# Vol 15, No 1 (2011)

**Year:**2011**Articles:**40**URL:**https://journals.eco-vector.com/1991-8615/issue/view/1258

### Preface

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):5-8

5-8

### Mathematical modeling of molecular "nano-machines"

#### Abstract

A new approach to mathematical modeling of "molecular machines", e.g. macromolecular structures which functional prototypes are the proteins, is presented. In the center of the approach lies the description of multi-scale fluctuation induced mobility of proteins by the ultrametric random processes. In order todemonstrate how p-adic equations of the reaction-diffusion type are described the molecular machine operation, a heuristic model is constructed in this article. It is shown that such multi-scale modeling allows to have an insight into unexpected resources that can be used in order to control the functional cycle.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):9-15

9-15

### On nonlocal cosmological equations on half-line

#### Abstract

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.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):16-27

16-27

### Some features of second kind fredholm equations kernels

#### Abstract

Kernels of Fredholm integral equations of the second kind with exceptions are analysed in this article. The equations under consideration have a meaning of magnetic field boundary condition and are used in problems of scattering on scatterers with finite thickness. It is shown that these kernels could be stated in a form of Dirac delta- functions. This mathematical formalization results in interesting physical effect that induced current calculated via physical optics equals the difference of face and back currents of the scatterer, calculated using method of integral equations.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):28-33

28-33

### Mathematical questions for theory of nonlinear pseudodifferential equations with p-adic string

#### Abstract

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.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):34-41

34-41

### Cauchy problem for the wave equation on non-global hyperbolic manifolds

#### Abstract

We consider Cauchy problem for wave equation on two types of non-global hyperbolic manifolds: Minkowski plane with an attached handle and Misner space. We prove that the classical solution on a plane with a handle exists and is unique if and only if a finite set of point-wise constraints on initial values is satisfied. On the Misner space
the existence and uniqueness of a solution is equivalent to much stricter constraints for the initial data.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):42-46

42-46

### Weak and strong convergence of solutions to linearized equations of low compressible fluid

#### Abstract

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.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):47-52

47-52

### The estimates of the solution of the dirichlet problem with boundary function from Lp for a second-order elliptic equation

#### Abstract

We study the solvability of the Dirichlet problem for a second-order elliptic equation with measurable and bounded coefficients. Assuming that coefficients of equation are Dini-continued on the boundary, it is established that there is the unique solution of the Dirichlet problem with boundary function from $L_p$, $p > 1$. We prove the estimate of the analogue of area integral.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):53-67

53-67

### Generalized functions asymptotically homogeneous along the unstable degenerated node

#### Abstract

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.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):68-82

68-82

### Method of general Сoule-Hopf substitutions in theory of finite-dimensional dynamical systems

#### Abstract

We consider the results of applying the method of generic Cole-Hopf substitutions to integration of finite-dimensional dynamical systems. Dynamical systems are represented in the form of matrix ordinary differential equations with specific matrix algebra of finite dimension. The Cole-Hopf type substitutions are applied to matrix equations by using the differentiation on algebra in the form of commutator with a specific algebra element. Recurrent relations for Cole-Hopf substitutions were found. Particular cases of exactly integrable dynamical systems are presented. The algorithm of calculating the integrals of motion is shown.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):83-89

83-89

90-92

### The solution of the full matrix analogue of the generalized abel equation with constant coefficients

#### Abstract

The system of generalized integral Abel equations in the matrix form with constant coefficients on the segment. Was considered at the terms of the integral Riemann-Liouville operators of matrix order. It's reduction to the system of singular integral equations was founded. Solution of this system was found for the case of the commutative matrices of the simple structure in the explicit form.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):93-98

93-98

### Simple proof of the adiabatic theorem

#### Abstract

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.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):99-107

99-107

### Special solutions of matrix Gellerstedt equation

#### Abstract

Fundamental solutions for the Gellerstedt equation and its generalization were obtained in the distribution space using the method applied by I. M. Gelfand and J. Barros-Neto to the studying the Tricomi equation. The degenerating system of the mixed-type partial differential equations was considered, its special solutions were constructed in the regions bounded by the characteristics of these equations (in the hyperbolic halfplane). The elements of the theory of matrices, theory of the generalized functions and the special functions (hypergeometric series) were used for this construction.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):108-112

