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.     

Die Bestimmung von Methylpropan

Ohne Zusammenfassung     

Synthesis of substituted 5-aminomethyl tetrahydro-isoquinolines and dihydro-isoindoles

The synthesis of ten substituted aminomethylene tetrahydro-isoquinolines is described, proceeding in eight steps from 5-hydroxyisoquinoline via reductive amination of N-Boc tetrahydro-isoquinoline 5-carboxaldehyde. Likewise, reductive amination was used to prepare four substituted dihydro-isoindoles from the corresponding aldehyde. The dihydro-isoindole ring system was conveniently accessed via a 2+2+2 cycloaddition reaction.     

Comparison between biokinetics of inhaled plutonium nitrate and gadolinium oxide in humans and animals

Due to the paucity of human data after inhalation of different chemical forms of radionuclides, the implications for human exposure are often based on animal studies. This paper describes biokinetic studies of plutonium nitrate and gadolinium oxide in human volunteers and rats. The results, together with information from other studies with radionuclides, suggests that animal studies can be used with advantage for assessing the biokinetic behavior in humans, and for providing guidance on the assessment of intake and optimal monitoring regimens.     

A simple derivation of Vonnegut's equation for the determination of interfacial tension by the spinning drop technique

A simple derivation of Vonnegut's equation for use in the determination of interfacial tension by the spinning drop technique is described. The derivation involves a cylindrical approximation and calculates the kinetic energy of rotation of the system from a consideration of its moment of inertia.     

The determination of lead and nickel by atomic-absorption spectrometry with a flameless wire loop atomizer

In an extension of studies of flameless atomizers for atomic-absorption spectrometry, an electrically heated tungsten-rhenium alloy wire loop was examined. Reduction of metallic salts to ground-state metal atoms was accomplished with the high temperature produced by the loop. Lead and nickel were investigated. Experimental parameters such as wavelength, slit width, atomization temperature and sheathing gas flow rate were optimized. Absolute detection limits of 6.6·10^?10 and 1.2·10^?10, and absolute sensitivities of 7·10^?10 and 8·10^?11 g of lead, were established for unenclosed and enclosed cells, respectively. The interferences of twenty cations and sixteen anions were studied; foreign cations generally enhanced the lead absorption by retarding its vaporization, allowing the slow detection system to respond more efficiently. Nickel was investigated as a representative less volatile metal; an absolute detection limit of 1.6·10^?9 and an absolute sensitivity of 9·10^?10 g of nickel were established.     

Analysis of urinary nucleosides. I. Optimisation of high performance liquid chromatography/electrospray mass spectrometry

In order to optimise the analysis of urinary nucleosides by high performance liquid chromatography/mass spectrometry (HPLC/MS), the HPLC separation of these compounds was performed at different 'flow rates' and 0.2mL/min was found to give both a better separation and ionisation. The ionisation conditions were optimised to give the best intensity of the molecules quasi-molecular ions. The ion distribution profile and ionisation in both positive and negative mode were examined and the detection of the protonated molecule in positive mode chosen for further analysis. The limits of detection of the method developed are reported and representative LC/MS and LC/MS/MS spectra shown. Typical urinary nucleoside chromatograms are presented.     

Point-splitting regularization for gauge theories: Quantum electrodynamics

As a method of regularization, point splitting has played an essential role in the recent theoretical determination of the masses of the Higgs boson and the top quark. It is the purpose of this paper to put this pointsplitting regularization on a firm basis. The result turns out to be extremely simple: replace the usual vertex factor-ieγ ^µ in quantum electrodynamics by $$ - ie(\gamma _\mu - \frac{{\not p}}{{p \cdot \delta }}\delta _\mu ),$$ wherep is the momentum of the photon line, andδ ^µ is the distance for point splitting. No additional vertices are needed.     

An NMR study of methanol diffusion in polymer electrolyte fuel cell membranes

Methanol diffusion in two polymer electrolyte membranes, Nafion 117 and BPSH 40 (a 40% disulfonated wholly aromatic polyarylene ether sulfone), was measured using a modified pulsed field gradient NMR method. This method allowed for the diffusion coefficient of methanol within the membrane to be determined while immersed in a methanol solution of known concentration. A second set of gradient pulses suppressed the signal from the solvent in solution, thus allowing the methanol within the membrane to be monitored unambiguously. Over a methanol concentration range of 0.5–8 M, methanol diffusion coefficients in Nafion 117 were found to increase from 2.9 × 10⁻⁶ to 4.0 × 10⁻⁶ cm² s⁻¹. For BPSH 40, the diffusion coefficient dropped significantly over the same concentration range, from 7.7 × 10⁻⁶ to 2.5 × 10⁻⁶cm² s⁻¹. The difference in diffusion behavior is largely related to the amount of solvent sorbed by the membranes. Increasing the methanol concentration results in an increase in solvent uptake for Nafion 117, while BPSH 40 actually excludes the solvent at higher concentrations. In contrast, diffusion of methanol measured via permeability measurements (assuming a partition coefficient of 1) was lower (1.3 × 10⁻⁶ and 6.4 × 10⁻⁷ cm² s⁻¹ for Nafion 117 and BPSH 40 respectively) and showed no concentration dependence. The differences observed between the two techniques are related to the length scale over which diffusion is monitored and the partition coefficient, or solubility, of methanol in the membranes as a function of concentration. For the permeability measurements, this length is equal to the thickness of the membrane (178 and 132 μm for Nafion 117 and BPSH 40 respectively) whereas the NMR method observes diffusion over a length of approximately 4–8 μm. Regardless of the measurement technique, BPSH 40 is a greater barrier to methanol permeability at high methanol concentrations.     

An automatic controlled-potential coulometer

The design and operation of an automatic controlled-potential coulometer is described. It was primarily designed for use with mg quantities of uranium, neptunium and plutonium but can in principle be used on any element possessing more than one valency state in solution. It is capable of an accuracy and precision of ±0.25% and can analyse up to 20 samples on a set programme without operator attention.