## Vol 16, No 4 (2012)

**Year:**2012**Articles:**25**URL:**https://journals.eco-vector.com/1991-8615/issue/view/1249

Articles

In Memory of Vasilii S. Vladimorov

#### Abstract

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(4):8-16

Problem with shift for the third-order equation with discontinuous coefficients

#### Abstract

The unique solvability of boundary value problem with Saigo operators for the thirdorder equation with multiple characteristics was investigated. The uniqueness theorem with constraints of inequality type on the known functions and different orders of generalized fractional integro-differentiation was proved. The existence of solution is equivalently reduced to the solvability of Fredholm integral equation of the second kind.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(4):17-25

The problem with shift for the Bitsadze–Lykov equation

#### Abstract

The Bitsadze-Lykov equation is considered. The problem with shift containing the Kober–Erdélyi and M. Saigo operators in boundary condition is set for this equation. The questions of uniqueness (ununiqueness) of this problem solution with different functions and constants in boundary condition are investigated. The number of theorems is formulated and proved.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(4):26-36

The local solvability of a problem of determining the spatial part of a multidimensional kernel in the integro-differential equation of hyperbolic type

#### Abstract

The multidimensional inverse problem of determining spatial part of integral member kernel in integro-differential wave equation is considered. Herein, the direct problem is represented by the initial-boundary problem for this with zero initial data and Neyman’s boundary condition as Dirac’s delta-function concentrated on the boundary of the domain $(x, t) \in \mathbb R^{n+1}$, $z > 0$. As information in order to solve the inverse problem on the boundary of the considered domain the traces of direct problem solution are given. The signiﬁcant moment of the problem setup is such a circumstance that all given functions are real analytical functions of variables $x \in \mathbb R^n$. The main result of the work is concluded in obtaining the local unique solvability of the inverse problem in the class of continuous functions on variable $z$ and analytical on other spatial variables. For this, by means of singularity separation method, the inverse problem is replaced by the initial-boundary problem for the regular part of the solution of this problem. Further, direct and inverse problems are reduced to the solution of equivalent system of Volterra type integro-differential equations. For the solution of the latter, the method of Banach space scale of real analytical functions is used.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(4):37-47

The well-posedness of the local boundary value problem in a cylindric domain for the multi-dimensional wave equation

#### Abstract

This paper proves the unique solvability of the local boundary value problem in a cylindric domain for the multi-dimensional wave equation, which is the generalization of the Dirichlet and Poincare problems. We also obtain the criterion for the uniqueness of the regular solution.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(4):48-55

On a question of limiting distribution of series in random binary sequence

#### Abstract

Limiting forms of distribution of length of the maximum series of successes in random binary sequences, formed in Bernulli–Markov’s chain and in Polya’s scheme which is an equivalent to local trends of a time series of strictly stationary process are investigated. More simple and added proof of theorems of the law of the big numbers for series of both types is offered. For series of the second type the effect of the cyclic bimorphism of the limiting law with degeneration on one of the phases and the convergence according to the probability on set no more, than four consequent values of the natural series is established.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(4):56-71

Generalized stochastic model of creep and creep rupture beams in pure bending and its application to the estimation of reliability

#### Abstract

Generalized stochastic model of creep and creep rupture beams under pure bending in terms “generalized load”, “generalized displacement”, “time” is oﬀered. Beam is considered as a single entity (the speciﬁc model). The complete analogy between the curves of uniaxial creep sample under constant stress and generalized creep curves beams in the curvature of the beam coordinates “curvature beams – time” under the constant bending moment is determined. On the basis of this analogy the stochastic equation of state beam is formed. Method of reliability estimating of the beams bending under creep on parametric criteria of failure in a signiﬁcant scatter of the data is developed. Calculation results and recommendations for lifelength assigning are presented.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(4):72-86

The method for solving the boundary value problem of the beam’s creep and creep rupture strength condition of the pure bending based on the rod type structural model

#### Abstract

The method for solving the boundary value problem of the beam’s creep and creep rupture strength under condition of the pure bending based on the rod type structural model is proposed. Energy criterion of local element destruction is introduced. Comparative analysis of structural model calculated data and the quarter beam of D16T alloy at $T = \rm 250~^\circ C$ curvature value found by experiment is performed. Calculated data agree with those found by experiment. Correlation of the calculated data based on the proposed method with those based on phenomenological model of energy type creep is performed.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(4):87-96

Method of pure shear problem solving for stochastically inhomogeneous plane in a steady-state creep

