**
**### Математика

1. Alsykova A. A.

Nonlocal problems with integral conditions for Boussinesq equation
**Abstract.** We examine solvability of nonlocal problems with integral conditions for Boussinesq equation. The theorems of existence and uniqueness of the solution of these problems are proved. Keywords: nonlocal boundary value problem, Boussinesq equation, integral condition, existence and uniqueness of regular solution.

**Keywords:** nonlocal boundary value problem, Boussinesq equation, integral condition, existence and uniqueness of regular solution.

2. Anosov V. P.

Isomorphism of spaces of traces of vector functions in spaces of L. N. Slobodetskii
**Abstract.** The isomorphism of spaces of traces of vector functions in abstract spaces of L. N. Slobodetskii proved.

**Keywords:** : isomorphism of spaces, traces of vector functions, space.

3. Atlasova E. I.

Solvability of conjugate problems for quasiparabolic equations of third order
**Abstract.** We study solvability of conjugate problems (generalized diffraction problems) for quasiparabolic equations of odd order. We prove existence and uniqueness theorems for regular solutions to these problems.

**Keywords:** : quasiparabolic equation, conjugation problem, existence, uniqueness, a priori estimate, regular solution.

4. Dyshaev M. M., Fedorov V. E.

Symmetry analysis and exact solutions for a nonlinear model of the financial markets theory
**Abstract.** Group classification is obtained for the Sircar-Papanicolaou equations family with a free parameter that contains the Black-Scholes equation as the simplest partial case. The five-dimensional group of equivalence transformations is calculated and three-dimensional kernel of principal Lie algebras and four-dimensional principal Lie algebras in cases of two free element specifications are found. Optimal subalgebras systems and corresponding invariant solutions or invariant submodels are calculated for every Lie algebra. Invariant solutions are included in more general multiparameter solutions families that are invariant with respect to the whole Lie algebra.

**Keywords:** nonlinear partial differential equation, Black–Scholes equation, Sircar– Papanicolaou model, pricing options, group analysis, invariant solution, invariant submodel, dynamic hedging, feedback effects of hedging.

5. Ivanova A. O.

Tight description of 4-paths in 3-polytopes with minimum degree 5
**Abstract.** Back in 1922, Franklin proved that every 3-polytope P5 with minimum degree 5 has a 5-vertex adjacent to two vertices of degree at most 6, which is tight. This result has been extended and refined in several directions. In particular, Jendrol' and Madaras (1996) ensured a 4-path with the vertex degree-sum at most 23. The purpose of this note is to prove that every P5 has a (5, 6, 6, 6)-path or (5, 5, 5, 7)-path, where all parameters are tight.

**Keywords:** planar graph, plane map, structural properties, 3-polytope, 4-path.

6. Nikiforov D. V.

The structure of neighborhoods of 5-vertices in normal plane maps with minimum degree 5
**Abstract.** In 1940, Lebesgue described the neighborhoods of vertices of degree 5 in normal plane maps with minimum degree 5 (M5), presenting only an idea of the proof but not the details. The paper presents a detailed scheme of a complete proof of Lebesgue's description with improving two of its parameters without worsening the others. Moreover, it is present a scheme of the proof of the height of a 5-star (the maximum degree of its vertices) in an M5, which improves the result of O.V.Borodin, A.O.Ivanova, T.R.Yensen (2013).

**Keywords:** plane graph, normal plane maps, structure, neighborhood.

7. Petrushko I. M., Petrushko M. I.

On the first problem for the degenerate parabolic equations with changing time direction
**Abstract.** We study the properties of solutions of parabolic equations with changing time direction. We prove that the Riesz and Littlewood-Paley conditions for these solutions are equivalent. We demonstrate the unique solvability of the first mixed problem with boundary and initial functions of the space type and also establish the existence of limits in with the weight of the decisions on the sections of the border, which are free from the initial conditions.

**Keywords:** degenerate equations, changing time direction, functional spaces, integral identities, first mixed problem, solvability.

8. Popov N. S.

On the solvability of boundary value problems for multidimensional parabolic equations of fourth order with nonlocal boundary condition of integral form
**Abstract.** We investigate solvability of the initial-boundary value problem for linear parabolic equations of fourth order with the boundary conditions connecting the values of solution or conormal the derivative of the solution with values of a certain integral operator from the solution. We prove the theorem of existence and uniqueness of regular solutions.

**Keywords:** parabolic equation of fourth order, Sobolev space, initial-boundary value problem, continuation method the parameter, a priori estimates, regular solutions.

9. Khludnev A. M., Popova T. S.

On the hierarchy of thin delaminated inclusions in elastic bodies
**Abstract.** We consider models of thin delaminated inclusions in elastic bodies. The delamination means a presence of a crack between the inclusion and the matrix. Inequality type boundary conditions are imposed at the crack faces to prevent a mutual penetration. This approach leads to free boundary problem formulations. Connections between different mathematical models are discussed. Passages to limits with respect to inclusion rigidity parameters are analyzed.

**Keywords:** thin inclusion, elastic body, crack, non-linear boundary conditions, rigidity parameter, limiting models.

10. Chernikov P. V.

On convergence of D-limits
**Abstract.** Some statements about the convergence of D-limits are proved.

**Keywords:** finally compact topological space, countably complete ultrafilter.