# Vol 19, No 3 (2015)

**Year:**2015**Articles:**12**URL:**https://journals.eco-vector.com/1991-8615/issue/view/1217

### Oleg Igorevich Marichev (On the occasion of his 70th birthday)

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(3):407-414

407-414

### The Ising model with long-range interactions

#### Abstract

The phase transition in the two-dimensional and three-dimensional Ising models with long-range spin interactions are studied with the Monte-Carlo method. The interaction region between spins is characterized by the radius $R$. Results based on numerical simulations have shown the critical temperature $T_c$ dependence from the spin interaction radius $R$. Analytical function $T_c (R)$ approximating this dependence is designed.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(3):415-424

415-424

### Gauge group contraction of electroweak model and its natural energy limits

#### Abstract

The low and higher energy limits of the Electroweak Model are obtained from ﬁrst principles of gauge theory. Both limits are given by the same contraction of the gauge group, but for the diﬀerent consistent rescalings of the ﬁeld space. Mathematical contraction parameter in both cases is interpreted as energy. The very weak neutrino-matter interaction is explained by zero tending contraction parameter, which depends on neutrino energy. The second consistent rescaling corresponds to the higher energy limit of the Electroweak Model. At the inﬁnite energy all particles lose masses, electroweak interactions become long-range and are mediated by the neutral currents. The limit model represents the development of the early Universe from the Big Bang up to the end of the ﬁrst second.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(3):425-440

425-440

### D-meson production at large hadron collider in the regge limit of QCD

#### Abstract

We study the inclusive production of $D^0$, $D^+$, $D^{\star+}$, and $D_s^+$ mesons in proton-antiproton collisions at the Tevatron and in proton-proton collisions at LHC at leading order in the parton Reggeization approach endowed with universal fragmentation functions fitted to $e^+e^-$ annihilation data from CERN LEP1. We have described $D$-meson transverse momentum distributions measured in the central region of rapidity by the CDF Collaboration at Tevatron ($|y|<1$) and ALICE Collaboration at LHC ($|y|<0.5$) within uncertainties and without free parameters, using Kimber-Martin-Ryskin unintegrated gluon distribution function in a proton. The $2\to1$ hard subprocess of gluon production via a fusion of two Reggeized gluons in the PRA framework is proposed for the first time in the case of $D$-meson fragmentation production and proved to be a dominant one. We found our results for $D$-meson central-rapidity production are in the good agreement with experimental data from the LHC and with large-transverse-momenta Tevatron data. The achieved degree of agreement for the LHC exceeds the one obtained by NLO calculations in the conventional collinear parton model and previous LO calculations in $k_T$-factorization without taking into account the $2\to1$ subprocess. The predictions for the $D$-meson production in the central rapidity region for the expected LHC energy of $\sqrt S=14$ TeV are also presented.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(3):441-461

441-461

### The complete solution of the Yang-Mills equations for centrally symmetric metric in the presence of electromagnetic field

#### Abstract

Previously, we found the complete solution of Yang-Mills equations for a centrally symmetric metric in 4-dimensional space of conformal torsion-free connection in the absence of the electromagnetic ﬁeld. Later, in another article, we found a solution of the Yang-Mills equations for the same metric in the presence of an electromagnetic ﬁeld of a special type, suggesting that its components depend not on the four, but only on two variables. There we compared the resulting solutions with the well-known Reissner-Nordstrom solution and indicated the reason why these solutions do not match. In this paper, we do not impose any prior restrictions on the components of the electromagnetic ﬁeld. This greatly complicates the derivation of the Yang-Mills equations. However, all computational diffculties were overcome. It turned out that the solutions of these equations all the same depend only on two variables and new solutions, in addition to previously obtained, do not arise. Consequently, we have found all the solutions of the Yang-Mills equations for a centrally symmetric metric in the presence of an arbitrary electromagnetic ﬁeld, agreed with the Yang-Mills equations in the torsionfree space (i.e., without sources). These solutions are expressed in terms of the Weierstrass elliptic function.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(3):462-473

