Vol 15, No 1 (2011)

A system of nonlocal cosmological equations where the time variable runs over a half-line is considered. These equations are more suitable for description of the Universe than the previously discussed cosmological equations on the whole line since the Friedmann metric contains a singularity at the beginning of time. Definition of the exponential operator includes a new arbitrary function which is absent in the equations on the whole line. It is shown that this function could be choosen in such a way that one of the slow roll parameters in the chaotic inflation scenario can be made arbitrary small. Solutions of the linearized nonlocal equations on the half-line are constructed.

This work is devoted to the mathematical description of the dynamics of tachyons of open, closed and open-closed p-adic strings. The questions of existence and nonexistence of continuous solutions and their properties, as well zero structure of solutions is discussed. New multidimensional nonlinear equations of ultraparabolic type are obtained. Some unsolved problems are listed.

Initial-boundary value problem for linearized equations of viscous barotropic low compressible fluid in a bounded domain is considered. Convergence of solutions of this problem at withincompressible limit approaching to zero is studied. Sufficient conditions for the weak and strong convergence of this problem for uncompressible liquid are given.

The generalized functions which have quasiasymptotics along the trajectories of one-parametric group are called asymptomatically homogeneous. The corresponding limit functions are homogeneous with respect to this group. In this paper we give the full description of asymptotically homogeneous generalized functions along the trajectories of unstable degenerated node. The obtained results are applied for description of ho-mogeneous generalized functions for such trajectories in two dimensional case.

Spaces of p-adic BMO and VMO functions are considered. It is proved that locally constant functions are dense in VMO space under BMO norm.

Simple proof of the adiabatic theorem is given in a finite dimensional case for nondegenerate as well as degenerate states. The estimate is obtained for the deviation of the norm of the solution of the Shchrodinger equation which is uniform on the parameter in the Hamiltonian.

We discuss the family of metrics in multimensional p-adic spaces. For a metric under consideration the set of balls differs from the set of balls for the standard multidimensional metric. Moreover, we it is possible to consider the metrics for which the form of the sets of balls depends on the scale and position. As the example of the introduced metric spaces we study the 2-adic parametrization of the genetic code. We show that the degeneracy of the genetic code is described by the metric space from the considered family, i.e. the map of the genetic code is constant for the balls with respect to the metric from the introduced class.

One of modern approaches to a problem of the coordination of classical mechanics and the statistical physics - the functional mechanics is considered. Deviations from classical trajectories are calculated and evolution of the moments of distribution function is constructed. The relation between the received results and absence of paradox of Poincare-Zermelo in the functional mechanics is discussed. Destruction of periodicity of movement in the functional mechanics is shown and decrement of attenuation for classical invariants of movement on a trajectory of functional mechanical averages is calculated.

In this paper we prove the unique solution of the problem with a shift to a Lykov's type system of differential equations of first order. The proof is given for different values of the generalized operators of fractional integro-differentiation included in the boundary condition.

We propose a procedure for obtaining the Boltzmann equation from the Liouville equation in a non-thermodynamic limit. It is based on the BBGKY hierarchy, the functional formulation of classical mechanics, and the distinguishing between two scales of space-time, i.e., macro- and microscale. According to the functional approach to mechanics, a state of a system of particles is formed from the measurements, which have errors. Hence, one can speak about accuracy of the initial probability density function in the Liouville equation. Let's assume that our measuring instruments can observe the variations of physical values only on the macroscale, which is much greater than the characteristic interaction radius (microscale). Then the corresponfing initial density function cannot be used as initial data for the Liouville equation, because the last one is a description of the microscopic dynamics, and the particle interaction potential (with the characteristic interaction radius) is contained in it explicitly. Nevertheless, for a macroscopic initial density function we can obtain the Boltzmann equation using the BBGKY hierarchy, if we assume that the initial data for the microscopic density functions are assigned by the macroscopic one. The H-theorem (entropy growth) is valid for the obtained equation.

In this work we consider the comparison of planar and planar parquet approximations for zero-dimensional hermitian matrix models. We discuss how the parquet approach reproduces planar one for matrix model φ4, multi-trace model, two-matrix model and the Goldstone matrix model.

The two-dimension model of the deformation of asymmetrical elastic solid is considered in the article. The exact solution of flat asymmetrical elasticity's problem about simple shear in the plate which is weakening by hole is demonstrated. Comparative analysis of the obtained solution with the classical ones are given.

The present paper is devoted to a study of a natural 12-dimensional symmetry algebra of the three-dimensional hyperbolic differential equations of the perfect plasticity, obtained by D. D. Ivlev in 1959 and formulated in isostatic coordinates. An optimal system of one-dimensional subalgebras constructing algorithm for the Lie algebra is proposed. The optimal system (total 187 elements) is shown consist of a 3-parametrical element, twelve 2-parametrical elements, sixty six 1-parametrical elements and one hundred and eight individual elements.

The solution of nonlinear stochastically boundary value problem of creep of a thin plate under plane stress is developed. It is supposed that elastic deformations are insignificant and they can be neglected. Determining equation of creep is taken in accordance with nonlinear theory of viscous flow and is formulated in a stochastic form. By applying the method of small parameter nonlinear stochastic problem reduces to a system of three linear partial differential equations, which is solved about fluctuations of the stress tensor. This system with transition to the stress function has been reduced to a single differential equation solution of which is represented as a sum of two series. The first row gives the solution away from the boundary of the body without boundary effects, the second row represents the solution boundary layer, its members quickly fade as the distance increases from plate boundary. Based on this solution, the statistical analysis random stress fields near the boundary of the plate was taken.

Second order orthogonal finite functions are studied, their compact supports are local sets of grid triangles. The theorem on approximating the properties of sequences of sets of such functions is formulated. The applications in algorithms of mixed numerical methods of boundary value problems solution, in algorithms of linear approximation ofsurfaces are considered.

One-dimensional generalized rheologic model of viscoelastic body with Riemann- Liouville derivatives is considered. Instead of derivatives of order α > 1 there are employed in defining relations derivatives of order 0 < α < 1 from integer derivatives. It's shown, that the differential equation for the deformation with given dependence of the tension from the time with classical initial conditions of Cauchy are reduced to the Volterra integral equations. Some variants of the generalized fractional Voigt's model are considered. Explicit solutions for corresponding differential equation for the deformation are found out. It's indicated, that these solutions coincide with the classical ones when the fractional parameter vanishes.

The article represents the numerical solution of the problem of the increasing wedging crack in the disturbed material. Solution of problem was achieved by the method of discontinuous displacements. Analytical dependences of the stress intensity factors of the first type on the normal displacements of the crack was determined.

Three-dimensional axially symmetrical exactly soluble model of quantum ring has been considered in the constant magnetic field. Potential, which restricts particle movement in the system in two directions, has been used. By applying method of splitting into physical processes, an electron wave functions and a quasi-stationary energy levels values have been obtained. Also a qualitative description of wave packet movement has been given for the case of quantum ring threading by alternating magnetic field.

A gauge model with chiral color symmetry of quarks is considered and possible effects of the color G -boson octet predicted by this symmetry are investigated. The contributions of the G -boson to the cross section σtt and to the forward-backward asymmetry App of tt production at the Tevatron are calculated and analysed in dependence on two free parameters of the model, the mixing angle θG and G mass mG , in comparision with the new Tevatron data on App . The G -boson contributions to σtt and App are shown to be consistent with the Tevatron data on σtt and App , the allowed region in the -θG consistency is found.