# Vol 18, No 2 (2014)

**Year:**2014**Articles:**19**URL:**https://journals.eco-vector.com/1991-8615/issue/view/1244

### On Leibniz-Poisson special polynomial identities

#### Abstract

In this paper we study Leibniz-Poisson algebras satisfying polynomial identities. We study Leibniz-Poisson special and Leibniz-Poisson extended special polynomials. We show that the sequence of codimensions $\{r_n({\bf V})\}_{n\geq 1}$ of every extended special space of variety ${\bf V}$ of Leibniz-Poisson algebras over an arbitrary field is either bounded by a polynomial or at least exponential. Furthermore, if this sequence is bounded by polynomial then there is a polynomial $R(x)$ with rational coefficients such that $r_n({\bf V}) = R(n)$ for all sufficiently large n. It follows that there exists no variety of Leibniz-Poisson algebras with intermediate growth of the sequence $\{r_n({\bf V})\}_{n\geq 1}$ between polynomial and exponential. We present lower and upper bounds for the polynomials $R(x)$ of an arbitrary fixed degree.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(2):9-15

9-15

### On the lowest by $x$-variable terms influence on the spectral properties of dirichlet problem for the hyperbolic systems

#### Abstract

We made the comparison study and characterize the spectral properties of differential operators induced by the Dirichlet problem for the hyperbolic system without the lowest terms of the form $$ \cfrac{\partial^2{u^1}}{\partial{t}^2}+\cfrac{\partial^2{u^2}}{\partial{x}^2} = \lambda{u^1}+f^1, \quad \cfrac{\partial^2{u^2}}{\partial{t}^2}+\cfrac{\partial^2{u^1}}{\partial{x}^2} = \lambda{u^2}+ f^2, $$ and for the hyperbolic system with the lowest terms of the form $$ \cfrac{\partial^2{u^1}}{\partial{t}^2}+\cfrac{\partial^2{u^2}}{\partial{x}^2}+\cfrac{\partial{u^2}}{\partial{x}} =\lambda{u^1}+f^1, \quad \cfrac{\partial^2{u^2}}{\partial{t}^2}+\cfrac{\partial^2{u^1}}{\partial{x}^2}+\cfrac{\partial{u^1}}{\partial{x}} = \lambda{u^2}+ f^2, $$, which are considered in the closure $V_{t,x}$ of the bounded domain $\Omega_{t,x}=(0;\pi)^2$ in Euclidean space $\mathbb{R}^2_{t,x}.$ The spectral properties of the boundary value problems for the systems of linear differential equations of the hyperbolic type are investigated in the Hilbert space $\mathcal{H}_{t,x}$ in the terms of spectral closed operators $L:\mathcal{H}_{t,x}\to\mathcal{H}_{t,x}$. We study the spectra of the closed differential operators $L:\mathcal{H}_{t,x}\to\mathcal{H}_{t,x},$ induced by the Dirichlet problem for the second order hyperbolic systems: $C\sigma{L}=R\sigma{L}$ - empty set; point spectrum $P\sigma{L}$ is in the real straight line of the complex plane $\mathbb{C}$. The operator $L$ eigen vector functions generate the orthogonal basis for the hyperbolic system without the lowest terms. For the hyperbolic system with the lowest terms the operator $L$ eigen vector functions generate the Riesz basis, nonorthogonal in the Hilbert space $\mathcal{H}_{t,x}.$ The theorems on the structure of the induced by the Dirichlet problem operator $L$ spectrum $\sigma L$ are formulated.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(2):16-21

16-21

### Boundary value problem with shift for one partial differential equation containing partial fractional derivative

#### Abstract

We investigate a nonlocal boundary value problem for the equation of special type. For $y > 0$ it is the equation of fractional diffusion, which contains partial fractional derivative of Riemann-Liouville. For $y < 0$ it is the hyperbolic type equation with two perpendicular lines of degeneracy. The conditions of existence and uniqueness of the solution of the boundary value problem are formulated. The uniqueness of the solution of the problem is proved using the extremum principle and the use of generalized operator of fractional integro-differential in M. Saygo sense. The existence of a solution is reduced to the solvability of differential equations of fractional order, which solution is written out explicitly.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(2):22-32

