**I.**** Vechtomov E.M.** **Structure of semifields**

This work is an analytic scientific review over the theory of semifields, sponsored by grant RFBR № 08-01-11000-ано.

**II.**** Efimov D.B., Kostyakov I.V., Kuratov V.V.** **On exact representations for the group motions of Galilean plane**__Text__

The Pimenov algebra with two generators D_{2} is defined and some of its properties are shown. Some exact two- and three-dimensional matrix D_{2}-representations for the group motions of Galilean plane (the Galilean group) are considered. A geometric interpretation of them is giving. We consider also a exact representation of the Galilean group by elements of Grassmann algebra.

**III.**** Kostyakov I.V., Kuratov V.V.** **Massive Yang-Mills fields, translation and nonsemisimple gauge symmetry**

Gauge fields of semisimple groups of internal symmetries are massless and require special techniques for their mass. Massive mechanisms usually contain translational transformations specific to nonsemisimple groups. We show that under the localization nonsemisimple internal symmetry gauge fields corresponding to generators of translation, are massive. In addition, we introduce nonlinear generalizations of well-known models, with local translational symmetry and as a result, the massive gauge fields. Thus, the local Galilean symmetry is realized on a special pair of scalar fields, leading to massive electrodynamics, and the localization of the Euclidean group leads to massive non-Abelian theory without matter fields. We propose a simple interpretation of the Stueckelberg mechanism.

**IV.**** Tikhomirov A.N.** **On the circular law of random matrices**

In review the new results on proving the circular law for random matrices are given. Among them are the results obtained by the author together with F.Gotze for rared non-Hermitian matrices of large dimension.

**V.**** Malozemov V.N., Solov’eva N.A.** **On the frame matrix**

The authors consider the following problem. Given positive-definite Hermitian matrix S and positive numbers a_{1}, … , a_{m}, m ≥ n, find a frame {φ_{1}, … , φ_{m}} in C^{n} such that S is frame matrix and equations IIφ_{1}II = a_{1}, … , IIφ_{m}II = a_{m} hold. The authors give new proof of the theorem on necessary and sufficient conditions for existence of such frame.

**VI.**** Podorov A.E., Sakovnich D.Y.** **Platform for testing methods of solving linear cutting problem**

Common ideas of building applications for solving cutting problems were developed. Using these ideas, platform for testing algorithms was built. This platform was applied for compare some algorithms.

**VII.**** Belyaeva N.A., Nikonova N.N.** **Structural model of extrusion with usage of the generalised model of Newton**

The mathematical model of Solid-Phase is presented extrusion of a porous viscoelastic material with a condition of constancy of speed plunger the press. The results confirming rightfulness of replacement of the equation of movement on the equation of equilibrium are received. For the description of considered stream Lagrangian (mass) coordinates are used.

**VIII.**** Belyaeva N.A., Spiridonov A.V.**** ****The structural models of deformation processes**

The mathematical model of tverdofaznoy extrusion of porous viscoelastic material is presented with the condition of constancy of speed of plunzhera the press. Got results, confirm legitimacy of replacement of equalization of motion on equalization of equilibrium in the works before executed on this subject. For a chosen type of flow specification lagranzh(mass) coordinates are used.** **

**IX.**** Vasilyev ****А****.****А****., Koroleva ****А****.N.**** ****Some applications of computational geometry to linear programming problems**

The work considers one of the approaches to the solution of linear-programming problem with two variables: the one of computational geometry and its generalization on the problem of the least covering circle. Numerical approbation of the given method is carried out, the algorithm of the decision is constructed.

**X.**** Nikitenkov V.L., Podorov A.E.**** ****Modifications of waste cutting problem**

The problem of cutting materials with various length and limited stores is considered as the waste cutting problem. Five it’s modifications are offered. In some of them methods of a finding admissible inverse matrix are given. Results of numerical experiments are discussed.

**XI.**** Yakovlev V.D., Afonin R.E.** **The hunting on numbers**

In this article the history of search of amicable pairs since times of ancient Greeks and up to now is shown. Also current outcomes of search of aliquot sequences and Mersenne numbers are given.