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Classification of pattern recognition problems is offered. This classification is founded on the basic properties of pattern recognition problems. It is shown, a choice of methods of decisions must be coordinated with features of classes of pattern recognition problems.

In the article, the concept of metastructural identification of a modeled system is formalized as the construction of a pair consisting of a neighborhood structure (graph) and the type of interactions between the nodes of this structure. In the language of metagraphs, two types of interactions are defined: vertex type, when the equations of the model correspond to the nodes of the structure, and the relational type, when the equations correspond to the edges of the structure. Structural identification of the modeled system, as a rule, can be divided into two stages.

In this paper the class of simplest not rough $\Omega$-stable flows on a sphere isconsidered. We call simplest not rough $\Omega$-stable flow an $\Omega$-stable flow with least number of fixed points, a single separatrix connecting saddle points and without limit cycles. For such flows we design the Morse energy function.

The set of all quadratic irrationalities ($s$-discriminants) with decomposition $[q_{0},\overline{q_{1},q_{2},\ldots,q_{2},q_{1},sq_{0}}]$ $(s \ge 2$ — parameter) are described. Theory of Pell $s$-equation is constructed. The inverse problem (reconstruction of $s$-discriminant with the help from continued fraction period’s symmetric part) is solved.

Keywords: $s$-discriminants, Pell $s$-equation, partial periodic continuesfraction, quadratic irrationalities.

In this paper optimization of profitability functions of queueing systems is considered. It is shown that as a rule such functions are unimodal.

In this paper the finite element method is analyzed for nonlinear elliptic variational problem which is formally equivalent to a two-dimensional nonlinear elliptic boundary problem with mixed nonhomogeneous boundary conditions. The given problem is analyzed under the maximum angle condition and is solved in the case of a bounded domain $\Omega$ whose boundary $\delta \Omega$ consists of two circles $\Gamma_{1}$, $\Gamma_{2}$; of the same centre $S_{0}$. These circles have the radii $R_{1}$, $R_{2} = R_{1} +\varrho$, where $\varrho \ll R_{1}$.

Three concrete examples of strongly nonregular polynomial pencils of ordinary differential operators of the third order generated on $[0, 1]$ by a linear differential expression with constant coefficients, polynomially depending on spectral parameter $\lambda$, and by two-point not semisplitting boundary conditions are considered, namely:

$1)$ the pencil $L^{1}_{0}(\lambda)$ of the form

$y'''−\lambda y''+ \lambda^{2}y' − \lambda^{3}y=0$,

$y(0)+y(1)=y'(0)+iy'(1)=y''(0)−y''(1)=0$;

$2)$ the pencil $L^{2}_{0}(λ)$ of the form

Taurida University was officially opened in the Crimea on October 14, 1918. It was established in the crucial period and fully experienced all the difficulties and trials on the way of its development. On the eve of the centennial of the University awareness comes of a historic role it had played in the establishment of scientific and cultural traditions and of the entire system of education in Crimea.

Let $V$ be a domain in $R^{n}$, $n>2$. A set A is a regular simplex, whose edge is $\sqrt{2}$, in four-dimensional space. Some problems about functions is locally integrable on a set $V$ with vanishing integrals over all images $\lambda A\subset V$ , $\lambda \in M(n)$ of a fixed compact set $A\subset R^{n}$ are studied in the present paper. If the only function is locally integrable on a set $V$ and satisfying this condition is $f=0$ then the set $A$ is called a Pompeiu set in $V$.

Zero sets of solutions of the hyperbolic Darboux equation Volchkov V. V. and Volchkov Vit. V.
A hyperbolic analog of the generalized Darboux equation is considered. We investigate the structure of zero sets of its solutions for the case where the solution is a radial function of second variable. We show that every solution vanishing on some annulus must be zero in some other annulus containing the first one.

Keywords: Darboux equation, hyperbolic plane, zero sets, uniqueness theorems, transmutation maps.

The optimization problem with precedent (training sample) initial information is considered. Some approaches for reconstruction of the target function of such optimization problem are proposed. The open problems that must be solved to obtain better quality solutions of this problem are highlighted.