108-112

### Multidimensional p-adic metric and genetic code

#### Abstract

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.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):113-117

113-117

### Dirac-Yang-Mills model equations with a spinor gauge symmetry

#### Abstract

In the developed model where spin 1/2 fermions acquire masses by an interaction with (spin 1) gauge field with spinor symmetry. Particle mass is determined by the constant interaction of the particle with the gauge field.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):118-123

118-123

### The functional mechanics: evolution of the moments of distribution function and the poincare recurrence theorem

#### Abstract

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.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):124-133

124-133

### On classical and functional approachs to mechanics

#### Abstract

In this paper the relevance of the classical trajectory of the anharmonic oscillator and the average trajectory obtained within the functional approach is considered. Dependence of threshold time of divergence of trajectories on dispersion of initial values is derived.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):134-139

134-139

### Nonlocal boundary value problem for a Lykov's type system of first-order

#### Abstract

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.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):140-150

140-150

### Solution in explicit form of non-local problem for differential equation with partial fractional derivative of Riemann-Liouville

#### Abstract

A non-local problem for a mixed type equation with partial fractional derivative of Riemann-Liouville is studied, boundary condition of which contains generalized operator of fractional integro-differentiation. Unique solution of the problem is then proved.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):151-157

151-157

### Boltzmann equation and H-theorem in the functional formulation of classical mechanics

#### Abstract

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.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):158-164

158-164

### Theorem on the norm of elements of spinor groups

#### Abstract

In this article we consider Clifford's algebra over the field of real numbers of finite dimension. We define the operation of Hermitian conjugation for the elements of Clifford's algebra. This operation allows us to define the structure of Euclidian space on the Clifford algebra. We consider pseudo-orthogonal group and its subgroups - special pseudo-orthogonal, orthochronous, orthochorous and special orthochronous groups. As known, spinor groups are double covers of these orthogonal groups. We proved a theorem that relates the norm of element of spinor group with the minor of matrix of the orthogonal group.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):156-171

156-171

### Matrix models and parquet approximation

#### Abstract

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.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):172-178

172-178

### On stability of solutions of equations of interaction between elastic walls of channels and affluent liquid

#### Abstract

In this article the dynamical stability of elastic walls of plane channels under flowing of the perfect liquid is investigated on the basis of mathematical models. Either the law of pressure change or the potential of liquid velocity or the longitudinal component of liquid velocity are applied on the input and output from the channels. The sufficient conditions of stability are obtained. These conditions impose a constraint on the velocity of the liquid homogeneous stream, on the compressing (decompressing) force and other parameters of mechanical system.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):179-185

179-185

### Investigation of the stress-strain state of the asymmetrical elastic plates

#### Abstract

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.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):186-190

186-190

### Сoncentrated force acting near the tip of an interface crack with a rigid overlay on its side

#### Abstract

Plane stress state near the tip of an interface crack induced by specified concentrated force is considered. One of the crack faces is partially reinforced by a rigid straight line overlay. The complex potentials, the stress intensity factors at the crack-tip are found, corresponding plots are presented.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):191-195

191-195

### An optimal system of one-dimensional subalgebras for the symmetry algebra of three-dimensional equations of the perfect plasticity

#### Abstract

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.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):196-220

