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We considered the problem of obtaining upper bounds for the mathematical expectation of the $q$-norm ($2\leqslant q \leqslant \infty$) of the vector which is uniformly distributed on the unit Euclidean sphere. We finish the paper with numerical experiments illustrating our results., Comment: in Russian, 10 pages, 2 figures

The paper introduces and investigates partial orders that are finer than the minus partial order and are based on inverses along an element and other specific outer inverses. It turns out that in this way a number of classical partial orders can be equivalently defined.

Let $A_j,B_j$ $(j=0,1,\ldots)$ be $m \times m$ matrices, whose elements are complex numbers, $A_j$ are selfadjoint matrices and $B_j^{-1}$ exist. We study the deficiency index problem for minimal closed symmetric operator $L$ with domain $D_L$, generated by the Jacobi matrix $\textbf{J}$ with entries $A_j,B_j$ in the Hilbert space $l_m^2$ of sequences $ u=(u_0,u_1, \ldots), u_j \in C^m$ by mapping $u \rightarrow \textbf{J}u$, i.e. by the formula $Lu=lu$ for $u \in D_L$, where $lu=((lu)_0,(lu)_1, \ldots)$ and $$ (lu)_0:=A_0u_0+B_0u_1, \quad (lu)_j:=B^*_{j-1}u_{j-1}+A_ju_j+B_ju_{j+1}, \;\; j=1,2, \ldots $$ It is well known that the case of the minimal deficiency numbers of the operator $L$ corresponds to the determinate case, and the case of the maximal deficiency numbers of this operator corresponds to the completely indeterminate case of the matrix power moment problem. In this paper we obtain new conditions of the minimal, maximal and not maximal deficiency numbers of the operator $L$ in terms of the entries of the matrix $\textbf{J}$. The special attention is paid to the case $m=1$, i.e. we present some conditions on the elements of the numerical tridiagonal Jacobi matrix, which ensure the realization of the determinate case of the classical power moment problem., 9 pages, in Russian

Zaremba's conjecture (1971) states that every positive integer number d can be represented as a denominator of a finite continued fraction b/d = [d1,d2,...,dk], with all partial quotients d1,d2,...,dk being bounded by an absolute constant A. Several new theorems concerning this conjecture were proved by Bourgain and Kontorovich in 2011. The easiest of them states that the set of numbers satisfying Zaremba's conjecture with A = 50 has positive proportion in natural numbers. In 2014 I. D. Kan and D. A. Frolenkov proved this result with A = 5. In this paper the same theorem is proved with A = 4., in Russian

International audience; http://mech.math.msu.su/~fpm/ps/k14/k143/k14303.pdfIn this paper, a computerized model for morphological analysis of languages with word-formation based on affixation processes is proposed. The main idea consists in defining structural patterns of words and corresponding lists of suffixes. First, a detaileddescription of a stemming algorithm, its modification, and the technique of determining grammatical characteristics of word-forms are given. The next part of this work focuses on the application of the proposed algorithms for the French language. Finally, some results of execution of these algorithms are provided.; http://mech.math.msu.su/~fpm/ps/k14/k143/k14303.pdfВ работе предлагается метод структурных схем в качестве модели морфологического анализа словоформ естественного языка с развитым аффиксальным словообразованием и словоизменением. Дано описание алгоритма выделения псевдоосновы, его модификация, а также алгоритм восстановления грамматических характеристик словоформ. Описано применение предложенного метода для анализа словоформ французского языка. Представлены результаты работы предложенных алгоритмов.