Formation of coherent spatial structures in the reaction-diffusion atmospheric system under the action of a planetary wave

On the example of the effect of an atmospheric wave with given amplitude on the density oscillations of minor gas constituents near the mesopause (at heights 80–90 km, we consider the influence of the quasi-resonant periodic spatial transport on a distributed system responding subharmonically to a periodic external action (diurnal sunlight variations). Numerical modeling revealed the formation of a coherent horizontal distribution of the dynamical variables (i.e., the concentrations of small constituents). Application of this effect to the mesopause conditions is discussed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 47, No. 9, pp. 739–752, September, 2004.     

Adaptive regulation for deterministic systems

For the linear deterministic system with unknown orders and coefficients adaptive controls are given so that the closed-loop system is stabilized and the unknown parameters are consistently estimated. Moreover, if the parameter estimation is ignored, then the system input and output can be reduced to zero with an exponential rate.This project is supported by the National Natural Science Foundation of China.In Commemoration of the 15th Anniversary of the Acta Mathematicae Applicatae Sinica.     

Attractors of the sine-Gordon equation in the field of a quasiperiodic external force

The well-known sine-Gordon equation, supplemented with small damping and small quasiperiodic external force, is studied under the zero Dirichlet boundary conditions at the endpoints of a finite interval. The main assumption is that all frequencies of the external force are in 1:1 resonance with certain eigenfrequencies of the unperturbed equation; i.e., the socalled fundamental multifrequency resonance is observed. It is shown that in this case, by an appropriate choice of the parameters of the external force, one can make it so that the boundary value problem has a stable invariant torus of any finite dimension that bifurcates from zero on any preassigned finite set of spatial modes. It is also shown (by numerical analysis) that in a number of cases the above-mentioned torus coexists with a chaotic attractor.     

Fully-angular polyhex chains with minimal total π-electron energy

By Hückel molecular orbital (HMO) theory, the calculation of the total energy of all π-electrons in conjugated hydrocarbons can be reduced to that of E(G)=|λ₁|+|λ₂|+?+|λn|, where λi are the eigenvalues of the corresponding graph G. Denote by Ψn the set of all fully-angular polyhex chains with n hexagons. In this paper, we show that Hn has the minimum total π-electron energy among chains in Ψn, where Hn is the helicene chain.     

Comparison of the methods for computing interference elastic wave fields in thin-layered media. 1

A nonstationary problem concerning the propagation of low-frequency waves for a model that consists of a packet of elastic layers lying on an elastic half-space is considered, and exact mathematical solutions in the form of repeated integrals are constructed in three basic forms. It is proved that these solutions satisfy initial conditions. The representation of solutions as a superposition of multiple waves is treated. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 220–239.     

Sharp condition of global existence for second-order derivative nonlinear Schrödinger equations in two space dimensions

This paper discusses a class of second-order derivative nonlinear Schrödinger equations which are used to describe the upper-hybrid oscillation propagation. By establishing a variational problem, applying the potential well argument and the concavity method, we prove that there exists a sharp condition for global existence and blow-up of the solutions to the nonlinear Schrödinger equation. In addition, we also answer the question: how small are the initial data, the global solutions exist?     

A Liouville type theorem for polyharmonic elliptic systems

In this paper, we consider the polyharmonic system m(−Δ)U=Vq,m(−Δ)V=Up in RN, for m>1, N>2m, with p?1, q?1, but not both equal to 1, where m(−Δ) is the polyharmonic operator. Set α=2m(q+1)/(pq−1), β=2m(p+1)/(pq−1), for α,β∈[(N−2)/2,N−2m), we prove the nonexistence of positive solutions.     

Topological K-groups of two-dimensional local fields

A complete two-dimensional local field K of mixed characteristic with finite second residue field is considered. The existence of a completely ramified extension L of K such that L is a standard field is assumed. It is proved that the rank of the quotient U(1)K ₂ ^top K/T_K, where T_K is the closure of the torsion subgroup, is equal to the degree of the constant subfield of K over ℚ_p. I. B. Zhukov constructed a set of generators of this quotient in the case where K is a standard field. In this paper, two natural generalizations of this set are considered, and it is proved that one of them generates the entire group and the other generates its subgroup of finite index. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 343, 2007, pp. 206–221.     

Fabrication of bamboo-shaped GaN nanorods

Bamboo-shaped GaN nanorods were formed through a simple sublimation method. They were characterized by means of X-ray powder diffraction (XRD), transmission electron microscopy (TEM), high-resolution transmission electron microscopy (HRTEM) and selected-area electron diffraction (SAED). The TEM image showed that the nanorods were bamboo-like. XRD, HRTEM and SAED patterns indicated that the nanorods were single-crystal wurtzite GaN. Received: 8 January 2001 / Accepted: 28 April 2001 / Published online: 20 December 2001