卷 42, 编号 4 (2018)

The classical two-dimensional Fuller problem is considered. The boundary value problem of Pontryagin’s maximum principle is considered. Based on the central symmetry of solutions to the boundary value problem, the Pontryagin maximum principle as a necessary condition of optimality, and the hypothesis of the form of the switching line, a solution to the boundary value problem is constructed and its optimality is substantiated. Invariant group analysis is in this case not used. The results are of considerable methodological interest.

An iterative procedure is proposed for calculating the number of k-valued functions of n variables such that each one has an endomorphism different from any constant and permutation. Based on this procedure, formulas are found for the number of three-valued functions of n variables such that each one has nontrivial endomorphisms. For any arbitrary semigroup of endomorphisms, the power is found of the set of all three-valued functions of n variables such that each one has endomorphisms from a specified semigroup.

The problem of selecting the optimum system of models for forecasting short-term railway traffic volumes is considered. The historical data is the daily volume of railway traffic between pairs of stations for different types of cargo. The given time series are highly volatile, noisy, and nonstationary. A system is proposed that selects the optimum superpositioning of forecasting models with respect to features of the historical data. A model of sliding averages, exponential and kernel-smoothing models, the ARIMA model, Croston’s method, and LSTM neural networks are considered as candidates for inclusion in superpositioning.