22-32

### Problems of optimal and hard control over solutions of special type of nonstationary Sobolev type equations

#### Abstract

Sobolev type equations now constitute a vast area of nonclassical equations of mathematical physics. Those called nonclassical equations of mathematical physics, whose representation in the form of equations or systems of equations partial does not ﬁt within one of the classical types (elliptic, parabolic or hyperbolic). In this paper we prove the existence of a unique optimal and hard control over solutions of Showalter-Sidorov problem for nonstationary operator-differential equations unresolved with respect to the time derivative. In this case, one of the operators in the equation is multiplied by a scalar function of the time-variable, besades stationary equation has a strong continuous degenerate resolving semigroup of operators. Apart from the introduction and bibliography article comprises two parts. The ﬁrst part provides the necessary information regarding the theory of p-radial operators, the second contains the proof of main results of this article.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(2):33-38

33-38

### A double inverse problem for Fredholm integro-differential equation of elliptic type

#### Abstract

In this paper the double inverse problem for partial differential equations is considered. The method of studying the one value solvability of the double inverse problem for a Fredholm integro-differential equation of elliptic type with degenerate kernel is offered. First the method of degenerate kernel designed for Fredholm integral equations is modiﬁed and developed to the case of Fredholm integro-differential equation of elliptic type. The system of differential-algebraic equations is obtained. The inverse problem is called double inverse problem if the problem consisted to restore the two unknown functions by the aid of given additional conditions. The ﬁrst restore function is nonlinear with respect to the second restore function. In solving the inverse problem with respect to the ﬁrst restore function the inhomogeneous differential equation of the second order is obtained, which is solved by the method of variation of arbitrary constants with initial value conditions. With respect to the second restore function the nonlinear integral equation of the ﬁrst kind is obtained, which is reduced by the aid of special nonclassical integral transform into nonlinear Volterra integral equation of the second kind. Further the method of successive approximations is used, combined with the method of compressing maps.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(2):39-49

39-49

### Investigations of the numerical range of a operator matrix

#### Abstract

We consider a $2\times2$ operator matrix $A$ (so-called generalized Friedrichs model) associated with a system of at most two quantum particles on ${\rm d}$-dimensional lattice. This operator matrix acts in the direct sum of zero- and one-particle subspaces of a Fock space. We investigate the structure of the closure of the numerical range $W(A)$ of this operator in detail by terms of its matrix entries for all dimensions of the torus ${\bf T}^{\rm d}$. Moreover, we study the cases when the set $W(A)$ is closed and give necessary and sufficient conditions under which the spectrum of $A$ coincides with its numerical range.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(2):50-63

50-63

### The Stress-strain State of the Rubber-metall Seismic Bearing

#### Abstract

This work is devoted to elaboration of ﬁnite element approach for the numerical analysis of parameters of the stress-strain state of the rubber-metal seismic bearing under viscoelastic deformation in the presence of layers of porous rubber. Elastic characteristics of porous rubber were determined by self-consistency method for the spherical pores. The integral relations on the basis of Boltzmann-Volterra hereditary theory have been used for viscoelastic behavior modeling. The exponential core containing instant and long elastic characteristics of the material has been used as core of relaxation. The ﬁnite element model of deforming the construction with spatial discretization and time discretization was built on the basis of the variational principle. The resulting system of resolving equations contains the additional load vector modeling the rheological constituents of the deformation process; a modiﬁed Newton-Kantorovich method has been used to solve this system. For increasing the accuracy of numerical results the precise ﬁnite element moment scheme with cubic approximation of displacements has been applied. The numerical convergence of the ﬁnite element schemes has been studied on the example of solution of Lame problem for hollow viscoelastic cylinder made of porous rubber. The rubber-metal seismic bearing was calculated on the assumption of the relaxation of the shift module of porous rubber only. The basic parameters of the stress-strain state have been obtained depending on the time and the applicable stamps of rubber.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(2):64-81

