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The paper studies the quantum action for the five-dimensional real $\phi^3$-theory in the case of a general formulation using the background field method. The three-loop renormalization is performed with the usage of a cutoff regularization in the coordinate representation. The explicit form of the first three coefficients for the renormalization constants is presented. The absence of non-local singular contributions and partial results for the fourth correction are discussed., Comment: LaTeX, 20 pages, 5 figures. Firstly appeared in Russian, April 9, 2024, see https://www.pdmi.ras.ru/preprint/2024/24-05.html

The paper discusses an applicability criterion for a cutoff regularization in the coordinate representation in the Euclidean space with a dimension larger than two. It is shown that the set of functions satisfying the criterion is not empty. As an example, an explicit function is presented. It is proved by explicit construction that there are functions satisfying the criterion in a stronger formulation., Comment: LaTeX, 8 pages, 1 figure. Firstly appeared in Russian, March 5, 2024. Compared to the Russian version, some comments and minor corrections have been added. The forgotten imaginary unit in the Fourier transform has also been restored. See https://www.pdmi.ras.ru/preprint/2024/24-04.html

The paper studies the quantum action for the four-dimensional real $\phi^4$-theory in the case of a general formulation using the background field method. The three-loop renormalization is performed with the usage of a cutoff regularization in the coordinate representation. The absence of non-local singular contributions and the correctness of the renormalization $\mathcal{R}$-operation on the example of separate three-loop diagrams are also discussed. The explicit form of the first three coefficients for the renormalization constants and for the $\beta$-function is presented. Consistency with previously known results is shown., Comment: LaTeX, 71 pages, 8 figures. Firstly appeared in Russian, February 6, 2024, see https://www.pdmi.ras.ru/preprint/2024/pr2024.html

Formal asymptotics are substantiated that describe typical dropping cusp singularity of quasiclassical approximations to solutions of two cases of the integrable nonlinear Schr\"odinger equation $-i\varepsilon\Psi'_{t}=\varepsilon^2\Psi''_{xx}\pm2|\Psi| ^2\Psi$, where $\varepsilon$ is a small parameter. The substantiation uses the ideology and facts of the mathematical catastrophe theory and the part of the theorem of Yu. F. Korobeinik, concerning analytical as $h\to 0$ solutions $G(h,u)$ of the mixed type linear equation $hG''_{hh}=G''_{uu}$ to which the hodograph images of both cases of the systems of equations of these quasiclassical approximations are equivalent., Comment: In Russian, submitted to Differential Equations

This paper continues a series of papers in which 2-by-2 matrix joint solutions of two scalar evolutionary equations are constructed, which are analogs of the Schrodinger time equations. These equations correspond to the Hamiltonian system $H^{2+2+1}$, being one of the representatives of the hierarchy of degenerations of the isomonodromic Garnier system. The mentioned hierarchy was described by H. Kimura in 1986. In terms of solutions of linear systems of ordinary differential equations of the method of isomnodromic deformations, the compatibility condition of which are the Hamiltonian equations of the system $H^{2+2+1}$, the constructed joint matrix solutions of analogs of the time equations of the Schrodinger equations in this paper will be written out explicitly., Comment: in Russian language

For the Potts model on Cayley trees, a very wide class of new Gibbs measures is given. We give a review of all known Gibbs measures of the Potts model on trees and compare them with our new measures., Comment: 18 pages, in Russian language, 4 figures

In the work we consider a boundary value problem for a second order equation with variable coefficients in a multi-dimensional domain perforated by small cavities closely spaced along a given manifold. We assume that the linear sizes of all cavities are of a same order of smallness, while their shapes and distributions are arbitrary. The boundaries of the cavities are subject to a nonlinear Robin condition. We prove that the solution of the perturbed problem converges to that of the homogenized problem in norm $L_2$ and $W_2^1$ uniformly in $L_2$-norm of the right hand side in the equation. We also establish the estimates for the convergence rates., Comment: The paper is written in Russian

A complete description of two-periodic Gibbs measures on the Cayley tree of orders two and three for HC model with two states is obtained and using the reconstruction method, the extremality of these measures in the area of their existence is proved., Comment: in Russian

