are proposed in terms of the receiver position and the proportion of soft ground below the propagation path

a mountain pass theorem 143 in other words the mountain separating 0 and e may even be assumed of zero altitude and the conclusion is still true moreover as we shall show if c a the critical point can be chosen with llx ll r it is interesting to ask whether this extension of the ambrosetti rabinowitz theorem remains true in the infinite dimensional case we prove here that if 1 is strengthened a little to the form 1 there exist real numbers u r r such that 0 a for every x e a x e x y a moreover if c a the critical point can be chosen with r i x0 i r roughly speaking in brief the mountain pass theorem continues to hold for a mountain of zero altitude provided it also has non zero thickness in addition if c a the pass itself occurs precisely on the mountain i e satisfying r llroll r there are two interesting and immediate corollaries the first one says that a c function which has iwo local minimum points also has a third critical point the second states that a u periodic c function with a local minimum e has a critical point x e ku k 0 1 2 the precise statements of the above results will be given in the next sec tions the proof depends upon a lemma of clark 3 formulated here in a version suitable to our purpose in section 3 some applications are presen ted for the forced pendulum equation 2

long distance sound propagation over an impedance discontinuity abstract a calculation method is presented for sound propagation over an impedance discontinuity which is an improvement over previous methods being particularly efficient for long distances a homogeneous still atmosphere and flat ground are assumed the starting point is to make an approximation in a two dimensional boundary integral equation formulation of the problem this leads to an expression of the solution as the solution for homogeneous ground plus an integral over part of the boundary this integral is recognized as a generalized fourier integral to which standard techniques of asymptotic analysis can be applied an asymptotic approximation is found which can replace the part of the integral which is most expensive to evaluate by numerical quadrature the remaining integral is evaluated by numerical quadrature monofrequency excess attenuation results for propagation from a point source above rigid ground to a receiver above absorbing ground are discussed when the source and receiver are low enough very orderly trends are observed the results are applied with air attenuation and a weighting to a notional jet engine noise source simple trends are noted

almost square banach spaces abstract we single out and study a natural class of banach spaces almost square banach spaces in an almost square space we can find given a finite set x 1 x 2 x n in the unit sphere a unit vector y such that x i y is almost one these spaces have duals that are octahedral and finite convex combinations of slices of the unit ball of an almost square space have diameter 2 we provide several examples and characterizations of almost square spaces we prove that non reflexive spaces which are m ideals in their biduals are almost square we show that every separable space containing a copy of c 0 can be renormed to be almost square a local and a weak version of almost square spaces are also studied

weakly open sets in the unit ball of some banach spaces and the centralizer abstract we show that every banach space x whose centralizer is infinite dimensional satisfies that every non empty weakly open set in b y has diameter 2 where y n s x n fold symmetric projective tensor product of x endowed with the symmetric projective norm for every natural number n we provide examples where the above conclusion holds that includes some spaces of operators and infinite dimensional c algebras we also prove that every non empty weak open set in the unit ball of the space of n homogeneous and integral polynomials on x has diameter two for every natural number n whenever the cunningham algebra of x is infinite dimensional here we consider the space of n homogeneous integral polynomials as the dual of the space n s e x n fold symmetric injective tensor product of x endowed with the symmetric injective norm for instance every infinite dimensional l 1 satisfies that its cunningham algebra is infinite dimensional we obtain the same result for every non reflexive l embedded space and so for every predual of an infinite dimensional von neumann algebra

a wavenumber independent boundary element method for an acoustic scattering problem in this paper we consider the impedance boundary value problem for the helmholtz equation in a half plane with piecewise constant boundary data a problem which models for example outdoor sound propagation over inhomogeneous flat terrain to achieve good approximation at high frequencies with a relatively low number of degrees of freedom we propose a novel galerkin boundary element method using a graded mesh with smaller elements adjacent to discontinuities in impedance and a special set of basis functions so that on each element the approximation space contains polynomials of degree nu multiplied by traces of plane waves on the boundary we prove stability and convergence and show that the error in computing the total acoustic field is cal o n nu 1 log 1 2 n where the number of degrees of freedom is proportional to n log n this error estimate is independent of the wavenumber and thus the number of degrees of freedom required to achieve a prescribed level of accuracy does not increase as the wavenumber tends to infinity

