# Vol 64, No 4 (2018): Contemporary Problems in Mathematics and Physics

**Year:**2018**Articles:**9**URL:**https://journals.rudn.ru/CMFD/issue/view/1255**DOI:**https://doi.org/10.22363/2413-3639-2018-64-4

## Full Issue

## New Results

### A Discrete Analog of the Lyapunov Function for Hyperbolic Systems

#### Abstract

We study the diﬀerence splitting scheme for the numerical calculation of stable solutions of a two-dimensional linear hyperbolic system with dissipative boundary conditions in the case of constant coeﬃcients with lower terms. A discrete analog of the Lyapunov function is constructed and an a priori estimate is obtained for it. The obtained a priori estimate allows us to assert the exponential stability of the numerical solution.

**Contemporary Mathematics. Fundamental Directions**. 2018;64(4):591-602

591-602

### Restrictions on Parameters of Minimal Supersymmetric Standard Model

#### Abstract

Higgs boson with mass mh = 126 GeV was discovered at Large Hadron Collider in 2012. Its mass corresponds both to Standard Model of elementary-particles physics and to the mass of the most lightweight Higgs boson in the minimal supersymmetric Standard Model. In this paper, we consider the MSSM model not preserving CP-invariance that contain a large number of parameters to be variated. Using the experimental value of the Higgs boson mass, we obtain the restrictions on the parameters of the model, describe phenomenological scenarios, and analyze possible areas of the space of parameters.

**Contemporary Mathematics. Fundamental Directions**. 2018;64(4):603-615

603-615

### A Real-Time Iterative Projection Scheme for Solving the Common Fixed Point Problem and Its Applications

#### Abstract

In this paper, we are concerned with the Common Fixed Point Problem (CFPP) with demicontractive operators and its special instance, the Convex Feasibility Problem (CFP) in real Hilbert spaces. Motivated by the recent result of Ordon˜ ez et al. [35] and in general, the ﬁeld of online/real-time algorithms, e.g., [20, 21, 30], in which the entire input is not available from the beginning and given piece-by-piece, we propose an online/real-time iterative scheme for solving CFPPs and CFPs in which the involved operators/sets emerge along time. This scheme is capable of operating on any block, for any ﬁnite number of iterations, before moving, in a serial way, to the next block. The scheme is based on the recent novel result of Reich and Zalas [37] known as the Modular String Averaging (MSA) procedure. The convergence of the scheme follows [37] and other classical results in the ﬁelds of ﬁxed point theory and variational inequalities, such as [34]. Numerical experiments for linear and non-linear feasibility problems with applications to image recovery are presented and demonstrate the validity and potential applicability of our scheme, e.g., to online/real-time scenarios.

**Contemporary Mathematics. Fundamental Directions**. 2018;64(4):616-636

616-636

### The Schwarz Inequality and the Schwarz Formula for A-Analytic Functions

#### Abstract

In this paper, we study A-analytic functions. We consider main fundamental theorems of the theory of A-analytic functions and prove analogs of the Schwarz inequality, the Schwars formula, and the Poisson formula for A-analytic functions.

**Contemporary Mathematics. Fundamental Directions**. 2018;64(4):637-649

637-649

### On Boundedness of Maximal Operators Associated with Hypersurfaces

#### Abstract

In this paper, we obtain the criterion of boundedness of maximal operators associated with smooth hypersurfaces. Also we compute the exact value of the boundedness index of such operators associated with arbitrary convex analytic hypersurfaces in the case where the height of a hypersurface in the sense of A. N. Varchenko is greater than 2. Moreover, we obtain the exact value of the boundedness index for degenerated smooth hypersurfaces, i.e., for hypersurfaces satisfying conditions of the classical Hartman-Nirenberg theorem. The obtained results justify the Stein-Iosevich-Sawyer hypothesis for arbitrary convex analytic hypersurfaces as well as for smooth degenerated hypersurfaces. Also we discuss some related problems of the theory of oscillatory integrals.

**Contemporary Mathematics. Fundamental Directions**. 2018;64(4):650-681

650-681

### Computably Separable Models

#### Abstract

We state fundamental results of structural theory of computably separable models and consider applications of this theory to solution of some actual problems of the theory of eﬀective linear orders and theoretical informatics.

**Contemporary Mathematics. Fundamental Directions**. 2018;64(4):682-705

682-705

### On Complexiﬁcation of Real Spaces and Its Manifestations in the Theory of Bochner and Pettis Integrals

#### Abstract

This work is a continuation of our work [12] where we considered linear spaces in the following two situations: a real space admits a multiplication by complex scalars without changing the set itself; a real space is embedded into a wider set with a multiplication by complex scalars. We studied there also how they manifest themselves when the initial space possesses additional structures: topology, norm, inner product, as well as what happens with linear operators acting between such spaces. Changing the linearities of the linear spaces unmasks some very subtle properties which are not so obvious when the set of scalars is not changed. In the present work, we follow the same idea considering now Bochner and Pettis integrals for functions ranged in real and complex Banach and Hilbert spaces. Finally, this leads to the study of strong and weak random elements with values in real and complex Banach and Hilbert spaces, in particular, some properties of their expectations.

**Contemporary Mathematics. Fundamental Directions**. 2018;64(4):706-722

706-722

### Construction of Optimal Interpolation Formulas in the Sobolev Space

#### Abstract

In the present paper, using the discrete analog of the diﬀerential operator d2m/dx2m, optimal interpolation formulas are constructed in L2(4)(0, 1) space. The explicit formulas for coeﬃcients of optimal interpolation formulas are obtained.

**Contemporary Mathematics. Fundamental Directions**. 2018;64(4):723-735

723-735

736-748