64-81

### Potential fields of free energy at the stages of hardening and softening of the Hencky medium at nonpositivity of volume deformation

#### Abstract

The Hencky medium with softening under isothermal and quasistatic deformation is considered. It is believed that volume deformation is not positive. In this case softening is characterized by the part of union curve with negative slope. For aforementioned conditions function of free energy is presented. For all stages of deformation in the space “volume deformation - intensity of shear’s deformation” level lines of the free energy are constructed. It is established that level lines are ellipses in hardening and function of free energy increases with distance from their centers, while in softening hyperbolas are level lines and function of free energy decreases with distance from their centers. Obtained results indirectly conﬁrm the change in type of boundary value problem from elliptic to hyperbolic under material transition to the softening.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(2):82-88

82-88

### A longitudinal stability of a ribbed cover in a multimodulus elastic medium

#### Abstract

The stability of a longitudinal compressed hinge-supported cylindrical cover stiffened by stringers and located on the border of two Winkler’s ambiences is considered. The derivation of the equations was carried out under the assumptions: using a simpliﬁed theory of Donnell-Vlasov, axisymmetric deformation of a cover, only normal load acts on the shell. The problem is solved using a combined exhaustive search algorithm. This method includes full and local search of variants to search a form deﬂection and a critical force. Full search of variants is required to construct a form deﬂection of a shell. Local search of variants is necessary to clarify a critical force. As a result of numerical experiments we found out that increasing the number of stringers reinforces the shell. These results are consistent with the results obtained in the other works.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(2):89-95

89-95

### Boundary integral equation method in the modeling of nonlinear deformation and failure of the 3D inhomogeneous media

#### Abstract

The method of boundary integral equations is applied for solving the nonlinear problems of thermo-elastic-plastic deformation and fracture of inhomogeneous 3D bodies of the complex form. Solution is constructed on the basis of the generalized identity of Somigliana involving method of sequential linearization in the form of initial plastic deformations. The increments of plastic deformation are determined on the basis of the ﬂow theory of hardening elastoplastic media with the use of modifed Prandtl-Reus's relations. The cases of complex thermo mechanical loading of compound piecewise homogeneous media in contact are considered. For describing the processes of nonlinear deformation and fracture of the bodies with a complex geometry and local peculiarities a method of discrete domains (DDBIEM) is developed. The solutions of some practical signiﬁcant 3D non-linear problems of the mechanics of deformation and fracture are presented.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(2):96-114

96-114

### Creep theory inverse problem for non-work-hardening body

#### Abstract

The body formation by constant external forces in the conditions of the steady-state creep during set time problem is formulated and solved so that after removal of loadings the movements of points of a surface accepted preset values. The case of small deformations is considered. At certain assumptions and restrictions the uniqueness theorem for the solution of this task is proved. Applied questions of a problem of ﬁnding the external inﬂuences which are necessary for receiving a demanded shape of a body for set time in the conditions of rheological deformation after removal of external forces (taking into account elastic unloading) are analyzed. The analysis of a thin-walled isotropic plate for a case of a ﬂat tension is made in details. The solution for movements is searched in the form of an expansion in small parameter. The model solution for a round plate of single radius under the inﬂuence of constant external loadings which should have the set ﬁeld of movements after creep and elastic unloading is provided.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(2):115-124

