Volume 23, issue #3

Mathematics

1. Gadoev M. G. and Iskhokov F. S. On invertibility of a class of degenerate differential operators in the lebesgue space
Abstract.  We construct the right-hand regularizing operator for a class of partial differential operators in non-divergent form in an arbitrary (bounded or unbounded) domain in the n-dimensional Euclidian space with non-power degeneracy on the boundary. On its base we prove the existence of the inverse operator in the Lebesgue space.
Keywords:  partial differential operator, non-power degeneration, right-hand regularizing operator, inverse operator, partition of unity.
        
2. Grigorieva A. I. The conjugation problem for pseudoparabolic and pseudohyperbolic equations
Abstract.  We study solvability of a conjugation problem for pseudoparabolic and pseudohyperbolic equations. The equations are considered as equations with discontinuous coefficients. We prove the existence and uniqueness theorem using natural parameter continuation.
Keywords:  pseudoparabolic equation, pseudohyperbolic equation, discontinuous coefficient, conjugation problem.
      
3. Ivanova A. O. Description of faces in 3-polytopes without vertices of degree fr om 4 to 9
Abstract.  In 1940, Lebesgue proved that every normal plane map contains a face for which the set of degrees of its vertices is majorized by one of the following sequences:
(3, 6, ∞), (3, 7, 41), (3, 8, 23), (3, 9, 17), (3, 10, 14), (3, 11, 13),
(4, 4, ∞), (4, 5, 19), (4, 6, 11), (4, 7, 9), (5, 5, 9), (5, 6, 7),
(3, 3, 3, ∞), (3, 3, 4, 11), (3, 3, 5, 7), (3, 4, 4, 5), (3, 3, 3, 3, 5).
In this note prove that every 3-polytope without vertices of degree fr om 4 to 9 contains a face for which the set of degrees of its vertices is majorized by one of the following sequences: (3, 3, ∞), (3, 10, 12), (3, 3, 3, ∞), (3, 3, 3, 3, 3),
which is tight.
Keywords: planar graph, plane map, structure properties, 3-polytope, weight.
        
4. Karachanskaya E. V. and Petrova A. P. Non-random functions and solutions of langevin-type stochastic differential equations
Abstract.  We construct a solution of a Langevine-type stochastic differential equation (SDE) with a non-random function depending on its solution. We determine conditions for such non-random function to appear. Using the solution of a homogeneous SDE, we obtain a solution of the generalized Langevine-type SDE by reducing it to a linear one. We construct a stochastic process with non-random modulus in square which is not a solution to an Ito-type SDE.
Keywords:  Langevine-type equation, Brownian motion, stochastic differential equation, Ito’s formula, deterministic modulus in square for velocity, analytical solution.
      
5. Kozhanov A. I. and Potapova S. V. On a non-standard conjugation problem for elliptic equations
Abstract.  We investigate the regular solvability of the conjugation problem for elliptic equations with non-standard boundary conditions and sewing conditions on the plane x = 0. Let Q be a parallelepiped. On the bottom of Q we give a boundary condition for u(x, t, a) in the part where x > 0 and for ut(x, t, a) in the part where x < 0. On the plane x = 0 these conditions "intertwist", so on the top of Q we give a boundary condition for u(x, t, a) in the part where x < 0 and for ut(x, t, a) in the part wh ere x > 0. Combining the regularization method and natural parameter continuation, we prove the uniqueness and existence theorems for regular solutions of this non-standard conjugation problem.
Keywords:  conjugation problem, regular solution, sewing condition, elliptic equation, discontinuous boundary conditions.
      
6. Poiseeva S. S. On the structure of finite groups with large irreducible character degree p2q
Abstract.  We study a finite nontrivial group G with an irreducible complex character Θ degree Θ(1) = p2q such that |G| ≤ 2Θ(1)2, where p, q are primes.
Keywords:  finite group, character of a finite group, irreducible character degree of a finite group.

Mathematical modeling

7. Fatyanov A. G.The stable analytical solution for the wave fields in the sphere
Abstract. We investigate the well-known analytical solution to the problem of the wave fields in the sphere. It is shown that the use of the standard asymptotic behavior of the Bessel functions leads to interference in the solution. A new asymptotic expression for the Bessel functions is found which gives a stable analytical solution that allows one to obtain the exact solution. The homogeneous and inhomogeneous waves for the sphere are detected. We present some examples of analytical calculation of the full wave fields and the primary wave for the sphere.
Keywords:  mathematical modeling in the sphere, stable analytical solution, full wave field, primary wave, new asymptotic behavior of Bessel functions, homogeneous and inhomogeneous waves for the sphere.