a novel boundary integral formulation for three dimensional analysis of thin acoustic barriers over an impedance plane this article presents a three dimensional formulation for the analysis of acoustic barriers over an impedance plane as infinitely thin structures the barriers are therefore modeled as simple surfaces rather than volumetric structures using this approach the problems caused by near singular integrations and near degenerate systems of equations are averted and mesh generation is made easier a dual boundary element method is used in the analysis involving the simultaneous solution of standard and hypersingular boundary integral equations an optimization procedure is used to speed up the assembling of the system of equations increasing the applicability of the method to a wider range of frequencies

ground effect analysis surface wave and layer potential representations abstract this paper presents two kinds of analytical exact expressions of the sound field reflected by a plane boundary as obtained by using either surface wave or layer potentials representations both solutions are first expressed as a sum of integrals which have a form suitable for numerical computation then these integrals are expanded into convergent series which provide analytical approximations of the solution numerical techniques are proposed for computing a the surface wave representation and b the approximation deduced from the layer potentials representation this approximation and the classical one sum of the surface wave and the first two terms of the asymptotic series are compared with the exact solution several examples show that the approximate formulas established here are valid on a range much wider than the validity domain of the classical ones

geometry of spaces of polynomials abstract connections between the shape of the unit ball of a banach space and analytic properties of the banach space have been studied for many years in this article some geometric properties of spaces related ton homogeneous polynomials are considered in particular the rotundity and smoothness of spaces of continuousn homogeneous polynomials and its preduals are studied furthermore an inequality relating the product of the norms of linear functionals on a banach space with the norm of the continuousn homogeneous polynomial determined by the product of the linear functionals is derived this inequality is used to study the strongly exposed points of the predual of the space of continuous 2 homogeneous polynomials

geometric theory of spaces of integral polynomials and symmetric tensor products abstract we investigate the nature of extreme weak exposed and weak strongly exposed points of the unit ball of spaces of n homogeneous integral polynomials on and n fold symmetric products of a banach space e for the space of integral polynomials we show the set of extreme points is contained in the set n e 1 we give smul yan type theorems for spaces of n homogeneous polynomials and n fold symmetric tensors that characterise weak exposed resp weak strongly exposed points in terms of g teaux resp frechet differentiability of the norm on various spaces of tensor products and polynomials our study of the geometry of these spaces has many applications when e has the radon nikod m property we show that the spaces of n homogeneous integral and nuclear polynomials are isomerically isomorphic for each integer n when the dimensions of e and n are both at least 2 then the space of n homogeneous polynomials on e is neither smooth nor rotund for a certain class of reflexive banach space the space of n homogeneous approximable polynomials on e is either reflexive or is not isometric to a dual banach space we conclude with a choquet theorem for a space of homogeneous polynomials

octahedral norms in duals and biduals of lipschitz free spaces abstract we continue with the study of octahedral norms in the context of spaces of lipschitz functions and in their duals first we prove that the norm of f m is octahedral as soon as m is unbounded or is not uniformly discrete further we prove that a concrete sequence of uniformly discrete and bounded metric spaces k m satisfies that the norm of f k m is octahedral for every m finally we prove that if x is an arbitrary banach space and the norm of lip 0 m is octahedral then the norm of lip 0 m x is octahedral these results solve several open problems from the literature

near field computation for sound propagation above ground using complex image theory in this paper a very simple method is introduced for the efficient computation of the sound field in the near region above an impedance ground or an extended reaction ground the procedure is based on complex image theory using prony s method the slowly varying factor in the integrand of the sommerfeld integral is approximated by a short series in terms of complex exponential functions the result can be exactly evaluated by the sommerfeld identity a new closed form approximate expression of the sound field is obtained which is represented as the original source its quasistatic image and several images with complex amplitudes and positions avoiding numerical evaluation of the sommerfeld integral makes these closed form expressions computationally very efficient the representation of the field with complex images makes the formula useful in measurement of ground impedance or in other cases requiring repeated mapping of the sound field numerical results show that these closed form green s functions are very accurate for the computation of the near field

multiple solutions for a two point boundary value problem abstract in this paper using a recent result of ricceri we prove two multiplicity theorems for the problem u f u g x u u 0 u 1 0 extending a previous result that g bonanno obtained for 0