We construct the Lagrange-D'Alembert Principle as a pure mathematical theory that meets the modern standards of rigor. We note several new aspects of the Lagrange-D'Alembert principle but the article is mainly methodological by its substance. The article is addressed to specialists in ODE who are not experts in theoretical mechanics and to graduate students that specialize in dynamical systems., Comment: 10 pages in Russian

Appearing in 1921 as an equation for small-amplitude waves on the surface of an infinitely deep liquid, the Nekrasov equation quickly became a source of new results. This manifested itself both in the field of mathematics (theory of nonlinear integral equations of A.I. Nekrasov; 1922, later - of N.N. Nazarov; 1941), and in the field of mechanics (transition to a fluid of finite depth - A.I. Nekrasov; 1927 and refusal on the smallness of the wave amplitude - Yu.P. Krasovskii; 1960).The main task of the author is to find out the prehistory of the Nekrasov equation and to trace the change in approaches to its solution in the context of the nonlinear functional analysis development in the 1940s - 1960s. Close attention will be paid to the contribution of European and Russian mathematicians and mechanics: A.M. Lyapunov, E. Schmidt, T. Levi-Civita, A. Villat, L. Lichtenstein, M.A. Krasnoselskii, N.N. Moiseev, V.V. Pokornyi, etc. In the context of the development of qualitative methods for the Nekrasov equation investigating, the question of the interaction between Voronezh school of nonlinear functional analysis under the guidance of Professor M.A. Krasnoselskii and Rostov school of nonlinear mechanics under the guidance of Professor I.I. Vorovich., Comment: In Russian

For the Langevin model of the dynamics of a Brownian particle with perturbations orthogonal to its current velocity, in a regime when the particle velocity modulus becomes constant, an equation for the characteristic function $\psi (t,\lambda )=M\left[\exp (\lambda ,x(t))/V={\rm v}(0)\right]$ of the position $x(t)$ of the Brownian particle. The obtained results confirm the conclusion that the model of the dynamics of a Brownian particle, which constructed on the basis of an unconventional physical interpretation of the Langevin equations, i. e. stochastic equations with orthogonal influences, leads to the interpretation of an ensemble of Brownian particles as a system with wave properties. These results are consistent with the previously obtained conclusions that, with a certain agreement of the coefficients in the original stochastic equation, for small random influences and friction, the Langevin equations lead to a description of the probability density of the position of a particle based on wave equations. For large random influences and friction, the probability density is a solution to the diffusion equation, with a diffusion coefficient that is lower than in the classical diffusion model., Comment: 11 pages, preprint, in Russian

We consider fertile three-state hard core models with the activity parameter on an order-three Cayley tree. It is known that there exist four types of such models: in two of them the regions of extremality of the unique translation-invariant Gibbs measure were found., Comment: 13 pages, in Russian, 7 figures

It is declared that the aim of simplifying representations of coefficients of power series of classical statistical mechanics is to simplify a process of obtaining estimates of the coefficients using their simplified representations. The aim of the article is: to formulate criteria for the complexity (from the above point of view) of representations of coefficients of the power series of classical statistical mechanics and to demonstrate their application by examples of comparing the Ree-Hoover representations of virial coefficients (briefly -- the RH representations) with such representations of power series coefficients that are based on the conception of the frame classification of labeled graphs (the abbreviation -- FC). To solve these problems, mathematical notions were introduced (such as a basic product, a basic integral, a basic linear combination, a basic linear combination with coefficients of insignificant complexity(the abbreviation -- BLC with CIC) and the classification of representations of the coefficients of power series of classical statistical mechanics is proposed. In the classification, the class of BLC's with CIC is the most important. It includes all the above representations of the coefficients of power series of classical statistical mechanics. Three criteria are formulated for estimate the comparative complexity of BLC's with CIC. These criteria are ordered by their accuracy. Based on each of these criteria, a criterion for the comparative complexity of finite sets of BLC's with CIC is constructed. The constructed criteria are ordered by their accuracy. The application of all the constructed criteria is demonstrated by examples of comparing RH representations with the representations of the power series coefficients based on the concept FC. The obtained results are presented in the tables and commented., Comment: 42 pages, in Russian, 6 tables

