Vol. 45 No. 1 (224) (2011)

In the present article the first order linear differential operators with unbounded coefficients are investigated. The boundedness of the operators under consideration was proved.

In this paper we consider the solvability of the boundary value problem, connected with the anisotropic Helmholtz-Schrödinger equation, under the boundary conditions of the first and second type on the line $y=0$.

In this paper the non-characteristic singular integral equations on the semi-axis are discussed. The solution of these equations is reduced to the solution of one-dimensional pseudo-differential equations. Some examples of singular equations for which explicit solutions exist are provided.

In this paper the stability problem for frequency distribution in frame of natural parameterization is formulated and discussed. The case of finite number of independent parameters is characterized. A corresponding stability problem is investigated in terms of $l_p$-metric.

The differential several person game in case of aim sets is considered, the dynamics of which is described by a nonlinear differential equation. A balance set of strategies is constructed by means of the method of extreme aiming at the corresponding set.

In the present paper the type correctness problem is considered for functional programs without the type information. The aim of this research is to prove that there is no algorithm to reject all programs, during execution of which the type error would occur and accept all programs, during execution of which the type error would not occur.

In this paper the notion of transformation of untyped functional programs that preserves the main semantics of programs is presented. A transformation, representing programs by two equations, such that procedural semantics of programs that use interpretation algorithms based on substitution and normal form reduction remain the same, is introduced. It is proved also that there is no transformation, which represents programs with one equation, such that the procedural semantics of programs remain unchanged for all interpretation algo-rithms that are based on two operations: a substitution and a one-step β-reduction. 

The bound states of the electron in a quantum ring in the presence of different scalar potentials in two regions are considered. It is shown that the difference of scalar potentials lifts the degeneracy connected with the sign of orbital angular momentum. The same system with a magnetic flux threading the ring is also considered. The possibility of observing the oscillations of energy levels connected with the difference of scalar potentials and the intensity of the magnetic field that are similar to those of the bound states of the electron in a quantum ring threaded only by the magnetic flux is shown. 

Improvements in the “hybrid” imaging method, based on traditional Laser Scanning Microscopy in combination with the single-layer flat-coil-oscillator (SFCO) sensitive technique introduced by authors earlier, are discussed. This method is capable of imaging 2D-grained structure, as well as the normal-to-superconductive phase transition in flat thin high-Tc superconductive materials with 1–2 μm spatial resolution. It enabled to detect weakly expressed peculiarities of the heat capacity in YBa₂Cu₃O_y film (preceding the well-known specific heat “jump”), which are important for true understanding of the real nature of superconductivity phenomenon. The method uses a well-focused He-Ne laser beam as a probing signal, and the SFCO-technique as a high-resolution sensing tool.

The presence of a resonant medium in the capacitor core essentially changes the character of THz radiation of photoconductive antenna. As a result the coherent radiation at the frequency of elementary excitation (LO phonon or plasmon) is obtained. Though the power of radiation slightly differs from that provided by a usual THz source, the energy of pulse is significantly greater due to longer duration of the pulse. Besides, such a method of THz pulse generation is very useful for THz emission spectroscopy of many materials. 

Based on the similarity revealed between the processes occurring in the atmosphere and in semiconductor electron-hole plasma, the issues of controlling the atmospheric conditions in the surface layer have been considered. It was established that on the analogy with multistable semiconductor electron-hole plasma a new method for monitoring the atmospheric conditions and processes taking place in the layer may be developed. Analytical expressions for the control signal and potential formed in the atmosphere of dissipative structures were obtained. These expressions were shown to serve as a sufficient basis for construction of appropriate numerical methods.

A structural theorem is proved for pMqBM completely invariant subspaces of free associative algebras with a unit, having a countable number of free variables over the field of characteristic zero. In particular, it is shown that such spaces contain a Lie nilpotent polynomial.