The problem of realization of Boolean functions by generalized *α*-formulas is considered. The notion of a universal set of generalized *α*-formulas is introduced for a given set of Boolean functions. Universal sets of generalized *α*-formulas are constructed for the set of constant-preserving Boolean functions.

Closed classes of functions of three-valued logic whose generating systems include nonmonotone symmetric functions taking values in the set {0,1} are studied. It is shown that in some cases the problems of existence of a basis and existence of a finite basis can be reduced to a similar problem for reduced generated systems.

The problem of realization of Boolean functions by initial Boolean automata with constant states and *n* inputs is considered. Such automata are those whose output function coincides with one of *n*-ary constant Boolean functions 0 or 1 in all states. The exact value of the maximum number of *n*-ary Boolean functions, where *n* > 1, realized by an initial Boolean automaton with three constant states and *n* inputs is obtained.

The maximal inequality for the skew Brownian motion being a generalization of the wellknown inequalities for the standard Brownian motion and its module is obtained in the paper. The proof is based on the solution to an optimal stopping problem for which we find the cost function and optimal stopping time.

We present an example of a 6 × 6 matrix *A* such that rk *t* (*A*) = 4, rk *K* (*A*) = 5. This disproves the conjecture formulated by M. Chan, A. Jensen, and E. Rubei.

The geometry of level lines of harmonic functions being real parts of some Abelian integrals is studied. Such harmonic functions appear in the study of asymptotics of solutions to second-order differential equations, and the corresponding level lines are related both to the distribution of eigenvalues of a non-self-adjoint Sturm–Liouville problem and to location of trajectories of the corresponding quadratic differentials.

The problem of realization of Boolean functions by initial Boolean automata with two constant states and n inputs is considered. An initial Boolean automaton with two constant states and n inputs is an initial automaton with output such that in all states the output functions are n-ary constant Boolean functions 0 or 1. The maximum cardinality of set of n-ary Boolean functions, where n > 1, realized by an initial Boolean automaton with two constant states and n inputs is obtained.

This paper is focused on a multichannel queueing system with heterogeneous servers and regenerative input flow operating in a random environment. The environment can destroy the whole system and the system is reconstructed after that. The necessary and sufficient ergodicity condition is obtained for the system.

This paper is devoted to $M|GI|1|\infty$ queueing system with unreliable server and customer service times depending on the system state. Condition of ergodicity and generating function are found in the stationary state.

The trajectory of expedition to Mars and its satellite Phobos with return to the Earth is optimized. The attraction of the Sun, or the Earth, or Mars is considered at each part of the trajectory. The Earth and Mars positions correspond to ephemerides DE424, and the position of Phobos corresponds to ephemerides MAR097. Not more than 6 impulses are assumed on the trajectory. The spacecraft must start from the Earth in the period of 2020–2030 and stay at Phobos at least for 30 days. The total time of the expedition is limited to 1500 days. The characteristic velocity is minimized.

A combinatorial proof that the number of points of the space Mо,n(Fq) satisfies the recurrent formula for the Poincaré polynomials of the space Mо,n(C) is obtained.

Original Russian Text © N.Ya. Amburg, E.M. Kreines, and G.B. Shabat, 2017, published in Vestnik Moskovskogo Universiteta, Matematika. Mekhanika, 2017, Vol. 72, No. 4, pp. 20–27.

The local qualitative robustness of GM-tests against outliers in the autoregression model is studied in the paper. A local scheme of data contamination by independent outliers with the intensity *O*(*n*−1/2) is considered. The qualitative robustness in terms of power equicontinuity is obtained. The GM-tests asymptotically optimal in the maximin sense are constructed.

The problem of the complexity of word assembly is studied. The complexity of the word refers to the minimum number of concatenation operations sufficient to obtain this word on the basis of one-letter words over a finite alphabet $A$ (repeated use of the received words is permitted). Let $L_A^c(n)$ be maximum complexity of words of length~$n$ over a finite alphabet $A$. In this paper, we establish that $ L_A^c(n) = \frac n {\log_{|A|} n} \left( 1 + (2+o(1)) \frac {\log_2 \log_2 n}{\log_2 n} \right). $

The paper deals with properties of GM-estimators and GM-tests for linear hypotheses in AR(p)-processes when observations contain outliers. In particular, we obtain the marginal distribution of test statistics, which allows us to prove the robustness of these GM-tests. The scheme of data contamination by additive single outliers with the intensity *O*(*n*−1/2), where *n* is the data level, is considered.

The almost everywhere convergence condition similar to the Menchoff-Rademacher condition is obtained for functional series with some weak analogue of the orthogonality property. As a corollary, results related to almost everywhere convergence of series with respect to Riesz systems, Hilbert and Bessel systems, and frames are obtained.