In a wide class of propagators regularized by the $\varepsilon$-metric [1], the $R$-operation is formulated. It is proved that the limit of renormalized Feynman integrals exists and is covariant. Possible applications in gravity are discussed. (The paper is an English translation of the second of two articles in Russian published by the author in 1987-88: V.D. Ivashchuk, Regularization by $\varepsilon$-metric. II. Limit $\varepsilon = +0$, Izvestiya Akademii Nauk Moldavskoy SSR, Ser. fiziko-tekhnicheskih i matematicheskih nauk, No. 1, p. 10-20 (1988) [in Russian] .), Comment: 18 pages + 2 pages of additional references, MS WORD. The paper is an English translation of the second of two articles in Russian published by the author in 1987-88

In this paper we study Hard-Core model on a Cayley tree. For a normal divisor of index four new conditions for uniqueness and non-uniqueness of weakly periodic Gibbs measures are found., Comment: in Russian

We study random coloring of the hexagons of a honeycomb lattice into $2^{n-1}$ colors (that is the standard Potts model at infinite temperature). It may be considered as a generalization of percolation to $n$ pairwise independent, but mutually dependent liquids. We introduce a new observable that can be interpreted as the fraction of percolated liquids. An analogue of the central limit theorem for this observable is proved and several conjectures based on numeric experiments are proposed., Comment: In Russian, 10 pages, 5 figures

The one-dimensional Boltzmann equation $f_{t} + cf_{x} + (\mathcal {F} f)_{c} = 0$ is considered with the function $\mathcal{F}$ depending on $(t, x, c, f)$. In this paper we obtain a complete group classification of such equations in the class of point transformations of the entire set of variables $(t, x, c, f)$ under the conditions of invariance of a) the relations $dx = cdt$, $dc = \mathcal {F} dt$, b) the lines $ dt = dx = 0 $ and c) the form $fdxdc$ fixing the physical meaning of the variables and the relationships between them., Comment: in Russian

The paper derives the representation of the two-particle T-matrix scattering elements for the Coulomb interaction with respect to special bases without expansion in terms of partial waves. The results obtained are applicable to small-particle systems. The advantage of this expansion also arises in three-body problems when solving the Faddeev equation for three-particle systems. The main problem in solving the Faddeev equation is the approximate choice of approximation for the interaction potentials, at which the T-matrix scattering elements acquire a separable form. However, even in this case the solution to the Faddeev equation does not always become practical in view of the fact that the T-matrix elements themselves do not factor in the integral equations. Here we give the results with the T-matrix elements represented in the basis, for which there is an addition theorem and hence the integral Faddeev equations are reduced to a factored form., Comment: 9 pages, in Russian

Starting from the classical results of Shubnikov and Zamorzayev, computer models of shapes are implemented, which allow to visualize the action of discrete subgroups of continuous topological groups. The action is visualize by performing partitions of the shapes into the fundamental domains of the discrete symmetry groups of similarity, the definition of which was given by Zamorzaev. Particular attention is paid to the models of quasilattice, with the help of which such tiling of figures are constructed, that their multicolored coloring allows us to investigate, including the action of colored groups of symmetry of similarity. It is shown that for two-dimensional quasicrystals (quasi-lattices) the homothety coefficients are integers algebraic numbers of the quadratic expansion of the field of rational numbers. Some conformal mappings of crystal sets have been studied, which made it possible to establish the correspondence between the stereocyclic projection and the coordinate system of inverse radii., Comment: (In Russian) 38 pages, 15 figures, 19 references

The regularization of propagators by means of a complex metric is considered. (The paper is an English translation of the first of two articles in Russian published by the author in 1987-88: V.D. Ivashchuk, Regularization by {\epsilon}-metric. I, Izvestiya Akademii Nauk Moldavskoy SSR, Ser. Fiziko-tekhnicheskih i matematicheskih nauk, No 3, p. 8-17 (1987) [in Russian] .), Comment: 15 pages, 1 figure, MS WORD, The paper is an English translation of the article: V.D. Ivashchuk, Regularization by {\epsilon}-metric. I, Izv. Akad. Nauk Mold. SSR, Ser. fiz.-tekhn. i mat. nauk, No 3, p. 8-17 (1987) [in Russian]

In this paper for the Potts-SOS model on a Cayley tree under some conditions the existence of at least one periodic (non translation-invariant) Gibbs measure is proved., Comment: in Russian

