A hierarchy of Sturm–Liouville problems

Sturm–Liouville equations will be considered where the boundary conditions depend rationally on the eigenvalue parameter. Such problems apply to a variety of engineering situations, for example to the stability of rotating axles. Classesof these problems will be isolated with a rather rich spectral structure, for example oscillation, comparison and completeness properties analogous to thoseof the ‘usual’ Sturm–Liouville problem which has constant boundary conditions.In fact it will be shown how these classes can be converted into each other, andinto the ‘usual’ Sturm–Liouville problem, by means of transformations preserving all but finitely many eigenvalues. Copyright © 2003 John Wiley & Sons, Ltd.     

The linear polarization constant of R n

Summary He present work deals with estimations of the n-th linear polarization constant c(H)_n of an n-dimensional real Hilbert space H. We provide some new lower bounds on the value of sup_║y║=1 │1,y> ... n,y>│, where x₁, ... ,x_n are unit vectors in H. In particular, the results improve an earlier estimate of Marcus. However, the intriguing conjecture c(H) _n= n^n/2 remains open.     

The Interaction of Shocks with Dispersive Waves II. Incompressible-Integrable Limit

This is the second in a two-part series of articles in which we analyze a system similar in structure to the well-known Zakharov equations from weak plasma turbulence theory, but with a nonlinear conservation equation allowing finite time shock formation. In this article we analyze the incompressible limit in which the shock speed is large compared to the underlying group velocity of the dispersive wave (a situation typically encountered in applications). After presenting some exact solutions of the full system, a multiscale perturbation method is used to resolve several basic wave interactions. The analysis breaks down into two categories: the nonlinear limit and the linear limit, corresponding to the form of the equations when the group velocity to shock speed ratio, denoted by ε, is zero. The former case is an integrable limit in which the model reduces to the cubic nonlinear Schrödinger equation governing the dispersive wave envelope. We focus on the interaction of a “fast” shock wave and a single hump soliton. In the latter case, the ε=0 problem reduces to the linear Schrödinger equation, and the focus is on a fast shock interacting with a dispersive wave whose amplitude is cusped and exponentially decaying. To motivate the time scales and structure of the shock-dispersive wave interactions at lowest orders, we first analyze a simpler system of ordinary differential equations structurally similar to the original system. Then we return to the fully coupled partial differential equations and develop a multiscale asymptotic method to derive the effective leading-order shock equations and the leading-order modulation equations governing the phase and amplitude of the dispersive wave envelope. The leading-order interaction equations admit a fairly complete analysis based on characteristic methods. Conditions are derived in which: (a) the shock passes through the soliton, (b) the shock is completely blocked by the soliton, or (c) the shock reverses direction. In the linear limit, a phenomenon is described in which the dispersive wave induces the formation of a second, transient shock front in the rapidly moving hyperbolic wave. In all cases, we can characterize the long-time dynamics of the shock. The influence of the shock on the dispersive wave is manifested, to leading order, in the generalized frequency of the dispersive wave: the fast-time part of the frequency is the shock wave itself. Hence, the frequency undergoes a sudden jump across the shock layer.In the last section, a sequence of numerical experiments depicting some of the interesting interactions predicted by the analysis is performed on the leading-order shock equations.     

Analysis of semiconducting materials by high-resolution radiofrequency glow discharge mass spectrometry

The analytical capabilities of a high-resolution mass spectrometer in combination with a 13.56 MHz glow discharge ion source for the analysis of semiconducting materials (silicon carbide and gallium arsenide) were studied. It was shown that single positively charged ions of sample material have about 10 eV higher average energy than the ions of the discharge and residual gas. Therefore effective energy separation of the ions of analyte from the ions of the discharge and residual gas was achieved by adjusting the ion transfer optics (breadth and position of energy slit), which improves the analytical capabilities of the developed method.Some analytical applications are presented to illustrate the performance of r.f. GDMS for the bulk analysis of semiconducting materials. The results of the trace element analysis of gallium arsenide and silicon carbide samples are compared with data of independent methods (LIMS, ICP-AES, SIMS).Dedicated to Professor Dr. rer. nat. Dr. h.c. Hubertus Nickel on the occasion of his 65th birthdayOn leave from the Institute of Inorganic Chemistry, 630090 Novosibirsk, Russia     

Minimizing makespan on a single machine subject to random breakdowns

We investigate optimal sequencing policies for the expected makespan problem with an unreliable machine, where jobs have to be reprocessed in their entirety if preemptions occur because of breakdowns. We identify a class of uptime distributions under which LPT minimizes expected makespan.     

Diblock copolymers based on allyl methacrylate: Synthesis,characterization, and chemical modification

Different diblock copolymers constituted by one segment of a monomer supporting a reactive functional group, like allyl methacrylate (AMA), were synthesized by atom transfer radical polymerization (ATRP). Bromo‐terminated polymers, like polystyrene (PS), poly(methyl methacrylate) (PMMA), and poly(butyl acrylate) (PBA) were employed as macroinitiators to form the other blocks. Copolymerizations were carried out using copper chloride with N,N,N′,N″,N″‐pentamethyldiethylenetriamine (PMDETA) as the catalyst system in benzonitrile solution at 70 °C. At the early stage, the ATRP copolymerizations yielded well‐defined linear block copolymers. However, with the polymerization progress a change in the macromolecular architecture takes place due to the secondary reactions caused by the allylic groups, passing to a branched and/or star‐shaped structure until finally yielding gel at monomer conversion around 40% or higher. The block copolymers were characterized by means of size exclusion chromatography (SEC), ¹H NMR spectroscopy, and differential scanning calorimetry (DSC). In addition, one of these copolymers, specifically P(BA‐b‐AMA), was satisfactorily modified through osmylation reaction to obtain the subsequent amphiphilic diblock copolymer of P(BA‐b‐DHPMA), where DHPMA is 2,3‐dihydroxypropyl methacrylate; demonstrating the feasibility of side‐chain modification of the functional obtained copolymers. © 2007 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 45: 3538–3549, 2007     

Application of N,N‐bis(diphenylphosphino)aniline palladium(II) complexes as pre‐catalysts in Heck coupling reactions

Palladium(II) complexes with N,N‐bis(diphenylphosphino)aniline ligands catalyse the Heck reaction between styrene and aryl bromides, affording stilbenes in good yield. The structures of two of the complexes used as pre‐catalysts have been determined by single‐crystal X‐ray diffraction. Copyright © 2007 John Wiley & Sons, Ltd.