A new method of numerical evaluation of singular integrals occurring in two-dimensional BIEM

A new quadrature rule for integrands having logarithmical singularities has been developed. This rule proved to be efficient especially in the context of the BIEM.     

Low-Energy Collision-Induced Dissociation Mass Spectra of Protonated p-Toluenesulfonamides Derived from Aliphatic Amines

Collision-induced fragmentation of protonated N-alkyl-p-toluenesulfonamides primarily undergo either an elimination of the amine to form CH₃-(C₆H₄)-SO₂ ⁺ cation (m/z 155) or an alkene to form a cation for the protonated p-toluenesulfonamide (m/z 172). To comprehend the fragmentation pathways, several deuterated analogs of N-decyl-p-toluenesulfonamides were prepared and evaluated. Hypothetically, two mechanisms, both of which involve ion-neutral complexes, can be envisaged. In one mechanism, the S–N bond fragments to produce an intermediate [sulfonyl cation/amine] complex, which dissociates to afford the m/z 155 cation (Pathway A). In the other mechanism, the C–N bond dissociates to produce a different intermediate complex. The fragmentation of this [p-toluenesulfonamide/carbocation] complex eliminates p-toluenesulfonamide and releases the carbocation (Pathway B). Computations carried out by the Hartree-Fock method suggested that the Pathway B is more favorable. However, a peak for the carbocation is observed only when the carbocation formed is relatively stable. For example, the spectrum of N-phenylethyl-p-toluenesulfonamide is dominated by the peak at m/z 105 for the incipient phenylethyl cation, which rapidly isomerizes to the remarkably stable methylbenzyl cation. The peaks for the carbocations are weak or absent in the spectra of most of N-alkyl-p-toluenesulfonamides because alkyl carbocations, such as the decyl cation, rearrange to more stable secondary cations by 1,2-hydride and alkyl shifts. The energy freed is not dissipated, but gets internalized, causing the carbocation to dissociate either by transferring a proton to the sulfonamide or by releasing smaller alkenes to form smaller carbocations. The loss of the positional integrity in this way was proven by deuterium labeling experiments.

Self-assembled growth of nanostructural Ge islands on bromine-passivated Si(111) surfaces at room temperature

We have deposited relatively thick (∼60 nm) Ge layers on Br-passivated Si(111) substrates by thermal evaporation under high vacuum conditions at room temperature. Ge has grown in a layer-plus-island mode although it is different from the Stranski-Krastanov growth mode observed in epitaxial growth. Both the islands and the layer are nanocrystalline. This appears to be a consequence of reduction of surface free energy of the Si(111) substrate by Br-passivation. The size distribution of the Ge nanoislands has been determined. The Br-Si(111) substrates were prepared by a liquid treatment, which may not produce exactly reproducible surfaces. Nevertheless, some basic features of the nanostructural island growth are reasonably reproducible, while there are variations in the details of the island size distribution.     

A Petrov-Galerkin method with quadrature for elliptic boundary value problems

We propose and analyse a fully discrete Petrov–Galerkinmethod with quadrature, for solving second-order, variable coefficient,elliptic boundary value problems on rectangular domains. Inour scheme, the trial space consists of C² splines of degreer 3, the test space consists of C⁰ splines of degree r –2, and we use composite (r – 1)-point Gauss quadrature.We show existence and uniqueness of the approximate solutionand establish optimal order error bounds in H², H¹ and L² norms.     

Matrix decomposition algorithms for the finite element Galerkin method with piecewise Hermite cubics

Matrix decomposition algorithms (MDAs) employing fast Fourier transforms are developed for the solution of the systems of linear algebraic equations arising when the finite element Galerkin method with piecewise Hermite bicubics is used to solve Poisson’s equation on the unit square. Like their orthogonal spline collocation counterparts, these MDAs, which require O(N ²logN) operations on an N×N uniform partition, are based on knowledge of the solution of a generalized eigenvalue problem associated with the corresponding discretization of a two-point boundary value problem. The eigenvalues and eigenfunctions are determined for various choices of boundary conditions, and numerical results are presented to demonstrate the efficacy of the MDAs. Weiwei Sun was supported in part by a grant from City University of Hong Kong (Project No. CityU 7002110).     

A sinc quadrature rule for Hadamard finite-part integrals

Summary A Sinc quadrature rule is presented for the evaluation of Hadamard finite-part integrals of analytic functions. Integration over a general are in the complex plane is considered. Special treatment is given to integrals over the interval (–1,1). Theoretical error estimates are derived and numerical examples are included.     

Negative refraction

Light bends the wrong way in materials where both ε and μ are negative as was pointed out in 1968, but the absence of natural materials with this property led to neglect of the subject until 1999 when it was shown how to make artificial materials, metamaterials, with negative μ. The rapid advance of the subject since that date, both in theory and experiment, is reflected in the exponential growth of publications now at the 200 per year level and still growing. This interest is explained by the sudden availability of a qualitatively different class of electromagnetic materials combined with the quite startling properties which these materials appear to have; all of which provokes debate as each new facet of their behaviour is revealed. Experiment has been vital to resolution of controversy and has chiefly been in the microwave region of the spectrum though there is potential in the optical region currently being explored by several groups.     

Alternating direction implicit orthogonal spline collocation on some non-rectangular regions with inconsistent partitions

The alternating direction implicit (ADI) method is a highly efficient technique for solving multi-dimensional dependent initial-boundary value problems on rectangles. Earlier we have used the ADI technique in conjunction with orthogonal spline collocation (OSC) for discretization in space to solve parabolic problems on rectangles and rectangular polygons. Recently, we extended applications of ADI OSC schemes to the solution of parabolic problems on some non-rectangular regions that allow for consistent nonuniform partitions. However, for many regions, it is impossible to construct such partitions. Therefore, in this paper, we show how to extend our approach further to solve parabolic problems on some non-rectangular regions using inconsistent uniform partitions. Numerical results are presented using piecewise Hermite cubic polynomials for spatial discretizations and our ADI OSC scheme for parabolic problems to demonstrate its performance on several regions.