On the Neumann problem for the Helmholtz equation in a plane angle

We prove that the solution of the Neumann problem for the Helmholtz equation in a plane angle Ω with boundary conditions from the space H_−1/2(Γ), where Γ is the boundary of Ω, which is provided by the well‐known Sommerfeld integral, belongs to the Sobolev space H₁(Ω) and depends continuously on the boundary values. To this end, we use another representation of the solution given by the inverse two‐dimensional Fourier transform of an analytic function depending on the Cauchy data of the solution. Copyright © 2000 John Wiley & Sons, Ltd.     

Phase behaviour of blends with mesophase components

Temperature transitions and structural transformations were studied for blends of two thermotropic mesophase cyclolinear polymethylsiloxanes with linear PDMS of various molecular weights by means of DSC, optical polarizing microscopy, and optical interferometry. Compatibility of the components which depends on the chemical structure of cyclolinear polymethylsiloxanes, molecular weight of PDMS, composition, and temperature affect formation of mesophase in cyclolinear polymethylsiloxanes. The most interesting aspect of the phase behaviour consists in the fact that it is possible to reach compatibility of the components in the mesomorphic state for the blends of two cyclolinear polymethylsiloxanens due to various annealing regimes in one‐phase molten state.     

Learning the integral concept by constructing knowledge about accumulation

We propose an approach to the integral concept for advanced high school students and provide evidence for the potential of this approach to support students in acquiring an in-depth proceptual view of the integral. The approach is based on the mathematical idea of accumulation. A ten-lesson unit has been implemented with four pairs of students. The students’ learning processes were micro-analysed using the methodological–theoretical framework of Abstraction in Context. In this paper, we focus on the lessons in which the notions of approximation and accumulation are introduced. The work of one student pair is analysed in detail, and the work of the other pairs is summarized. Our results show that most of the students reached a proceptual understanding of the integral that prepared them for the next step in the curriculum, namely the Fundamental Theorem of Calculus.     

Multiplicative arithmetic of finite quadratic forms over Dedekind rings

Letq(X) be a quadratic form in an even numberm of variables with coefficients in a Dedekind ringK. Let us assume that the setsR(q,a) = {N∈K ^m ;q(N) = a} of representations of elementsa ofK by the formq are finite. Then certain multiplicative relations are obtained by elementary means between the setsR(q,a) andR(q,ab), whereb is a product of prime elementsρ ofK with finite coefficientsK/ρK. The relations imply similar multiplicative relations between the numbers of elements of the setsR(q,a), which formerly could be obtained only in some special cases like the case whenK = ℤ is the ring of rational integers and only by means of the theory of Hecke operators on the spaces of theta-series. As an application, an almost elementary proof of the Siegel theorem on the mean number of representations of integers by integral positive quadratic forms of determinant 1 is given. Dedicated to the memory of Professor K G Ramanathan     

Mass spectra of some high molecular weight tetra(alkyl)silanes

The mass spectra of tetra(alkyl)silanes having one short and three long alkyl groups were recorded (C₂₀? C₃₂). These silanes did not give molecular ions, and the dominant course of fragmentation involved the loss of alkyl groups, followed by elimination of alkenes. The loss of alkyl groups, followed by elimination of alkenes. The loss of alkyl groups followed the sequence iso-Pr>Et, Pr,… C₁₄H₂₉» Me. With alkyl groups longer than hexyl, a rearrangement was observed in which n-alkanes were eliminated from the base ion. When the alkyl group was hexyl, the alkane eliminated was methane; when the alkyl group was heptyl, ethane was expelled; with octyl, it was propane; and with decyl, n-pentane was expelled. Deuterium labeling showed that at least two modes of rearrangement were occurring.     

Method of Best Successive Approximations for Nonlinear Operators

We present a new approach to the approximation of nonlinear operators in probability spaces. The approach is based on a combination of the specific iterative procedure and the best approximation problem solution with a quadratic approximant. We show that the combination of these new techniques allow us to build a computationally efficient and flexible method. The algorithm of the method and its application to the optimal filtering of stochastic signals are given.     

Surface patterning of silica nanostructures using bio-inspired templates and directed synthesis

Natural systems excel in directing the synthesis of inorganic materials for various functional purposes. One of the best-studied systems is silica synthesis, as occurs in diatoms and marine sponges. Various biological and synthetic polymers have been shown to template and catalyze silica formation from silicic acid precursors. Here, we describe the use of poly-L-lysine to promote the synthesis of silica in neutral, aqueous solution and when immobilized onto a silicon support structure under similar conditions. Either reagent jetting or conventional photolithography techniques can be used to pattern the templating polymer. Spots created by reagent jetting led to the creation of silica structures in the shape of a ring that may be a result of the spotting process. Photolithographically defined poly-L-lysine spots led to thin laminate structures after exposure to a dilute aqueous silicic acid solution. The laminate structures were nanostructured and highly interconnected. Photolithographic patterning of (3-aminopropyl)trimethoxysilane, a reagent that mimics the lysine functional group, led to similar silica coatings even though low-molecular-weight materials do not rapidly promote silica synthesis in solution. This result highlights the importance of functional-group arrangement for templating and promoting the synthesis of inorganic materials. The described surface-patterning techniques offer a route to integrate conventional silicon-patterning technologies with biologically based material synthesis. Such combined fabrication techniques enable controlled assembly over multiple length scales and an approach to understanding interfacial silica synthesis, as occurs in natural systems.     

Synthesis and molecular structure of solvated hydride complexes of aluminium and di-η5-cyclopentadienylyttrium

A series of hydride complexes of di-η⁵-cyclopentadienyl-yttrium and -aluminium solvated by a Lewis base, viz. Cp₂YAlH₄NEt₃ (I), Cp₂YAlH₄THF (II), Cp₂YAlH₄OEt₂ (III) and (Cp₂YAlH₄)₂OEt₂ (IV), (Cp = η⁵-cyclopentadienyl) was synthesized. An X-ray study of compounds I and IV was performed, yielding the coordinates of the heavy atoms. The positions of the hydrogen atoms were determined accordingly from the IR data and also by comparison with the analogous hydridohalide complexes (Cp₂YCl)₂AlH₃L. The structures of I and IV may be represented by the following formulae: [(η⁵-C₅H₅)₂Y(μ₃-H)AlH₂NEt₃]₂ (I) and {[(η⁵-C₅H₅)₂Y]₂(μ₃-H)}[(μ₂-H)₂AlH₂][(μ₂-H)₂AlHOEt₂] (IV).