Vol 18, No 2 (2014)

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.

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.

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.

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.

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.

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.

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.

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.

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.