196-220

### Propagation thermoelastic impulse through a cylindrical waveguide under sidewall heat interchanging

#### Abstract

The paper is devoted to a study of coupled harmonic thermoelastic impulse guided propagation through an infinite circular cylinder. Heat interchanging is supposed to take place between sidewall of the waveguide and environment. The analysis is carried out according to the principles of coupled generalized thermoelasticity theory of type III (GNIII-thermoelasticity). This theory combines thermodiffusion and wave mechanisms of heat transfer in solids including as limiting cases both the theories: classical thermoelasticity (GNI/CTE) and the theory of hyperbolic thermoelasticity (GNII ). The latter permits field-theoretic formulation and leads to the field equations of hyperbolic analytical type. Closed solution of the coupled GNIII-thermoelasticity equations satisfying the boundary conditions on the surface of waveguide is obtained by separation of variables. The analysis of frequency equation is given and wave numbers and modes of coupled thermoelastic waves of arbitrary order are obtained. The problems of coupled thermal and dynamic impulse propagation in the form of plane and normal waves in a free from tractions thermoisolated waveguide have been studied in our previous papers.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):221-227

221-227

### Сonstruction of analitical solution of 2D stochastically nonlinear boundary value problem of steady creep state with respect to the boundary effects

#### Abstract

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.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):228-235

228-235

### Progective algorithm of boundary value problem for inhomogeneous Lame's equation

#### Abstract

The method of boundary value problem solution for the stationary inhomogeneous Lame's equation is considered. An appointed vector-function space splitting is used that leads to inhomogeneous biharmonic equation and Poisson's equation problems for components of required vector field. The basic potentials method is proposed to solve these problems.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):236-240

236-240

### On orthogonal second order finite functions associated with triangular meshes and their application in mathematical modeling

#### Abstract

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.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):241-243

241-243

### Static bending and vibrations of multilayer orthotropic rectangular plate with simply supported edges

#### Abstract

Bending and vibrations of multilayer rectangular plate are considered. The plate is composed from N orthotropic layers of arbitrary thickness. Edges of the plate are simply supported. Possibility of closed-form solution is discussed. Numerical solution method is developed and implemented. Results are presented in table.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):244-254

244-254

### Rheological model of viscoelastic body with memory and differential equations of fractional oscillator

#### Abstract

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.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):255-268

255-268

### Method of states on the basis of Kilchevskiy's equations for analysis of 3d steady-state oscillation

#### Abstract

General solution of N. A. Kilchevskiy's equations for oscillating three-dimensional solids is constructed. Method of states for analysis of boundary-value problem about oscillations is proven, it is based on the following terms: states of medium (internal and boundary), space of states, scalar product, gilbert isomorphism.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):269-275

269-275

### Mathematical model of the development of the wedging crack in the disturbed material

#### Abstract

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.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):276-282

276-282

### Simulation of high-temperature shorteningof cylinders under different material creepequations

#### Abstract

A subsidence of circular cylinders between rigid stamps at creep conditions is studied. There are three different material models. Appearance of barrel shape form is allowed (this decision is realized on the basis of package LS-DYNA). Calculations of all principal parameters were carried out at two loading programs: at constant speed of the cylinders' bases coming together and at constant compressing force. Calculations have shown that the energy, spent on subsidence of the cylinder during the first loading variant isless, than in the second variant (this difference in certain processes is equal to 8-10%).

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):283-290

283-290

### Time dynamics of electron wave functions of 3d quantum ring in alternating magnetic field

#### Abstract

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.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):291-296

291-296

### Thermal equations of the radiation dominated universe evolution

#### Abstract

The equations of the evolution of the Universe in Friedmann model are considered using thermodynamic functions and temperature as function of time. The exit for frameworks of the standard model is carried out, the influence of dark energy is considered.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):297-304

297-304

### The chiral color symmetry and g -boson mass limit from new tevatron data on tt production

#### Abstract

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.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):305-312

305-312

### The $n {-} \bar n$ oscillations in neutron fluxes running from Sun to Earth

#### Abstract

The process of $n {-} \bar n$ oscillations in the solar cosmic-rays is considered. It is shown that it has high intensity with respect to the analogic processes on the Earth $(I_{c{-}r}/I_{Earth} \propto 10^8)$, because magnetic field strongly suppresses $n {-} \bar n$ oscillations. Energetic dependence of the $\bar n$ and $\bar p$ fluxes at the Earth is also found. Results obtained are argument for searching the $n {-} \bar n$ transitions in experiments with the solar cosmic-rays.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2011;15(1):231-317

231-317