The author studies structure of space $\mathbf{L}_{2}(G)$ of vectors - functions, which are integrable with a square of the module on the bounded domain $G $of three-dimensional space with smooth boundary, and role of the gradient of divergence and curl operators in construction of bases in its orthogonal subspaces $\mathcal{A}$ and $\mathcal{B}$. The ${\mathcal{A}}$ and ${\mathcal{B}}$ are contain subspaces ${\mathcal{A}_{\gamma}}(G)\subset{\mathcal{A}}$ and $\mathbf{V}^{0}(G)\subset{\mathcal{B}}$. The gradient of divergence and a curl operators have continuations in these subspaces, their expansion $\mathcal{N}_d$ and $S$ are selfadjoint and convertible,and their inverse operators $\mathcal{N}_{d}^{-1}$ and $S^{-1}$ are compact. In each of these subspaces we build ortonormal basis. Uniting these bases, we receive complete ortonormal basis of whole space $\mathbf{L}_{2}(G)$, made from eigenfunctions of the gradient of divergence and curl operators . In a case, when the domain $G$ is a ball $B$, basic functions are defined by elementary functions. The spaces $\mathcal{A}^{s}_{\mathcal{K}}(B)$ are defined. Is proved, that condition $ \mathbf{v}\in\mathcal{A}^{s}_{\mathcal {K}}(B)$ is necessary and sufficient for convergence of its Fourier series (on eigenfunctions of a gradient of divergence)in norm of Sobolev space $ \mathbf{H}^{s}(B)$. Using Fourier series of functions $ \mathbf{f}$ and $ \mathbf{u}$, the author investigates solvability(in spaces $\mathbf{H}^{s}(G)$)boundary value problem: $\nabla\mathbf{div}\mathbf{u}+ \lambda\mathbf{u}=\mathbf{f}$ in $G$, $\mathbf{n}\cdot\mathbf{u}|_{\Gamma}=g$ on boundary, under condition of $\lambda\neq0$. In a ball $B$ a boundary value problem: $\nabla\mathbf{div}\mathbf{u}+ \lambda\mathbf{u}=\mathbf{f}$ in $B$, $\mathbf{n}\cdot\mathbf{u}|_S=0$, is solved completely and for any $\lambda$., Comment: in Russian

The author studies the structure of space $ \mathbf {L} _ {2} (G) $ of vector-valued functions that are square integrable in a bounded connected domain $ G $ of the three-dimensional space with a smooth boundary and the role of gradient divergence operators and the rotor in the construction of bases in subspaces $ {\mathcal {{A}}} $ and $ {\mathcal {{B}}} $. The self-adjointness of the extension $ \mathcal {N} _d $ of operator $ \nabla \mathrm {div} $ to the subspace $ \mathcal {A} _ {\gamma} \subset {\mathcal {{A}}} $ and the basicity system of its own functions. Written explicit formulas for solving the spectral problem in a ball and the conditions for the decomposition vector-functions in a Fourier series in eigenfunctions gradient of divergence. The solvability of the boundary tasks: $ \nabla \mathrm {div} \, \mathbf {u} + \lambda \, \mathbf {u} = \mathbf {f} $ in $ G $, $ (\mathbf {n} \cdot \mathbf {u}) | _ {\Gamma} = g $ in Sobolev spaces $ \mathbf {H} ^ {s} (G) $ of order $ s \geq 0 $ and in subspaces. In passing, similar results for the operator of the rotor and its symmetric extension $ S $ to $ \mathcal {B} $., Comment: in Russian

The effect of the small dispersion on the self-focusing of solutions of the equations of nonlinear geometric optics in one-dimensional case is investigated. In the main order this influence is described by means of the universal special solution of the nonlinear Schr\"odinger equation, which is isomonodromic. Analytic and asymptotic properties of this solution are described., Comment: in Russian

Considered typical processes of rod bending under strong longitudinal compression. The dynamic equation of bending correspponds to a perturbation of the two- dimensional Laplace equation. It is established that, for these processes, expanding of do- mains of rapid increasing of bending begins in small neighborhoods of singularity points of solutions of the limiting Laplace's equation. The initial stages of these increases are described using the Hardy integral., Comment: 1) in russian 2) english version in RJMP (A.A.Ershov and B.I.Suleimanov Russ. J. Math. Phys. (2017) 24: 216. doi:10.1134/S1061920817020078)