462-473

### Hyperfine structure of S-states of muonic deuterium

#### Abstract

On the basis of quasipotential method in quantum electrodynamics we calculate corrections of order $\alpha^5$ and $\alpha^6$ to hyperfine structure of $S$-wave energy levels of muonic deuterium. Relativistic corrections, effects of vacuum polarization in first, second and third orders of perturbation theory, nuclear structure and recoil corrections are taken into account. The obtained numerical values of hyperfine splitting $\Delta E^{hfs}(1S)=50.2814$ meV ($1S$ state) and $\Delta E^{hfs}(2S)=6.2804$~meV ($2S$ state) represent reliable estimate for a comparison with forthcoming experimental data of CREMA collaboration. The hyperfine structure interval $\Delta_{12}=8\Delta E^{hfs}(2S)-\Delta E^{hfs}(1S)={-0.0379}$~meV can be used for precision check of quantum electrodynamics prediction for muonic deuterium.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(3):474-488

474-488

### Discrete and continuous cases for the problem of propagating waves for inhomogeneous medium with memory

#### Abstract

The article is devoted to the study of the wave equation for medium with memory. This equation is obtained in the process of considering the homogenized models of combined mediums. It describes one-dimensional case of the Kelvin-Voight’s viscoelastic oscillations law of homogenized models. The problem is to ﬁnd the function which describes the average offset of the material. The formula of propagating waves is used for this purpose. It allows to construct a solution using the general solution of the ﬁrst order system in which each equation is the equation of the transfer along the corresponding characteristics. The main result consists of two theorems for discrete and continuous modiﬁcation of the equation. Furthermore the article contains descriptive considerations which lead to the construction of the classical solution of the equations.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(3):489-503

489-503

### Residual stresses relaxation in surface-hardened half-space under creep conditions

#### Abstract

We developed the method for solving the problem of residual stresses relaxation in surface-hardened layer of half-space under creep conditions. At the ﬁrst stage we made the reconstruction of stress-strain state in half-space after plastic surface hardening procedure based on partial information about distribution for one residual stress tensor component experimentally detected. At the second stage using a numerical method we solve the problem of relaxation of self-balanced residual stresses under creep conditions. To solve this problem we introduce the following Cartesian system: x0y plane is aligned with hardened surface of half-space and 0z axis is directed to the depth of hardened layer. We also introduce the hypotheses of plane sections parallel to x0z and y0z planes. Detailed analysis of the problem has been done. Comparison of the calculated data with the corresponding test data was made for plane specimens (rectangular parallelepipeds) made of EP742 alloy during T = 650 °C after the ultrasonic hardening with four hardening modes. We use half-space to model these specimens because penetration's depth of residual stresses is less than specimen general size in two digit exponent. There is enough correspondence of experimental and calculated data. It is shown that there is a decay (in modulus) of pressing residual stresses under creep in 1.4-1.6 times.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(3):504-522

504-522

### Solution of 3D heat conduction equations using the discontinuous Galerkin method on unstructured grids

#### Abstract

The discontinuous Galerkin method with discontinuous basic functions which is characterized by a high order of accuracy of the obtained solution is now widely used. In this paper a new way of approximation of diffusion terms for discontinuous Galerkin method for solving diffusion-type equations is proposed. The method uses piecewise polynomials that are continuous on a macroelement surrounding the nodes in the unstructured mesh but discontinuous between the macroelements. In the proposed numerical scheme the spaced grid is used. On one grid an approximation of the unknown quantity is considered, on the other is the approximation of additional variables. Additional variables are components of the heat ﬂux. For the numerical experiment the initial-boundary problem for three-dimensional heat conduction equation is chosen. Calculations of three-dimensional modeling problems including explosive factors show a good accuracy of offered method.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(3):523-533