115-124

### Simulation of the process of entrainment of a powder particles by explosive shock waves

#### Abstract

The coating on the surface of metals and alloys is used to increase strength and durability of materials. There are many different technological schemes of this process. It is of interest to use the explosion energy to create a stream of particles deposited on the surface of metals. The article is based on a simple physical model describing the process of interaction between the products of detonation of the explosive substance with the particles of a powder in an explosive spraying of wear-resistant coatings. The entrainment of particles occurs due to inelastic collisions of molecules of detonation products with particles of a powder. An equation for determining the velocity of particles in the wave front, formed during the explosion of a spherical explosive charge, is received. The equation of motion of the particles can be written for the case of an explosion of a cylindrical charge of explosive. The solving algorithms of the obtained equation are analyzed depending on the dynamic characteristics of detonation products. The obtained results can be used for designing the technological schemes of explosive deposition and making the theoretical analysis of the process of superdeep penetration of particles of a powder in a metallic target.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(2):125-129

125-129

### Analytical solutions of the quasistatic thermoelasticity task with variable physical properties of a medium

#### Abstract

The high-precision approximate analytic solution of the nonlinear quasi-static problem of thermoelasticity for an inﬁnite hollow cylinder with variable along the radial coordinate physical properties is obtained using the orthogonal Bubnov-Galyorkin method developed by the construction of systems of coordinate functions exactly satisfying inhomogeneous boundary conditions in any approximation. The mathematical formulation includes non-linear equations for the unknown function of displacement and inhomogeneous boundary conditions. The desired solution is supposed to precisely satisfy the boundary conditions in advance. The exact fulﬁllment of the boundary conditions is achieved using the coordinate functions of special design. The unknown coefficients are found by constructing the disparity of original differential equation, that should be orthogonal to all the coordinate functions. Hence, the unknown coefficients of solution yields a system of linear algebraic equations, which number is equal to the number ofapproximations of the solution. It is shown that the solution accuracy increases substantially with increasing the number of approximations. Thus, already in the ninth approximation the disparity of original differential equation is zero almost the entire range of the spatial variable. The maximum disparity in the sixth approximation is $\varepsilon = 5\cdot 10^{-4}$.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(2):130-135

130-135

### Solving the classification problem by using neural fuzzy production based network models of Mamdani-Zadeh

#### Abstract

The article considers solving the problem of object recognition of intersected classes using fuzzy inference systems and neural networks. New multi-output network of Wang-Mendel is compared to a new architecture of neural fuzzy production network based on the model of Mamdani-Zadeh. Learning results of these models are given in the interpretation of logical operations provided by Godel, Goguen and Lukasiewicz algebras. New Wang-Mendel’s network can use minimum or sum-based formula as T-norm operation in accordance with an appropriate algebra rather than the standard multiplication only. Mamdani-Zadeh's network is designed as a cascade of T-norm, implication and S-norm operations deﬁned by selected algebra. Moreover defuzziﬁcation layer is not presented in Mamdani-Zadeh’s network. Both networks have several outputs in accordance with the number of subject area classes what differs them from the basic realizations. Compliance degrees of an input vector to deﬁned classes are formed at the network outputs. To compare the models the standard Fisher’s irises and Italian wines classiﬁcation problems were used. This article presents the results calculated by training the networks by backpropagation algorithm. Classiﬁcation error analysis shows that the use of these algebras as interpreting fuzzy logic operations proposed in this paper can reduce the classiﬁcation error for both multi-output network of Wang-Mendel and a new network of Mamdani-Zadeh. The best learning results are shown by Godel algebra, but Lukasiewicz algebra demonstrates better generalizing properties while testing, what leads to a less number of classiﬁcation errors.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(2):136-148

136-148

### Influence of dislocation density of nickel on dissolution kinetics in acidic chloride electrolyte

#### Abstract

An inﬂuence of dislocation density of nickel’s anode on the density of anode current and the homogeneity of anode's dissolution along the surface in the acidic chloride electrolyte was studied. To create the dislocation density of about 109 cm -2, nickel was annealed at the temperature of 900 °С for 0.5 hour. To raise the dislocation density up to 1010 cm -2, nickel was deformed by 15 % through forging. It was detected that an increase of dislocation density of one order of magnitude enlarged the density of anode current by several times over. An electrochemical etching of annealed nickel was occurring fairly even along the surface of sample revealing the well-formed grain structure. Dissolution of deformed nickel was uneven along the surface, and the grain structure was not discovered.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(2):149-155

