Mass,momentum and energy conservation laws in zero-pressure gas dynamics and δ-shocks: II

We study δ-shocks in a one-dimensional system of zero-pressure gas dynamics. In contrast to well-known papers (see References) this system is considered in the form of mass, momentum and energy conservation laws. In order to define such singular solutions, special integral identities are introduced which extend the concept of classical weak solutions. Using these integral identities, the Rankine–Hugoniot conditions for δ-shocks are obtained. It is proved that the mass, momentum and energy transport processes between the area outside the of one-dimensional δ-shock wave front and this front are going on such that the total mass, momentum and energy are independent of time, while the mass and energy concentration processes onto the moving δ-shock wave front are going on. At the same time the total kinetic energy transforms into total internal energy.     

Internal gravity–capillary solitary waves in finite depth

We consider a two‐dimensional inviscid irrotational flow in a two layer fluid under the effects of gravity and interfacial tension. The upper fluid is bounded above by a rigid lid, and the lower fluid is bounded below by a rigid bottom. We use a spatial dynamics approach and formulate the steady Euler equations as a Hamiltonian system, where we consider the unbounded horizontal coordinate x as a time‐like coordinate. The linearization of the Hamiltonian system is studied, and bifurcation curves in the (β,α)‐plane are obtained, where α and β are two parameters. The curves depend on two additional parameters ρ and h, where ρ is the ratio of the densities and h is the ratio of the fluid depths. However, the bifurcation diagram is found to be qualitatively the same as for surface waves. In particular, we find that a Hamiltonian‐Hopf bifurcation, Hamiltonian real 1:1 resonance, and a Hamiltonian 0²‐resonance occur for certain values of (β,α). Of particular interest are solitary wave solutions of the Euler equations. Such solutions correspond to homoclinic solutions of the Hamiltonian system. We investigate the parameter regimes where the Hamiltonian‐Hopf bifurcation and the Hamiltonian real 1:1 resonance occur. In both these cases, we perform a center manifold reduction of the Hamiltonian system and show that homoclinic solutions of the reduced system exist. In contrast to the case of surface waves, we find parameter values ρ and h for which the leading order nonlinear term in the reduced system vanishes. We make a detailed analysis of this phenomenon in the case of the real 1:1 resonance. We also briefly consider the Hamiltonian 0²‐resonance and recover the results found by Kirrmann. Copyright © 2016 John Wiley & Sons, Ltd.     

Gauge Modules for the Lie Algebras of Vector Fields on Affine Varieties

Algebras and Representation Theory - For a smooth irreducible affine algebraic variety we study a class of gauge modules admitting compatible actions of both the algebra $mathcal {A}$ of functions...     

Students’ Reasoning About One-Object Stochastic Phenomena in an ICT-Environment

This paper focuses on the different ways in which students in lower secondary school (14–16 year olds) experience compound random events, presented to them in the form of combined junctions. A carefully designed ICT environment was developed enabling the students to interact with different representations of such structures. Data for the analysis was gathered from two interview sessions. The analysis of the interaction is based on constructivist principles on learning; i.e. we adopted a student-oriented perspective, taking into consideration the different ways students try to make sense of chance encounters. Our results show how some students give priority to geometrical and physical concerns, and we discuss how seeking causal explanations of random phenomena may have encouraged this. With respect to numerically oriented models a division strategy appears to stand out as the preferred one.     

Single-frequency, single-mode, plane-polarized ytterbium-doped fiber master oscillator power amplifier source with 264 W of output power

We present a single-frequency, single-mode, plane-polarized ytterbium-doped all-fiber master oscillator power amplifier source at 1060 nm generating 264 W of continuous-wave output power. The final-stage amplifier operated with a high gain of 19 dB and a high conversion efficiency of 68%. There was no evidence of rollover from stimulated Brillouin scattering even at the highest output power, and the maximum output was limited only by the available pump power.     

Amplification reduction in fiber amplifiers due to up-conversion at Er3 concentrations below 1000 ppm

The parasitic effect of two-ion up-conversion in Er³⁺ -doped fibers have been investigated. Maximum amplification for eight different Er³⁺ concentrations have been measured and it is shown that up-conversion drastically reduces amplification for Er³⁺ concentrations in the range of 140-1680 ppm by weight. We have also measured an up-conversion induced luminescence at 980 nm. By comparison of experimental and simulated characteristics, we obtain an up-conversion coefficient of 10^-22 m³/s.     

2,3,5,6-Tetramethylmorpholine: II—1H n.m.r. structural studies of the isomers of 2,3,5,6-tetramethylmorpholine

The structures of the six isomers of 2,3,5,6-tetramethyl-morpholine have been determined by means of proton nuclear magnetic resonance studies at 100 MHz. The spectra have been analysed in terms of spin–spin coupling constants and population distributions of the possible conformers at low temperatures. Also included in this paper are results from n.m.r. studies on the six 4-benzyl-2,3,5,6-tetramethylmorpholine derivatives, which give information about the stereochemistry of the methyl groups α to the amine group.     

Barriers to internal rotation in 1,3,5-trineopentylbenzenes—II: Unsymmetrically halogen-substituted compounds

The temperature dependence of the three different methylene AB spectra due to the 1-, 3- and 5-neopentyl groups in 2-chloro-4-iodo, 2-bromo-4-iodo and 2-bromo-4-chloro-1,3,5-trineopentyl-benzene was studied by proton magenetic resonance measurements at 60 MHz. The identification of the lines of the AB quartets at low temperature was carried out on an HA-100 spectrometer by means of the INDOR technique. The activation parameters for all three barriers in each compound were found to be approximately the same, which perhaps reflects the interdependence of the rotations, with the smaller of the two halogens determining the size of the barrier. An explanation in terms of magnetic nonequivalence induced in the 1- and 5-methylenes by the 3-neopentyl group is considered to be most plausible.     

An yttrium Mξ source for ESCA

A new source for producing YMξ radiation in an ESCA spectrometer is described. The system makes use of continuous evaporation of yttrium on a rotating anode. It is demonstrated that this scheme allows YMξ-excited electron spectra to be recorded without gradual loss of resolution and intensity due to anode oxidation. Electron spectra of Ar, Hg and N₂ excited by YMξ radiation are studied. Relative photoelectric cross-sections of the 5d_{$\text{5}\text{2}$}, 5d_{$\text{5}\text{2}$} and 6s orbitals in Hg and the four valence orbitals in N₂ are measured. The valence electron shake-up spectrum of N₂ is discussed.