523-533

### Mathematical foundations of epistemology based on experiments

#### Abstract

The paper deals with basic prerequisites for the development of epistemology, which uses information concerning real experiments in the real world (with real objects). Such experiments are conducted by “formal-technological” analogs of Turing Machines. These analogs are called “universal synthesizersanalyzers”. They can perform syntheses and analyses of various objects or constructions (obtained by conjunctions of ﬁnite number of smaller objects called basic elements) with the help of various algorithmic systems having some restrictions. Such algorithmic systems are called Formal Technologies. They have formal structures that are very similar to the formal structure of Maltsev’s algebraic systems. This formal closeness allows us, ﬁrst, to set up a hypothesis concerning algorithmic basis of almost all surrounding physical processes, as understandable as well as till non-understandable ones, that partially explains the wide applicability of mathematics to the outer world; second, this closeness allows one to formulate and prove some theorems (called assertions) concerning features and peculiar properties of cognitive algorithms in one-, two- or three-dimensional surroundings for various formal technological systems, including a so called “acquired knowledge effectiveness theorem”. The theorem (assertion) can be applied to a very wide class of formal technologies which use an equality predicate for objects analyses. In the paper various cognitive algorithms are listed and proved. These algorithms have different sets of technological operations resembling syntheses and decompositions, as well as different sets of analytical operations including equality predicates, “random stationary mapping” operations (which use unknown algorithms to obtain stationary results, therefore these operations are very similar to oracles in Turing Machines), operations that deﬁne object shapes, and so on. The structure of automatic cognitive devices called “cognizers” is described.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(3):534-558

534-558

### Convergence of the matrix method of numerical integration of the boundary value problems for linear nonhomogeneous ordinary differential second order equations with variable coefficients

#### Abstract

The problems of stability and convergence of previously proposed matrix method of numerical integration of boundary value problems with boundary conditions of the ﬁrst, second and third kinds of nonhomogeneous linear ordinary differential second order equations with variable coeffcients are considered. Using of the Taylor polynomials of arbitrary degrees allowed to increase the approximation order of the method to an arbitrary natural number and to refuse from the approximation of derivatives by ﬁnite differences. When choosing the second degree Taylor polynomials the equation of the method coincided with the known equations of the traditional method of numerical integration of the boundary value problems where the derivatives are approximated by ﬁnite differences. It was shown that a suffcient criterion of stability when used in the method of Taylor polynomials of the third degree and more coincides with the suffcient criterion of stability of the traditional grid method for the numerical integration of boundary value problems with boundary conditions of the ﬁrst, second and third kind. Theoretically, it is established that the degree of convergence of the matrix method for integration of boundary value problems with boundary conditions of the ﬁrst kind is proportional to the degree of the used Taylor polynomials in the case, when the degree is even, and is proportional to the number that is one less than the degree if it is odd; when integrating the boundary value problems with boundary conditions of the second and third kind the degree of convergence of the method is proportional to the degree of the used Taylor polynomials regardless of its parity and one less than it. The obtained theoretical results are conﬁrmed by numerical experiments.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(3):559-577

559-577

### Method of additional boundary conditions in the problem of heat transfer for non-newtonian fluid moving in laminar mode in circular pipe

#### Abstract

Taking into account the dissipation of mechanical energy the problem of heat transfer is formulated for non-Newtonian ﬂuid moving in stable laminar mode in circular pipe. Two variants were considered: 1) non-stationary problem taking into account the diffusion component of heat transfer along the pipe; 2) the stationary problem without taking into account the longitudinal diffusion component of heat transfer in ﬂuids. The synthesis of method of initial functions and method of complementary boundary conditions were used for the approximate solving of problems that was possible to reduce by one the dimensionality of the problem by spatial variables. In the stationary case, due to the additional boundary conditions it was able to obtain a higher degree of approximation of the temperature ﬁeld than in the nonstationary case. Different methods of approximation of boundary conditions for the temperature of the liquid were studied at the entrance to the pipe with coordination and without coordination for the wall temperature. Calculations of temperature ﬁelds were conducted for melting of high-pressure polyethylene in accounting and neglect of dissipation of mechanical energy in the polymer. Comparison with calculations on the basis of other approximate method, previously developed, different from the one proposed in this study, was performed.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2015;19(3):578-600

578-600