The properties of curl and gradient of divergence operators in the domain $G$ of three-dimensional space are described. The self-conjugacy of these operators in the subspaces $\mathbf{L}_{2}(G) $ and the basis property of the system of eigenfunctions are discussed. Exact formulas are founded for solving boundary value problems in a ball and the conditions for the decomposition of vector functions into Fourier series in eigenfunctions of the curl and the gradient of divergence operators., Comment: in Russian

The spectral series of the Schr\"odinger operator with a delta-potential on a three-dimensional compact spherically symmetric manifold in the semiclassical limit as $h\to0$ are described., Comment: 17 pages, 2 figures, in Russian, English version can be found at http://link.springer.com/article/10.1134/S1061920813030072

In preprint we consider and compare different definitions of generalized solution of the Cauchy problem for 1d-scalar quasilinear equation (conservation law). We start from the classical approaches goes back to I.M. Gelfand, O.A. Oleinik, S.N. Kruzhkov and move to the modern finite-difference approximations approaches belongs to A.A. Shananin and G.M. Henkin. We discuss the conditions that provide definitions to be equivalent., Comment: Preprint CCAS RAS, 2006 (3-d edition, 2016); 75 pages, in Russian

In this paper we study the extremality of translation-invariant Gibbs measure for the HC-model on a Cayley tree. It is known that for this model the translation-invariant measure is unique. We give a new proof of this statement and found regions of the extremality of this measure on the Cayley tree of order k., Comment: in Russian

This is author's Habilitation Thesis (Dr. Sci. dissertation) submitted at the beginning of September 2004. It is written in Russian and is posted due to the continuing requests for the manuscript. The content: 1. Introduction, 2. Nonlinear chains with the discrete time and their self-similar solutions, 3. General theory of theta hypergeometric series, 4. Theta hypergeometric integrals, 5. Biorthogonal functions, 6. Elliptic hypergeometric functions with |q|=1, 7. Conclusion, 8. References. It contains an outline of a general heuristic scheme for building univariate special functions through self-similar reductions of spectral transformation chains, which allowed construction of the differential-difference q-Painleve equations, as well as of the most general known set of elliptic biorthogonal functions comprising all classical orthogonal polynomials and biorthogonal rational functions. One of the key results of the thesis consists in the discovery of genuinely transcendental elliptic hypergeometric functions determined by the elliptic hypergeometric integrals. The whole theory of such integrals can be built from the univariate elliptic beta integral -- the most complicated known definite integral with exact evaluation, which generalizes the ordinary binomial theorem and its q-extension, Euler's beta integral, the measure for Askey-Wilson polynomials, and many other previously established results on ordinary and q-hypergeometric functions., Comment: 218 pp, Habilitation Thesis, in Russian, JINR, Dubna, 2004

The paper considers the overdetermined system of PDE that describes a special type of two-dimensional motion of an ideal fluid. The analysis of the compatibility of the system was performed. All solutions of the system are obtained in a closed form., Comment: 20 pages, in Russian

Presentation of the probability as an intrinsic property of the nature leads researchers to switch from deterministic to stochastic description of the phenomena. The procedure of stochastization of one-step process was formulated. It allows to write down the master equation based on the type of of the kinetic equations and assumptions about the nature of the process. The kinetics of the interaction has recently attracted attention because it often occurs in the physical, chemical, technical, biological, environmental, economic, and sociological systems. However, there are no general methods for the direct study of this equation. Leaving in the expansion terms up to the second order we can get the Fokker-Planck equation, and thus the Langevin equation. It should be clearly understood that these equations are approximate recording of the master equation. However, this does not eliminate the need for the study of the master equation. Moreover, the power series produced during the master equation decomposition may be divergent (for example, in spatial models). This makes it impossible to apply the classical perturbation theory. It is proposed to use quantum field perturbation theory for the statistical systems (the so-called Doi method). This work is a methodological material that describes the principles of master equation solution based on quantum field perturbation theory methods. The characteristic property of the work is that it is intelligible for non-specialists in quantum field theory. As an example the Verhulst model is used because of its simplicity and clarity (the first order equation is independent of the spatial variables, however, contains non-linearity). We show the full equivalence of the operator and combinatorial methods of obtaining and study of the one-step process master equation., Comment: in Russian; in English

