**I.**** Poroshkin A.A., Poroshkin A.G.** **Three counter-example in analysis**

It is represented the examples of the continuous functions on the metric spaces for which the classical theorems of Weierstrass (about boundedness and about achievement of the face) and the theorem of Kantor (about uniformly continuity) are not true.

**II.**** Sidorov V.V.** **Structure lattice isomorphisms of semirings generated by a one nonnegative function**

In this paper we describe isomorphisms of lattices *A***_{f}** and

*A***of all subalgebras with unit of semirings of functions [f] and [g] generated by nonnegative real-valued functions f and g, respectively. It is proved that any isomorphism of lattices**

_{g}

*A***and**

_{f}

*A***is generated by an isomorphism of semirings [f] and [g]. A techniqe of unigenerated subalgebras is applied.**

_{g}**III.**** ****Grytczuk A.** **On the Diophantine equation x**^{2}** – dy**^{2}** = z**^{n}^{}

In this Note we remark that there is some duality connected with the problem of solvability of the Diophantine equation

**(*)** x^{2} – dy^{2} = z^{n}.

Namely, we prove that the equation (*) has no solution in positive integers *x,y* for every pime *z = q** ^{*}* generated by an arithmetic progression and for every odd positive integer

*n*if

*d*is squarefree positive integer such that

*p*|

*d*, where

*p*is an odd prime.

**IV.**** Afonin R.E., Malozemov V.N., Pevnyi A.B.** **Delsarte bounds for the number of elements of the spherical design**

The proof of the Delsarte’s theorem for lower bound for cardinality of spherical design is given. The exposition is closed, all auxiliary theorems are proved.

**V.**** Belyaeva N.A., Dovzhko E.S.** **Model of the formation of spherical products with the nonzero critical depth conversion of the material**

The mathematical model of the solidification of the spherical product in the mode of spread of the bilateral front. At the boundaries of the fronts are into account the conditions of coexistence of solid and liquid layers formed products. The results of numerical analysis.

**VI.**** Belyaeva N.A., Kuznetsov K.P.** **The dissipative structure and domain of anomaly structural liquid Couette flow in a flat clearance**

The bifurcation study of structural liquid Couette flow in a flat clearance in the superanomaly area was conducted. Bifurcation diagrams and the values of parameters corresponding to the superanomaly area were obtained. Bifurcation method allowed to obtain an analytical approximation of the stationary inhomogeneous solution in the neigborhood of the bifurcation point. A numerical simulation of the flow was conducted.

**VII.**** Belyayev Yu.N.** **Wave scattering continuosly stratified elastic media**

Method of calculating elements of the second order matrix, which characterizes the elastic continuously layered media is proposed. The representation of reflection and transmission coefficients of the layer through elements of characteristic matrix is given. General solution to the plane wave reflection and transmission in a periodic continuously stratified medium is found.

**VIII.**** Kotelina N.O.** **Methods of estimating kissing numbers**

Methods for estimating of kissing numbers based on linear programming, corresponding grid problems of linear programming and results of calculations in Matlab are given. The table of best known upper bounds for kissing numbers is also given.

**IX.**** Belyaeva N.A., Istomina M.N.** **Computing System “Bifurcation method in nonlinear models Mechanics”**

Computing system includes programs for the branching method in nonlinear mechanics models. The article discusses the general structure of the complex and a description of its constituent programs.

**X.**** Mikhailovskii E.I., Mironov V.V., Podorov V.R.** **Contact free boundary problem for beams and discrete elastic foundation**

The influence of the accounting of transverse shifts on the solution of contact problem for beams and supports of the unilateral action. A generalization to the case of beams, bent on the theory of Timoshenko, the method of enumeration of sets of active supports, based on the proof of the uniqueness of solutions of the nonlinear contact problem and the equations of the analytical version of the so-called theorem of three moments.

**XI.**** Pevnyi A.B., Istomina M.N.** **A modification of Delsarte’s theorem for the estimation of kissing numbers**

A modification of Delsarte’s theorem is proved.

**XII.**** Odyniec W.P.** **Two hundred years from the date of the birth of the creators of mechanical computers recommended for Demidov Prize H. Slonimsky and H. Kummer**

Some materials of the creation of calculating gadgets by H. Slonimsky, H. Kummer and H. Ioffe is considered. In details the Theorem by H.Slonimsky which was the base of these gadgets is presented. This Theorem, devoted to a property of the Farey sequence, is now widely applied in informatics.

**XIII. ****Professor Alexandr Grigiorievich Poroshkin: 60-th year in mathematics and education**

**XIV****.Валерьян Николаевич Исаков (к 65-летию со дня рождения)**