No 2 (2015)

In relative elliptic theory or “Sobolev” problem as B. Yu. Sternin named it in his works one is required to construct a Fredholm elliptic theory and find an index formula in the category of smooth pairs of manifolds (M,X), where X is a submanifold in M. From the point of view of (pseudo)differential equations the Sobolev problem deals with the comparison Du ≡ f(modX), where D is a pseudodifferential operator, while the sign “ ≡” means that the left and right hand sides are equal modulo distributions supported on X. Obviously, if the dimension of the submanifold is greater than one, the comparison written above does not define a Fredholm operator, since its kernel is infinite-dimensional. It turns out, that if we add to the comparison some operators B defined on X, which are related by an algebraic condition (of coercitivity type) with operator D, then the obtained operator (D,B) is already Fredholm in appropriate Sobolev spaces. Remarkably, this condition can be formulated invariantly as an ellipticity condition of some operator, which is induced by the problem on the submanifold X. Hence, the ellipticity conditions of operators D and (D,B) together give us a Fredholm operator. This theorem and the corresponding index formula were proved by B.Yu. Sternin. Note that all operators appearing in this theory are pseudodifferential. In particular, (D,B) is a pseudodifferential operator, meanwhile, this enabled one to define its ellipticity. We have a quite different situation, if the manifold M is endowed with an additional structure, for example, if it carries a Lie group action. In this case, (D,B) is in general no longer a pseudodifferential operator and, hence, the question of its ellipticity, formally speaking, can not even be rised. However, in our work, under certain conditions, we can examine the resulting operator (D,B), define its symbol and prove its Fredholm property. Moreover, we give an index formula in this more general situation. This is the subject of this work.

A problem in the mathematical teletraffic theory of passive optical networks (PON) with time division multiple access (TDMA) is to divide timing budget between optical network units to optimize the network performance. For the task we examine a segment of the passive optical network with wavelength division multiplexing (WDM) and time division multiple access technologies. An extend model of the segment is developed and an allocation problem of wavelength finite number between optical network units is resolved in the article. The optical network unit does not receive data from optical line terminal and send data to optical line terminal being in passive state. The algorithm for calculating of the probability of being each optical network unit in passive state is proposed. Finally, a problem for further research is formulated. It is the algorithm for calculation of call blocking probability in upstream traffic models for WDM-TDMA PON.

The objective of this paper is the construction and analysis of the model of the wireless LTE network (Long Term Evolution) as a multiserver queuing system where losses are caused by a lack of resources required to service requests. Adopted by the service application takes a random amount of resources given to several types of distribution functions. Random vectors describing the requirements of applications to resources, processes do not depend on input flow and service distribution are jointly independent and identically distributed. L independent Poisson flows of requests enter the system, and there are N identical devices. Service times are distributed exponentially. The functioning of the system is described by the semi-Markov process, which takes into account the number of serviced requests, their types and amounts of resources they occupy. Explicit expressions for the stationary distribution of the semi-Markov process, and the theorem on product form solution are main results of the paper. Further studies suggest checking the hypothesis of invariance with respect to the form of the stationary distribution of the distribution of the service time and the development of numerical methods for the analysis of probability measures of the system.

Solutions of the Schrödinger equation for complex values of the energy describe the quasistationary state. The energy spectrum of quasistationary states is quasidiscrete and consists of a series of broadened levels, width of which determines the lifetimes of the respective states. The introduction of quasistationary states only makes sense if the width of the respective quasidiscrete levels is small compared with the distances between levels. An investigation of the solutions of quasistationary states is carried out for the quasipotential equation with piecewise-constant potentials at various values of the parameter of the equation , included in the equation and the potential parameters. A comparative analysis of the solutions of the quasipotential equation for the different values of with the solutions of the Schrödinger equation is performed. It was found that at → 0 the solutions of quasipotential equation tend to the solutions of the Schrödinger equation. With the increasing of the parameter the lifetime of quasilevels for the quasipotential equation increases as compared with the results obtained for the Schrödinger equation except the level that is close to the edge of the barrier. For comparison, the wave functions for the Schrödinger equation and the quasipotential equation for fixed values of the potential parameters are shown.

The stationary Schrödinger equation depending on spatial coordinates has been considered. The problem of obtaining a differential relationship for the wave function squared was posed. By extracting Schrödinger’s equation itself from this relationship a differential equation for a physically interpretable quantity, i.e. the probability density (wave function squared), has been formulated. As an example the one-dimensional case admitting a simple analytic solution was considered. The solution obtained is shown to be a solution squared of the corresponding nonlinear differential equation for the probability density. In the final section a more general non-stationary case was considered for the potential involving a time-dependent term, such potentials are found in the non-stationary perturbation theory. The constant in separating the variables remains real. Thus the procedure considered proves to be similar to that presented above for the stationary equation.

For the first time, the single-domain (without twinning) crystals of LiCu 2O 2 were used to measure anisotropy of DC and AC conductivities along the principal crystal axes and low-frequency dynamics of charge transport in the temperature (4.2-295 K) and frequency (25-105 Hz) ranges. The temperature, frequency and field properties of the DC and AC conductivities reflect the strong localization of charge carriers as result of a local lattice distortion due to the structural, compositional (extra oxygen content) defects and particularly electrical and magnetic polarization. Therefore the conductivity is by hopping transport between localized states near the Fermi level and its character (activated or variable-range hopping) depends on the temperature range and on relative direction to the crystal axes. The analysis of experimental data allows us to draw conclusions about the electronic energy structure near the Fermi level, about the anisotropic properties, low-frequency dynamics and mechanisms of charge transport.