We study spectral properties of Schr\"odinger operators with $\delta$-interactions on a semi-axis by using the theory of boundary triplets and the corresponding Weyl functions. We establish a connection between spectral properties (deficiency indices, self-adjointness, semiboundedness, discreteness of spectra, resolvent comparability etc.) of Schr\"odinger operators with point interactions and a special class of block Jacobi matrices., Comment: In Russian

In this paper is studied ferromagnetic three states Potts model on a Cayley tree of order three and we give explicit formulas for translation-invariant Gibbs measures. Furthermore, we show that under some conditions on the parameter of the antiferromagnetic Potts model with q-states with zero external field on the Cayley tree of order $k>2$, there are exactly 2(2^q-1) periodic (non translation-invariant) Gibbs measures., Comment: in Russian

The theory of Physical Structures (TPS) was put forward by Professor Yu.I. Kulakov for the sake of classifying the laws of Physics. The history of the development of that theory is given in his monograph [1]. A physical structure is a geometry of one or two sets whose metric function assigns a number to every pair of points. Its phenomenological symmetry, under Kulakov, means that for every collection of some finite number of points all of their reciprocal distances are functionally related. Such geometries are endowed with a group symmetry under Klein, which is equivalent to the phenomenological symmetry, and many of them have an essential physical interpretation. That is why they are to be defined precisely and explored as purely mathematical objects. In this monograph we treat the mathematical basics of the TPS and present the results of attempts at classification that have been obtained by now. We hope that the monograph will be of interest for research workers and teachers, senior and post graduate students, as well as to all those interested in algebra, geometry and theoretical physics who would like to use the TPS in their research projects or could want to contribute to the development of its mathematical apparatus., Comment: 142 pages, in Russian

The Hamiltonian formalism is extremely elegant and convenient to mechanics problems. However, its application to the classical field theories is a difficult task. In fact, you can set one to one correspondence between the Lagrangian and Hamiltonian in the case of hyperregular Lagrangian. It is impossible to do the same in gauge-invariant field theories. In the case of irregular Lagrangian the Dirac Hamiltonian formalism with constraints is usually used, and this leads to a number of certain difficulties. The paper proposes a reformulation of the problem to the case of a field without sources. This allows to use a symplectic Hamiltonian formalism. The proposed formalism will be used by the authors in the future to justify the methods of vector bundles (Hamiltonian bundles) in transformation optics., Comment: in Russian, in English

The aim of this note is to present simple proofs of the completeness of Manakov's integrals for a motion of a rigid body fixed at a point in $\mathbb R^n$, as well as for geodesic flows on a class of homogeneous spaces $SO(n)/SO(n_1)\times\dots\times SO(n_r)$., Comment: 11 pages, 1 figure, an extended version of the note will appear in: Fundamentalnaya i Prikladnaya Matematika (in Russian); Journal of Mathematical Sciences (in English)

The phase topology of the integrable Hamiltonian system on $e(3)$ found by V.V.Sokolov (2001) and generalizing the Kowalevski case is investigated. The generalization contains, along with a homogeneous potential force field, gyroscopic forces depending on the configurational variables. Relative equilibria are classified, their type is calculated and the character of stability is defined. The Smale diagrams of the case are found and the classification of iso-energy manifolds of the reduced systems with two degrees of freedom is given. The set of critical points of the complete momentum map is represented as a union of critical subsystems; each critical subsystem is a one-parameter family of almost Hamiltonian systems with one degree of freedom. For all critical points we explicitly calculate the characteristic values defining their type. We obtain the equations of the surfaces bearing the bifurcation diagram of the momentum map. We give examples of the existing iso-energy diagrams with a complete description of the corresponding rough topology (of the regular Liouville tori and their bifurcations)., Comment: LaTex, 28 pages, 6 figures, Russian, submitted to Rus.J.Nonlin.Dyn. 05.07.2015 Final published version