#### Abstract

The analytical method for nonlinear problem of steady-state creep solving for pure shear of stochastically inhomogeneous plane on the basis of the second approximation method of small parameter was developed. It is supposed that elastic deformations are insignificant and they can be neglected. Stochasticity was introduced into the determinative creep equation, which was taken in accordance with the nonlinear theory of viscous ﬂow, through a homogeneous random function of coordinates. By using the decomposition technique of stress tensor components in a small parameter to the members of the second order of smallness, partial diﬀerential system of the ﬁrst and the second approximation of stress was obtained. This system was solved by the introduction of the stress function. The mathematical expectation and variances of the random stress ﬁeld were calculated. The analysis of the results in the ﬁrst and second approximations was obtained.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(4):97-105

Construction of material functions of aluminum alloy D16T inelastic deformation based on the results of tests of tension and torsion

#### Abstract

The tensor models of damage accumulation for isotropic materials are considered. The material functions of aluminum alloy D16T inelastic deformation based on the results of tests of tension, torsion, tension with torsion with diﬀerent relations of axial and shear deformations are deﬁned. Approximations of experimental data by analytical expressions are proposed. These expressions include the dependence on the ﬁrst and second invariants of the strain tensor. Comparison of experimental and theoretical dependences attests to adequacy of the proposed mathematical models.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(4):106-114

Low-frequency long-wave approximations of the 3D dynamic equations for the case of double-layered viscoelastic plate

#### Abstract

The asymptotic methods developed to obtain low-frequency long-wave approximations of the 3D dynamic equations for the case of double-layered viscoelastic plate are described. The 2D equations for the leading tangential and transverse approximations of stressstrain state are derived. The method of asymptotic integration of exact 3D equations is applied.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(4):115-121

Boundary value problem for the flexible axially loaded compound shells of revolution and beams systems

#### Abstract

The effective algorithm of static calculation of geometrically nonlinear compound thin structures is offered. Linear differential equations of moment theory are used. Nonlinearity is considered by assigning unknown initial angular displacement of each segment retaining the form of the generating line. Unknown values of algebraic equations resolving system are the arbitrary constants of the general solution and the initial angles of generating lines rotation. Linearization is realized by Newton–Raphson iterative method and provides the high precision of results.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(4):122-130

The use of hermite special functions for investigation of power properties of grubbs statistics

#### Abstract

We consider a normal sample with a single upper outlier. A distribution of studentized form of outlier’s deviation from the sample mean is obtained. This distribution uses Hermite special functions with negative integer-valued index. The integral relationships for David’s power measures of Grubbs criteria are obtained. We discuss the case, when Grubbs statistic is the likelihood-ratio statistic. We ﬁnd the maximal deviation of power function for Grubbs criteria from the probability that the contaminant is the outlier and it is identiﬁed as discordant. We receive the region of critical values of Grubbs statistic, where the second power measure of David equals to the third and forth power measures of David. We make calculations of power function for Grubbs criteria in the case of normal samples with a single upper outlier with the right shift. The results of calculations are similar to the theoretically expected facts.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(4):131-145

Percolation of the prolate ellipsoids of rotation in the continuum

#### Abstract

Continuum percolation of the hard prolate ellipsoids of rotation with permeable shell has been investigated. It is the model of phase transition sol–gel. Ellipsoids are located in the cube randomly. For each set of parameters 100 tests are spent. For each test the ﬁnding of the percolation cluster is the main task. The fraction of the packing for which the probability of the percolation cluster appearance is equal 0.5, is called a percolation threshold. Value of the percolation threshold corresponds to the gel point. Dependence of value of the percolation threshold on thickness of permeable shell and aspect ratio has been obtained. In addition to the percolation threshold the other characteristics of the model have been obtained, such as: the size distribution of clusters, the average cluster size, the strength and the fractal dimension of the percolation cluster, the average value and the distribution of neighbors of an element, the critical exponents.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(4):146-153

The reliability indicators calculation for the pipeline systems with crack-like defects and complicated service conditions

#### Abstract

The existing methods for pipeline systems reliability estimation, based on a normative approach, do not consider in an explicit form neither time as a factor, nor the probability nature of characteristics of load-carrying ability and loads that often leads to designation of unfairly high values for stress reserve factors. Methods based on probability approaches allow excluding these shortcomings, however, in spite of the fact that they are rather widely theoreti-cally justiﬁed, their practical applications do not exceed 1 %. In this paper the algorithms and methods for pipeline systems reliability indicators calculation for practical application which allow determining the values during design or service stage are presented. These methods are based on the theory of classical statistics.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(4):154-161

Polarization converter for higher-order laser beams based on the use of forming binary diffractive optical elements as beam splitter

#### Abstract

We propose a new approach to generating a pair of initial beams for a polarization converter that operates by summing up two opposite-sign circularly polarized beams. The conjugated pairs of vortex beams matched with laser modes are generated using binary diﬀractive optical elements. The same binary element simultaneously serves two functions: a beam shaper and a beam splitter. Two proposed optical arrangements are compared in terms of alignment complexity and energy eﬃciency. The diffractive optical elements in question have been designed and fabricated. Natural experiments that demonstrate the generation of vector higher-order cylindrical beams have been conducted.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(4):162-170

