**I.**** Vechtomov E.M., Chuprakov D.V.** **Pseudo-complements in a lattice of congruence of semirings of continuous functions**

We investigate the properties of lattice of congruencies of semirings and semifields of continuous nonnegative (positive) functions, set on an arbitrary topological space X. It’s proved that these lattices are lattices with pseudocomplements and every its element has not more than one complement. In terms of lattices of congruencies of semirings of continuous functions over X, the characterizations of some properties of space X are obtained.

**II.**** Eltsov N.P., Ogorodnikov V.A., Prigarin S.M.** **Analysis of cascade models of random fields**

The stochastic process with piecewise linear trajectories is considered. The process is based on models of stochastic walk on a sraight 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.

**III.**** Lovyagin Yu.N., Prazdnikova E.Wl.**** ****Elementar functions on the set of complex hyperrational numbers**

The basic elementary functions are enterd within the limits of the model-theoretic approach to the concept of complex hyperrational numbers. Their basic properties are resulted.

**IV.**** Odyniec W.P.** **One differential inequality and its application to the Pearson’s distribution densities**

We prove that for some large set of functions including the densities of Pearson distributions the condition 0<(f/f')'<1 is sufficient for differential inequality f''(t)*f(t)<(f'(t))^{2} (Theorem 1). The application of Theorem 1 to the densities of Reason distributions is given.

**V.**** Belyaeva N.A.** **Influence of the typical timeses on modes hardphase extrusion**

There are difined qualitative different extrusion regimes – stationary, quasistationary, transient modes – on base of the comparison of the typical times of the structuring, compactions and extrusion processes.

**VI.**** Bayborodina O.V., Sakovnich D.Y.** **Application of the complex algorithm for solving of size cutting problem in order to increase the efficiency of business processes of core production**

The possibility to increase the efficiency of business processes of spiral winding cores production by implementing the complex algorithm for solving of integer- valued problem of linear cutting is investigated. The results of the comparison analysis of MSY used calculation and the complex algorithm on the actual data of MSY are given.

**VII.**** Mikhailovskii E.I., Tulubenskaya E.V.** **The account of the transverse strain in the problem about stability of the cylindrical cover in the conditions constructive nonlinearity**

In work is considered the problem about stability of longitudinal compressed simply supported the cylindrical cover located at the border of two Winkler’s ambiences. The specified theory of Margera-Timoshenko type is used. The problem is solved by the means of the combined algorithm variant search.

**VIII.**** Tarasov V.N.** **Modeling the cell membrane potential of the sinoatrial node**

The work of the heart associated with a regular contractions of muscle forming the heart wall. The heart rate caused by the spread of the electrical action potential (AP) on heart fibers. Most of these fibers can self-excite. This is in contrast to AP in the nerve cells. For self-excitation AP in neuronal cells requires a signal (current initialization). Sector of the heart, giving the excitation maximum value – it is sinoatrial node. In this paper a mathematical model of electrical activity of cells of the sinotrial node is proposed.

**IX.**** Belyaeva N.A., Osipova V.V.** **The structural compaction model of viscoelastic composite material**

The structural model of solidphase plunger compaction of porous viscoelastic material is presented. Decisions of problems for case of the set speed and pressure on plunger are received. The program complex in the programming environment Delphi is developed. Results of numerical experiment are presented.

**X.**** Mikhailovskii E.I., Gintner V.V., Mironov V.V.** **Auxiliary operator method for solving linear boundary-value problem**

Linear boundary-value problem with homogeneous boundary condition under certain Green function may be considered as solved. But analytical construction of Green function is very problematical even for a homogeneous linear boundary-value problem with variable coefficients. At the same time one can select (auxiliary) differential operator with the same order, as operator of considered problem, the one for which Green function of the original problem with the boundary condition is fairly simply computed. This article shows that in these cases the original boundary-value problem is reduced to a Fredholm integral equation of the first or second kind. For its solution, for example, method of mechanical quadrature can be used.

**XI.**** Pevnyi A.B., Duriagin A.M.** **Spherical designs**

Definition of spherical t-design is made. The example of 4-design R^{3} is presented. The designs on a plane is in detail investigated, it is possible to name them designs.

**XII.**** Pevnyi A.B., Istomina M.N., Maksimenko V.V.** **Complementary tight frames**

For any tight frame in the space R^{n} consisting of m vectors, m>n, the authors construct a complimentary tight frame in the space of dimension n’=m-n.

**XIII.**** Vechtomov E.M.** **About problems of high professional education in Russia**

We are analyzing the state of affairs and exploring the contradictions in our high professional education. Our high education needs a modernization, but it should rely on reality in Russia and the traditions of fundamental education, not following western examples. Real education, moral upbringing and true science should be priorities of our government and society.