149-155

### Optimization of high-speed steels heat-resistance by volume and laser heat-treatments combination

#### Abstract

The peculiarities of the phase transformations by a pulse laser hardening in a solid state and from a melt, leading to the higher wear-and-heat-resistance of a laser inﬂuence zones, were studied for the cobaltic high-speed steel with a different initial structural-and-phase composition, created by the preliminary volume heat-treatment. The practical conclusions of this article show that laser treatment of a cutting instrument must be done with a short melting of the working edges surface for a guarantee of its wear-resistance higher increase. An instrument work after the laser hardening must be realized on heightened cutting speeds for the lowering of efficiency of weakening processes on the initial stage of its working edges heat.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(2):156-167

156-167

### Magnetostatic modes in tangentially magnetized ferrospinels films

#### Abstract

Absorption spectra of ferrospinel ﬁlms synthesized by the method of chemical transport reactions are considered. Using an EPR spectrometer, it is demonstrated that additional absorption peaks are recorded in the absorption spectra of manganese and magnesium-manganese ferrite ﬁlms magnetized parallel to their surface. Interpreting these peaks as magnetostatic modes, the wave number, group velocity, and extinction coefficient are retrieved from experimental values of the saturation magnetization, crystallographic anisotropy constant, and magnetic ﬁeld of the observed modes. The common effects typical of the ferrospinel ﬁlms are frequency oscillations of the surface mode extinction. Dependences of the extinction oscillation pattern on the surface anisotropy and wavelength of magnetic moment oscillations are established.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(2):168-179

168-179

### Gauge-invariant tensors of 4-manifold with conformal torsion-free connection and their applications for modeling of space-time

#### Abstract

We calculated basic gauge-invariant tensors algebraically expressed through the matrix of conformal curvature. In particular, decomposition of the main tensor into gaugeinvariant irreducible summands consists of 4 terms, one of which is determined by only one scalar. First, this scalar enters the Einstein’s equations with cosmological term as a cosmological scalar. Second, metric being multiplied by this scalar becomes gauge invariant. Third, the geometric point, which is not gauge-invariant, after multiplying by the square root of this scalar becomes gauge-invariant object - a material point. Fourth, the equations of motion of the material point are exactly the same as in the general relativity, which allows us to identify the square root of this scalar with mass. Thus, we obtained an unexpected result: the cosmological scalar coincides with the square of the mass. Fifth, the cosmological scalar allows us to introduce the gauge-invariant 4measure on the manifold. Using this measure, we introduce a new variational principle for the Einstein equations with cosmological term. The matrix of conformal curvature except the components of the main tensor contains other components. We found all basic gauge-invariant tensors, expressed in terms of these components. They are 1- or 3-valent. Einstein’s equations are equivalent to the gauge invariance of one of these covectors. Therefore the conformal connection manifold, where Einstein’s equations are satisﬁed, can be divided into 4 types according to the type of this covector: timelike, spacelike, light-like or zero.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(2):180-198

180-198

### Rising of efficiency of enciphering on the basis of summation of products

#### Abstract

The properties of the code numbers made on the basis of the sums with products of weight and free components are considered. Free components appear here, at ﬁrst, as equal powers of members of an arithmetical progression, secondly, as members of a geometrical progression, and, in the third, as members of sequence of the combined type. Besides, the structure of the speciﬁed properties includes character of a modiﬁcation of relative summarized residuals depending on a modiﬁcation of parameters of considered aspects of sequences. With respect to the introduction of a membership of parameters of considered sequences to set of real numbers the made code number also is characterized by the raised eﬃciency.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2014;18(2):199-207

199-207