Vol 40, No 8 (2019)

In this article we consider classes of harmonic and subharmonic functions introduced with using integral operators Riman-Liouville by Professor M. Djrbashyan when α > 0. These classes are significant generalizations of already well known classes of harmonic and subharmonic functions match up with them only in a particular case. In our article we consider angular and chordal limits of harmonic and subharmonic functions got by using Riman-Liouville integral operators. A set of the points at which, probably, these limits don’t exist are characterized by using a linear measure of zero.

A concept of the unitary hypergroup M_H over the group is introduced. If for a given set M and a group H all unitary hypergroups M_H over the group, up to isomorphism, are known, a method is proposed to construct all hypergroups M_H over the group, up to isomorphism. It reduces the problem of description of all hypergroups over the group, up to isomorphism, to the problem of description of all unitary hypergroups over the group, up to isomorphism, and can have many other applications. When M and H are finite, a formula is obtained, which connects the numbers of elements in the classes of isomorphism of all hypergroups over the group and of all unitary hypergroups over the group.

For an m-homogeneous hyperbolic (with respect to the vector N) operator P_m, and a weight function g: 1) we find the conditions on the lower order terms {Q}, operators {P_m(D) + Q(D)} to become g-hyperbolic with respect to any vector N¹ from a neighborhood O(N) of the vector N, 2) we show that the operators obtained by adding lower order terms have fundamental solutions whose supports are in the cone from upper half-space \(\overline {{H_N}} : = \{ (x,N) \ge 0\} \), 3) we show that if P(D):= (P_m + Q)(D), f ∈ G^ℜ (where G^ℜ is some Gevrey type space) and supp f ⊂ H_N:= {(x, N) > 0}, the equation P(D)u = f has a solution u ∈ G^ℜ such that supp \(u \subset \overline {{H_N}} \).

For weighted L^p-spaces of holomorphic functions in the unit matrix disc R_n2 the following problem is investigated: what type of function spaces in the unit bidisc are obtained when the mentioned spaces are restricted to the “diagonal” of the domain R_n2.

The algebra under study belongs to the class of operator algebras generated by a family of partial isometries, satisfying some relations on the initial and final projections. In turn, this family is uniquely determined by a self-mapping of a countable set. In the paper we consider a situation when isometry family generates an inverse semigroup. It is shown that in this (and only in this) case the corresponding C*-algebra has a nontrivial commutative AF-subalgebra, generated by a semi-lattice of projections of inverse semigroup. All invariant subspaces of the mentioned C*-algebra and its irreducible representations are described.

In the proposed paper, some last autor’s results of studies devoted to the acceleration of the convergence of truncated Fourier series is presented. The corresponding universal (traditional) and special adaptive algorithms are constructed. The main result (the phenomenon of over-convergence for an non-linear adaptive algorithm) states that the use of finite Fourier coefficients leads to an exact approximation for functions from certain infinite-dimensional spaces of quasipolynomials. The corresponding summation formula of truncated Fourier series for smooth functions has unprecedented accuracy.

For any mesh of triangles with fixed set of vertices we prove: the sum of cotangents of interior angles reaches his minimum for Delaunay triangulation. Using this extremal property, we obtain that for numerical solution of Maxwell equation of magnetic field the optimal mesh is Delaunay triangulation.

We discuss the capabilities of A. V. Tsirulsky’s method in the solution of direct problems of the logarithmic potential. We give classification of the Schwarz functions that arise in construction of the solutions and study the cases of finite-sheet and infinite-sheet Riemann surfaces.

Two-parameter Birnbaum-Saunders distribution has been widely studied in Reliability Theory due to its important Engineering applications. This article proposes a novel confidence intervals construction for the shape and scale parameters of the Birnbaum-Saunders distribution. We apply the following two methods: The generalized pivotal approach and the percentile bootstrap approach. The Monte Carlo simulations are used to evaluate the performance of the confidence intervals. We compare the coverage probability and average width of the proposed confidence intervals with already known. Simulation results have shown that the proposed confidence intervals perform well in terms of coverage probability and average length for various sample sizes. The illustrative example and some concluding remarks are finally presented.

The article presents a method of approximate evaluation of multistage hydraulic fracturing efficiency in homogeneous single porosity petroleum reservoir. The method considers various types of flow symmetry in certain zones of near-wellbore area. The problem is reduced to one-dimensional differential equation for pressure in fracture and a number of closing analytical formulae. Approximate solution is verified by 3D numerical modeling. Simplified method is applicable both for operational multistage hydraulic fracturing productivity estimation and for mathematical models of petroleum reservoir, especially in large-block modeling.

We provide a new procedure to generate random numbers that follow the three parameter Crack distribution. To generate Crack random numbers by the composition method, first we generate random numbers from two known distributions: Inverse Gaussian distribution and Length Biased Inverse Gaussian distribution. Finally, we derive Crack random numbers generation procedure.