Differential equations uniquely determined by algebras of point symmetries

We continue to investigate strongly and weakly Lie remarkable equations, which we defined in a recent paper. We consider some relevant algebras of vector fields on ℝ^k (such as the isometric, affine, projective, or conformal algebras) and characterize strongly Lie remarkable equations admitted by the considered Lie algebras. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 3, pp. 486–494, June, 2007.     

Runge–Kutta Methods for Itô Stochastic Differential Equations with Scalar Noise

A general class of stochastic Runge–Kutta methods for Itô stochastic differential equation systems w.r.t. a one-dimensional Wiener process is introduced. The colored rooted tree analysis is applied to derive conditions for the coefficients of the stochastic Runge–Kutta method assuring convergence in the weak sense with a prescribed order. Some coefficients for new stochastic Runge–Kutta schemes of order two are calculated explicitly and a simulation study reveals their good performance.     

One-pot diastereoselective synthesis of α-aminonitriles from aldehydes, chiral amines, and trimethylsilyl cyanide under solvent-free conditions

Treatment of aliphatic and aromatic aldehydes with chiral amines and trimethylsilyl cyanide in the absence or in the presence of Lewis acids (including lithium perchlorate) under solvent-free conditions afforded the corresponding α-aminonitriles in good yields and with a diastereoselectivity of 68 to 86%. Published in Russian in Zhurnal Organicheskoi Khimii, 2006, Vol. 42, No. 1, pp. 56–59. The text was submitted by the authors in English.     

Hydrogen-induced metallization on Ge(1 1 1) c(2 × 8)

We have studied hydrogen adsorption on the Ge(1 1 1) c(2 × 8) surface using scanning tunneling microscopy (STM) and angle-resolved photoelectron spectroscopy (ARPES). We find that atomic hydrogen preferentially adsorbs on rest atom sites. The neighbouring adatoms appear higher in STM images, which clearly indicates a charge transfer from the rest atom states to the adatom states. The surface states near the Fermi-level have been followed by ARPES as function of H exposure. Initially, there is strong emission from the rest atom states but no emission at the Fermi-level which confirms the semiconducting character of the c(2 × 8) surface. With increasing H exposure a structure develops in the close vicinity of the Fermi-level. The energy position clearly indicates a metallic character of the H-adsorbed surface. Since the only change in the STM images is the increased brightness of the adatoms neighbouring a H-terminated rest atom, we identify the emission at the Fermi-level with these adatom states.     

Linearisation and potential symmetries of certain systems of diffusion equations

We consider systems of two pure one-dimensional diffusion equations that have considerable interest in Soil Science and Mathematical Biology. We construct non-local symmetries for these systems. These are determined by expressing the equations in a partially and wholly conserved form, and then by performing a potential symmetry analysis on those systems that can be linearised. We give several examples of such systems, and in a specific case we show how linearising and hodograph-type mappings can lead to new solutions of the diffusion system.     

Raman spectroscopy of acoustic phonons in periodic and Fibonacci superlattices

Results of Raman scattering experiments on (a) periodic superlattices made up of GaAs/In_xGa_1−xAs layers with high indium concentrations, (b) GaAs/Ga_1−xAl_xAs Fibonacci superlattices, are presented. We discuss the observed peak positions and intensities using the continuum theory of acoustic wave propagation in layered media and the photo-elastic coupling model.