The influence of inhomogeneous exchange on the propagation of surface damon–eschbach waves in the ferrospinels films

#### Abstract

The law of magnetostatic waves dispersion in the manganese and magnesiummanganese ferrospinels ﬁlms was investigated using the moving-transducer method. The wave numbers, phase and group velocities of magnetostatic waves are deﬁned. The excitation of spin-wave waves in the surface layer of the ﬁlm is found. The wave numbers, spin pinning options, their frequency dependences and surface anisotropy constants are calculated.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(4):171-179

The influence of the pressed hollow billet of brass 59Cu–3,5Mn–2,5Al–0,5Fe–0,4Ni stracture on the fracture mode of destruction with following hot stamping

#### Abstract

The inﬂuence of the hot stamping temperature on the structural state of brass 59Cu−3,5Mn−2,5Al−0,5Fe−0,4Ni and the character of blocking rings destruction of the synchronizer of the car transfer change box have been investigated. It is shown that the lack of ductile $\alpha$-phase in the investigated alloy in combination with high temperature of heating ($780~^\circ \rm C$) before stamping leads to intergranular brittle fracture under the action of residual stresses. Decrease in temperature stamping to $700~^\circ \rm C$ with the same structural condition of the brass excludes cracking of the brass rings under the effect of residual stresses and changes the character of destruction under bending load with rock-like to pits-like one.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(4):180-187

The mechanism of the appearance of stochasticity in quantum mechanics

#### Abstract

The technique for the dynamic description of interaction between the quantum particle and the measuring instrument is oﬀered. This description allows to determine, that the statistic dispersion of measuring instrument characteristics is the cause of the results randomness of the quantum particle space localization.Space-time consideration of the macroscopic meter evolution, initiated by the quantum particle, allows to represent the mechanism of the appearance of probabilistic measure, expressed by the wave function modulus square.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(4):188-198

Enciphering on the basis of the sums with products of weight and free components as summands

#### Abstract

The purpose of the given paper is reviewing of mathematical resources of enciphering of the source text, allowing to ensure the simplicity of appropriate decryption; the source text is a sequence of integer weight coefficients. The composer of the cipher checks how the condition of separability of these coefficients is satisﬁed. The selection rule provides usage of operations of the lower or upper roundoff. The generated cipher is represented by the values of the ﬁnite sums.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(4):199-206

On the nilpotent Leibniz–Poisson algebras

#### Abstract

In this article Leibniz and Leibniz–Poisson algebras in terms of correctness of different identities are investigated. We also examine varieties of these algebras. Let $K$ be a base ﬁeld of characteristics zero. It is well known that in this case all information about varieties of linear algebras $V$ contains in its polylinear components $P_n (V )$, $n \in \mathbb N$, where $P_n (V )$ is a linear span of polylinear words of n different letters in a free algebra $K(X, V )$. In this article we give algebra constructions that generate class of nilpotent varieties of Leibniz algebras and also algebra constructions that generate class of nilpotent by Leibniz varieties of Leibniz–Poisson algebras with the identity $\{x_1 , x_2 \} \cdot \{x_3 , x_4 \} = 0$.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(4):207-211

Two problems for three-dimensional space analogue of the third order hyperbolic type equation

#### Abstract

For a complete hyperbolic equation of the third order with variable coefficients in the inﬁnite rectangle the problem with two integral conditions and conjugation on the characteristic plane (Problem I) is considered. As auxiliary Darboux problem is solved by Riemann method which is much simpliﬁed by the special presentation of one of the boundary conditions. Taking Darboux problem as a basis for the solution, authors reduce the Problem I to the uniquely solvable integral equation, which gives an explicit solution to the Problem I.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(4):212-217

Cauchy problem for the hyperbolic system with mixed derivatives

#### Abstract

Cauchy problem for the hyperbolic system with mixed derivative is considered. The given system is transformed to the triangular or diagonal form for the further equations separation. The Cauchy problem for each equation (homogeneous or inhomogeneous) is obtained.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(4):218-221

Mathematical simulation of the building – foundaition – soil systems

#### Abstract

This work addresses the problems of the deformed basis layer thickness assigning in the mathematical modeling of the building – foundation – soil systems on large foundation plates. It also contains the comparison of deformed soil layer thickness computation results for two methods: numerical and analytical.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(4):222-226

Boundary value problems solving method with the implicit use of the Taylor expansions

#### Abstract

Grid method for boundary value problems solving for partial differential equations based on high order Taylor expansions is suggested. Comparison of the proposed method with classical grid method is implemented. It is shown that the use of the Taylor expansion with speciﬁed partial differential equations allows to reduce the estimated faulty proportion of the numerical solution for a given constant sampling area by increasing the order of the expansion. A number of model boundary value problems is solved, the results of the estimated faulty proportion are given.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2012;16(4):227-232