We construct solutions of analogues of the nonstationary Schr\"odinger equation corresponding to the polynomial isomonodromic Hamiltonian Garnier system with two degrees of freedom. This solutions are obtained from solutions of systems of linear ordinary differential equations whose compatibility condition is the Garnier system. This solutions upto explicit transform also satisfy the Belavin --- Polyakov --- Zamolodchikov equations with four time variables and two space variables., Comment: 21 pages, in Russian

We consider fertile HC-models with four-states and the parametre activity on a Cayley tree. It is known that three types of such models exist. For each of these models we prove uniqueness of the translation-invarinat Gibbs measure on a Cayley tree of order two., Comment: in Russian

We present a review of the results obtained during the last fifty years in the problem of the motion of a heavy gyrostat under the conditions of the Kowalevski type. Hamad M. Yehia in 1986 has proved that the problem is complete integrable. Since then, a lot of works were devoted to different aspects of integrating this problem and of its topological investigation. The main idea of this work is to prove strictly all the facts of qualitative analysis and to fix different mistakes and inaccuracies. The work was supported by the grants of the RFBR No. 10-01-00043, 10-01-97001, 13-01-97025, and 14-01-00119., Comment: LaTex, 128 pages, 80 figures, in Russian, preprint

On base of differential biquaternions algebra and generalized functions theory the biquaternionic wave equation is considered under vector representation of its structural coefficient. Its generalized solutions are constructed, which describe nonstationary, time-harmonic and static twistors. Elementary harmohic twistors are determined, their biquaternions of energy-inpulse are calculated. By use of them polorized and unpolarized twistors field have been constructed., Comment: 12 pages in Russian

We consider fertile HC-models with three states on a Cayley tree. It is known that four types of such models exist. For these models we describe the translation-invarinat Gibbs measures on a Cayley tree of order three., Comment: in Russian

In this paper under some conditions on parameters of the Potts model with q-states on a Cayley tree of order k it is proved existence of the periodic (non translation-invariant)Gibbs measures. Also given a theorem about the number of these measures., Comment: in Russian

In this paper we show that under some conditions on the parameter of the Potts model with three states with zero external field on the Cayley tree of order $k>2$, there are exactly two periodic (non translation-invariant) Gibbs measures., Comment: in Russian

We study $p$-adic model of hard spheres with three states on the Cayley tree. We show that there exist three translation-invariant $p$-adic Gibbs measures and two periodic measures on a Cayley tree of oreder two., Comment: in Russian

For research in the field of transformation optics and for the calculation of optically inhomogeneous lenses the method of geometrization of the Maxwell equations seems to be perspective. The basic idea is to transform the coefficients of material equations, namely the dielectric permittivity and magnetic permeability in the effective geometry of space-time (besides the vacuum Maxwell equations). This allows us to solve the direct and inverse problems, that is, to find the permittivity and magnetic permeability for a given effective geometry (paths of rays), as well as finding an effective geometry on the dielectric permittivity and magnetic permeability. The most popular naive geometrization was proposed by J. Plebanski. Under certain limitations it is quite good for solving relevant problems. It should be noted that in his paper only the resulting formulas and exclusively for Cartesian coordinate systems are given. In our work we conducted a detailed derivation of formulas for the naive geometrization of Maxwell's equations, and these formulas are written for an arbitrary curvilinear coordinate system. This work is a step toward building a complete covariant geometrization of the macroscopic Maxwell's equations., Comment: in Russian; in English

In this paper we study the boundedness of the $p$-adic quasi Gibbs measures for the Vannimenus model on a Cayley tree of order two., Comment: in Russian

Consider a five-point discretization of a two-dimensional finite-gap for a fixed energy Schr\"{o}dinger operator. We construct the Green's function of the operator. In appears as the explicit formula in terms of the integral by the specific contour on the spectral curve of the differential which is constructed by the wave function and its double. The formula is parametrized by the point on the spectral curve and for almost every parameter value it gives the Green's function with known growth at infinity., Comment: 15 pages, in Russian

We are using the finite-gap approach for the construction of the Schr\"{o}dinger operator discretization on a quad graph. The latter is represented by a two-dimensional integer sublattice in a $d$-dimensional space. The Green's function of the operator can be posed explicitly as an integral of the differential built by the spectral data, calculated on contours of the special form. We also know the the asymptotics of the achieved function., Comment: 16 pages, in Russian