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.

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 R2 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.

Scattering by rough surfaces: the Dirichlet problem for the Helmholtz equation in a non-locally perturbed half-plane

We consider the Dirichlet boundary value problem for the Helmholtz equation in a non-locally perturbed half-plane, this problem arising in electromagnetic scattering by one-dimensional rough, perfectly conducting surfaces. We propose a new boundary integral equation formulation for this problem, utilizing the Green's function for an impedance half-plane in place of the standard fundamental solution. We show, at least for surfaces not differing too much from the flat boundary, that the integral equation is uniquely solvable in the space of bounded and continuous functions, and hence that, for a variety of incident fields including an incident plane wave, the boundary value problem for the scattered field has a unique solution satisfying the limiting absorption principle. Finally, a result of continuous dependence of the solution on the boundary shape is obtained.

Uniqueness results for direct and inverse scattering by infinite surfaces in a lossy medium

We consider the Dirichlet boundary-value problem for the Helmholtz equation, Au + x2u = 0, with Imx > 0. in an hrbitrary bounded or unbounded open set C c W. Assuming continuity of the solution up to the boundary and a bound on growth a infinity, that lu(x)l < Cexp (Slxl), for some C > 0 and S~< Imx, we prove that the homogeneous problem has only the trivial salution. With this resnlt we prove uniqueness results for direct and inverse problems of scattering by a bounded or infinite obstacle.

Approximate solution of second kind integral equations on infinite cylindrical surfaces

The paper considers second kind integral equations of the form $\phi (x) = g(x) + \int_S {k(x,y)} \phi (y)ds(y)$ (abbreviated $\phi = g + K\phi $), in which S is an infinite cylindrical surface of arbitrary smooth cross section. The “truncated equation” (abbreviated $\phi _a = E_a g + K_a \phi _a $), obtained by replacing S by $S_a $, a closed bounded surface of class $C^2 $, 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 - K_a $ is invertible and $(I - K_a )^{ - 1} $ is uniformly bounded for all sufficiently large a, and that $\phi _a $ converges to $\phi $ in an appropriate sense as $a \to \infty $. 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.

A high frequency $hp$ boundary element method for scattering by convex polygons

In this paper we propose and analyze a hybrid $hp$ boundary element method for the solution of problems of high frequency acoustic scattering by sound-soft convex polygons, in which the approximation space is enriched with oscillatory basis functions which efficiently capture the high frequency asymptotics of the solution. We demonstrate, both theoretically and via numerical examples, exponential convergence with respect to the order of the polynomials, moreover providing rigorous error estimates for our approximations to the solution and to the far field pattern, in which the dependence on the frequency of all constants is explicit. Importantly, these estimates prove that, to achieve any desired accuracy in the computation of these quantities, it is sufficient to increase the number of degrees of freedom in proportion to the logarithm of the frequency as the frequency increases, in contrast to the at least linear growth required by conventional methods.

X‑ray crystal structure of rac-[Ru(phen)2dppz]2+ with d(ATGCAT)2 shows enantiomer orientations and water ordering

We report an atomic resolution X-ray crystal structure containing both enantiomers of rac-[Ru(phen)2dppz]2+ with the d-(ATGCAT)2 DNA duplex (phen = phenanthroline; dppz = dipyridophenazine). The first example of any enantiomeric pair crystallized with a DNA duplex shows different orientations of the Λ and Δ binding sites, separated by a clearly defined structured water monolayer. Job plots show that the same species is present in solution. Each enantiomer is bound at a TG/CA step and shows intercalation from the minor groove. One water molecule is directly located on one phenazine N atom in the Δ-enantiomer only.

Self-assembly of three bacterially-derived bioactive lipopeptides

The self-assembly in aqueous solution of three lipopeptides obtained from Bacillus subtilis has been investigated. The lipopeptides surfactin, plipastatin and mycosubtilin contain distinct cyclic peptide headgroups as well as differences in alkyl chain length, branching and chain length distribution. Cryogenic transmission electron microscopy and X-ray scattering reveal that surfactin and plipastatin aggregate into 2 nm-radius spherical micelles, whereas in complete contrast mycosubtilin self-assembles into extended nanotapes based on bilayer ordering of the lipopeptides. Circular dichroism and FTIR spectroscopy indicate the presence of turn structures in the cyclic peptide headgroup. The unexpected distinct mode of self-assembly of mycosubtilin compared to the other two lipopeptides is ascribed to differences in the surfactant packing parameter. This in turn is due to specific features of the conformation of the peptide headgroup and alkyl chain branching.

Influence of elastase on alanine-rich peptide hydrogels

The self-assembly of the alanine-rich amphiphilic peptides Lys(Ala)6Lys (KA6K) and Lys(Ala)6Glu (KA6E)with homotelechelic or heterotelechelic charged termini respectively has been investigated in aqueous solution. These peptides contain hexa-alanine sequences designed to serve as substrates for the enzyme elastase. Electrostatic repulsion of the lysine termini in KA6K prevents self-assembly, whereas in contrast KA6E is observed, through electron microscopy, to form tape-like fibrils, which based on X-ray scattering contain layers of thickness equal to the molecular length. The alanine residues enable efficient packing of the side-chains in a beta-sheet structure, as revealed by circular dichroism, FTIR and X-ray diffraction experiments. In buffer, KA6E is able to form hydrogels at sufficiently high concentration. These were used as substrates for elastase, and enzyme-induced de-gelation was observed due to the disruption of the beta-sheet fibrillar network. We propose that hydrogels of the simple designed amphiphilic peptide KA6E may serve as model substrates for elastase and this could ultimately lead to applications in biomedicine and regenerative medicine.

