**I.**** Vechtomov E.M., Starostina O.V.** **Generalized Abelian-regular positive semirings**__Text__

Notions of generalized Abelian-regular positive semirings (arp-semirings) and generalized Boolean semirings are introduced. It is showed that the research of generalized arp-semirings reduces to arp-semirings. Functional representation of generalized arp-semirings and generalized Boolean semirings are obtained. Dauns-Hofmann’s theorem about representation of biregular rings is extended on biregular semirings.

**II.**** Lovyagin Yuri N.** **Hyperrational numbers as the basis of analysis**

Theory of hyperrational numbers as the basis of analysis is considered. The main aspects of the theory are adduced. Analogs of the classical differential and integral calculus of one-variable function theorems are stated.

**III.**** Odyniec W.P., Prophet M.P.** **On three forgotten results of S.Krein, N.Bogolyubov and V.Gurari with applications to bernstein operators**

The results of M. Frechet in 1934 about the largesteigenvalue of a stochastic matrix [6] attracted attentionto positive linear operators with norm 1. The study of compact linearoperators with stochastic kernel by N.M. Krylov and N.N.Bogolyubov during the 1930’s ([9], [10]) was generalized by S. Krein and N.N. Bogolyubov a decade later in [2]. These results contributed to the 1968 paper [8] of M. Krasnoselski in which the problem of determining minimal-normshape-preserving projections was present. Unfortunately, many of these papers are practically unknown, as they were published in the Ukrainian language. On the other hand, recent developments in the theory of minimal shape-preserving projections have been made using methodsthat are independent of Krasnoselski’s work (see [3] and [11]). In this paper, we attempt to connect these two directions by studying the (so called) Bernstein operators.

**IV.**** Prazdnikova El. Wl.** **Modelling the real analysis in the framework of axiomatic of hypernatural numbers**

In the paper formalized theory of Non-Standard theory of numbers and formalized theory of hyperrational numbers are stated. The main theorems of classical differential calculus are modelled.

**V.**** Savelev L.J., Ogorodnikov V.A., Sereseva O.V.** **Stochastic model of piecewise linear process**

The stochastic process with piecewise linear trajectories is considered. The process is based on models of stochastic walk on a straight line. The distributions of stochastic variables, forming this process, and, in particular, distribution of relative time expectations for Poisson flow of points are investigated. The appropriate mathematical expressions for these distributions, and also expressions for mean of process as time-varying function are received.

**VI.**** Mikhailovskii E.I.** **A non-linear theory of flexible shells Zhuravsky-type**

A non-linear theory of flexible shells has been built, taking into account transversal shears by D. I. Zhuravsky model, By introducing generalized forces and moments the equations are reduced to the kind formally coinciding whith equations in the theory of shells, based on Kirchhoff’s hypotheses. This allows to formulate an effective algorithm of accounting transversal deformations in some of Kirchhoff’s variations of the theory of shells and planes, The algorithm is illustrated by more exact specification of K. Margyerre’s non-linear theory of shallow shells.

**VII.**** Duriagin A.M.** **Real harmonic frames, toughness and redundancy**

It was proved that real harmonic frames possess maximal redundancy, i.e. if any m-n vectors are deleted then remaining n vectors form frame in R^{n} (in the general case, it is not tight frame). The fast frame expansion algorithm is offered.

**VIII.**** Kholmogorov D.V., Tarasov V.N.** **Influence of boundary conditions on the stability of a cylindrical shell**

In the work the problem of stability of the cylindrical shell, loaded with external normal pressure or undergoing axial compression, is examined. With a numerical study the tasks of displacement are approximated by splines. In contrast to the traditional trigonometric series the application of splines makes it possible to consider the real boundary condition, in particular, the displacement of the ends of shell as rigid whole is allowed.

**IX.**** Mikhailovskii E.I., Mironov V.V., Kuznetsova J.L.** **About one solution algorithm for solving of nonlinear boundary problem Karman-type**

On the basis of Green’s well-known function, an algorithm of edge problems with Karman-type non-linearity to the corresponding system of algebraic equations. The algorithm is realized on tne examples of simply supported opened cylindrical shell.

**X.**** Pevnyi ****А****.****В****., Istomina M.N.** **Recovery of the signal in the case when one frame coefficient is erased**

The authors consider Mercedes-Benz frame and frame expansions for every vector x ϵ R^{n}. In the case when one frame coefficient is erased the authors suggest a fast algorithm for the recovery of the vector x.

**XI.**** Malozemov V.N., Pevnyi A.B.** **Mercedes-Benz systems and tight frames**

The paper is for the section “Easy Reading for Professionals”. The authors study the tight frames in the space in R^{n}. A complete description of the tight frames in R^{n} consisting of n+1 vectors is given. An exact lover bound for the frame potential is proved.