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.

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.

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 from 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.

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 Itoˆ-type SDE.

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 u_t(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 u_t(x,t,a) in the part where 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.

We study a finite nontrivial group G with an irreducible complex character degree (1) = p²q such that |G| ≤ 2(1)², where p,q are primes.

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.