Experimental confirmation of transformation pathways between inverse double diamond and gyroid cubic phases

A macroscopically oriented double diamond inverse bicontinuous cubic phase (QIID) of the lipid glycerol monooleate is reversibly converted into a gyroid phase (QIIG). The initial QIID phase is prepared in the form of a film coating the inside of a capillary, deposited under flow, which produces a sample uniaxially oriented with a ⟨110⟩ axis parallel to the symmetry axis of the sample. A transformation is induced by replacing the water within the capillary tube with a solution of poly(ethylene glycol), which draws water out of the QIID sample by osmotic stress. This converts the QIID phase into a QIIG phase with two coexisting orientations, with the ⟨100⟩ and ⟨111⟩ axes parallel to the symmetry axis, as demonstrated by small-angle X-ray scattering. The process can then be reversed, to recover the initial orientation of QIID phase. The epitaxial relation between the two oriented mesophases is consistent with topologypreserving geometric pathways that have previously been hypothesized for the transformation. Furthermore, this has implications for the production of macroscopically oriented QIIG phases, in particular with applications as nanomaterial templates.

Electron doping and phonon scattering in Ti1+xS2 thermoelectric compounds

Bulk polycrystalline samples in the series Ti1+xS2 (x = 0 to 0.05) were prepared using high temperature synthesis from the elements and spark plasma sintering. X-ray structure analysis shows that the lattice constant c expands as titanium intercalates between TiS2 slabs. For x=0, a Seebeck coefficient close to -300 μV/K is observed for the first time in TiS2 compounds. The decrease in electrical resistivity and Seebeck coefficient that occurs upon Ti intercalation (Ti off stoichiometry) supports the view that charge carrier transfer to the Ti 3d band takes place and the carrier concentration increases. At the same time, the thermal conductivity is reduced by phonon scattering due to structural disorder induced by Ti intercalation. Optimum ZT values of 0.14 and 0.48 at 300K and 700K, respectively, are obtained for x=0.025.

A high frequency boundary element method for scattering by a class of nonconvex obstacles

In this paper we propose and analyse a hybrid numerical-asymptotic boundary element method for the solution of problems of high frequency acoustic scattering by a class of sound-soft nonconvex polygons. The approximation space is enriched with carefully chosen oscillatory basis functions; these are selected via a study of the high frequency asymptotic behaviour of the solution. We demonstrate via a rigorous error analysis, supported by numerical examples, that to achieve any desired accuracy it is sufficient for the number of degrees of freedom to grow only in proportion to the logarithm of the frequency as the frequency increases, in contrast to the at least linear growth required by conventional methods. This appears to be the first such numerical analysis result for any problem of scattering by a nonconvex obstacle. Our analysis is based on new frequency-explicit bounds on the normal derivative of the solution on the boundary and on its analytic continuation into the complex plane.

A frequency-independent boundary element method for scattering by two-dimensional screens and apertures

We propose and analyse a hybrid numerical–asymptotic hp boundary element method (BEM) for time-harmonic scattering of an incident plane wave by an arbitrary collinear array of sound-soft two-dimensional screens. Our method uses an approximation space enriched with oscillatory basis functions, chosen to capture the high-frequency asymptotics of the solution. We provide a rigorous frequency-explicit error analysis which proves that the method converges exponentially as the number of degrees of freedom N increases, and that to achieve any desired accuracy it is sufficient to increase N in proportion to the square of the logarithm of the frequency as the frequency increases (standard BEMs require N to increase at least linearly with frequency to retain accuracy). Our numerical results suggest that fixed accuracy can in fact be achieved at arbitrarily high frequencies with a frequency-independent computational cost, when the oscillatory integrals required for implementation are computed using Filon quadrature. We also show how our method can be applied to the complementary ‘breakwater’ problem of propagation through an aperture in an infinite sound-hard screen.

Acoustic scattering : high frequency boundary element methods and unified transform methods

We describe some recent advances in the numerical solution of acoustic scattering problems. A major focus of the paper is the efficient solution of high frequency scattering problems via hybrid numerical-asymptotic boundary element methods. We also make connections to the unified transform method due to A. S. Fokas and co-authors, analysing particular instances of this method, proposed by J. A. De-Santo and co-authors, for problems of acoustic scattering by diffraction gratings.

Synthesis and structure of cage-like mesoporous silica using different precursors

In this work the synthesis of cubic, FDU-1 type, ordered mesoporous silica (OMS) was developed from two types of silicon source, tetraethyl orthosilicate (TEOS) and a less expensive compound, sodium silicate (Na(2)Si(3)O(7)), in the presence of a new triblock copolymer template Vorasurf 504 (EO(38)BO(46)EO(38)). For both silicon precursors the synthesis temperature was evaluated. For TEOS the effect of polymer dissolution in methanol and the acid solution (HCl and HBr) on the material structure was analyzed. For Na(2)Si(3)O(7) the influence of the polymer mass and the hydrothermal treatment time were the explored experimental parameters. The samples were examined by Small Angle X-ray Scattering (SAXS) and Nitrogen Sorption. For both precursors the decrease on the synthesis temperature from ambient, -25 degrees C, to -15 degrees C improved the ordered porous structure. For TEOS, the SAXS results showed that there is an optimum amount of hydrophobic methanol that contributed to dissolve the polymer but did not provoke structural disorder. The less electronegative Br-ions, when compared to Cl-, induced a more ordered porous structure, higher surface areas and larger lattice parameters. For Na(2)Si(3)O(7) the increase on the hydrothermal treatment time as well as the use of an optimized amount of polymer promoted a better ordered porous structure. (C) 2011 Elsevier B.V. All rights reserved.