Summary The use of statistical methods for short-term forecasts of the underground water regime indicators (level, underground flow) assumes the existence of long-term series of monitoring observations. We performed forecasting of the underground flow to the Southern Bug River (Vinnytsia region) by the method of time series forecasting using the data of hydrogeological observations from 1980 to 2020. Neural networks (NM), which are computing systems capable of adaptation and learning by analyzing positive and negative influences, was used to forecast time series. Multilayer perceptrons (MLP) and radial basis function (RBF) were used as the network basis for prediction, and identical, logistic, and hyperbolic functions as the activation function of hidden neurons. Forecasts obtained using RBF networks turned out to be the best. The most successful was a network of twelve input and twenty hidden neurons. The identified tendency to increase groundwater flow during 2022–2023 is confirmed by the end of the 5–6 year cyclical period, characteristic of low-water years, and the increase in the annual amount of atmospheric precipitation in 2021–2022.

We consider a cooperative packing game in which the characteristic function is defined as the maximum number of independent simple paths of a fixed length included in a given coalition. The conditions under which the core exists in this game are established, and its form is obtained. For several particular graphs, the explicit form of the core is presented.

The dispersion dependences of electron excitations in crystalline graphite and single-layer graphene have been studied taking the electron spin into consideration. The correlations of the energy spectra of electron excitations and, for the first time, the compatibility conditions for two-valued irreducible projective representations characterizing the symmetry of spinor excitations in the indicated structures are determined, as well as the distributions of spinor quantum states over the projective classes and irreducible projective representations for all high-symmetry points in the corresponding Brillouin zones. With the help of theoretical symmetry-group methods for the spatial symmetry groups of crystalline graphite and single-layer graphene (in particular, the splitting of п-bands at the Dirac points), the spin-dependent splittings in their electron energy spectra are found. The splitting magnitude can be considerable, e.g., for dichalcogenides of transition metals belonging to the same spatial symmetry group. But it is found to be small for crystalline graphite and single-layer graphene because of a low spin-orbit interaction energy for carbon atoms and, as a consequence, carbon structures.

We consider a version of the secretary problem where elements may vanish during the selection and become unchoosable. We construct a selection strategy and identify the probability to select the best element, which turns out to be asymptotically maximal as number of elements increases indefinitely. As an auxiliary result of independent interest we establish large deviation probability estimates for sums of independent variables with distinct geometric distribution.

The dispersion dependences of electronic excitations in single-layer graphene and crystalline graphite have been studied taking the electron spin into consideration. Compatibility conditions for two-valued irreducible projective representations characterizing the symmetry of spinor excitations in the above structures and the distributions of spinor quantum states over projective classes and irreducible projective representations at all high-symmetry points in the corresponding Brillouin zones are determined for the first time. The principal existence of the spin-dependent splitting (or merging) of the electronic energy states, in particular, the electronic п-bands at the Dirac points, is established. The magnitude of spin-dependent splitting can be significant, e.g., for the transition-metal chalcogenides belonging to the same spatial symmetry group as crystalline graphite. However, because of the weak spin-orbit interaction for carbon atoms, it turns out small for all carbon structures including single-layer graphene and crystalline graphite.

The knapsack problem with indivisible items as agents is considered. Each agent has certain weight and utility and wants to be in a knapsack. Such situation is treated as a cooperative game with transferable utility. A characteristic function of this game generalizes the characteristic function associated with the bankruptcy problem but, in contrast to the latter case, it is not convex. Nevertheless, it turns out that the core of this game is non-empty. At the end of the paper some special cases of the knapsack problem are studied. For these cases, the Shapley value, the τ-value and also the nucleolus are found in the explicit form.

In this paper, we develop and study a game-theoretic model of mutual choice with two types of agents (groups) as follows. Each agent wants to make a couple with another agent from the opposite group. In contrast to classical best-choice models, two agents make a couple only by mutual agreement. We consider two setups, namely, natural mating (each agent acts in accordance with personal interests) and artificial selection (forced mating to maximize the average quality of couples). In the first case, the Nash equilibrium is determined; in the second case, an optimal selection procedure is designed. We analyze some modifications of the problem with different payoff functions and incomplete information.

The correlation between the vibrational and electron excitation modes in the energy spectra of single-layer graphene and crystalline graphite, as well as the dispersion dependences of those modes, has been studied. The methods of the theory of projective representations of the point and spatial symmetry groups are used for the first time in order to interpret those correlations. The correlations of vibrational and electron excitation spectra and the compatibility conditions for irreducible projective representations in the descriptions of quantum states of graphene and crystalline graphite at various points of their Brillouin zones are determined. For the projective representations of all projective classes belonging to the hexagonal system, standard factor-systems are constructed for the first time. In particular, the factor-systems for electron states are first determined. The results obtained are used to calculate, also for the first time, the correct spinor multiplication tables, i.e. the multiplication tables for elements in double symmetry groups. The developed method is applied to classify all high-symmetry points in the Brillouin zones of single-layer graphene and crystalline graphite with respect to the symmetry type of vibrational excitations.

A possibility to reveal the entanglement in generalized n-qubit two-parameter GHZ states, as well as in any n-qubit states, with the help of the Mermin and Ardehali inequalities from the collection generally called the Mermin-Ardehali-Belinskii-Klyshko inequalities has been studied. Formulas for the calculation of the Mermin and Ardehali correlation functions in any quantum n-qubit states are derived, and criteria of the violation of corresponding inequalities by specific states are obtained. A set of states that are absolutely insensitive to the Mermin and Ardehali operators is revealed. Modified Mermin and Ardehali operators are proposed, the set of which makes it possible to extend the class of n-qubit states, in which quantum correlations can be revealed.

In this paper we discuss decision procedures related to an important aspect of the backgammon game, namely a doubling. We focus on proper choice of the doubling time and optimal strategies on whether to accept or reject the doubling proposed by the opponent. There are two stages of the game that are most amenable to the analysis. The first one is called "the races", during this stage opponent's checkers do not block player's moves, the goal is to put all checkers into "the house" and then to take them off the board. In this case we calculate optimal doubling time as well as the optimal strategy for acceptance and rejection. Another case is the so-called two-steps game, the situation in which each player has at most one turn before the game ends. This situation is analyzed using the concept of complex rational behavior.

The best choice problem (also known as "the secretary problem") is one of the classical in stochastic optimization. In this paper, we consider a modification of the classical secretary problem by adding the second player, who can either help the first player to find the best element by a hint or precludes him by imposing some restrictions on the search. Nash equilibrium has been found in the explicit form of mixed strategies for three different types of the game. The asymptotic behavior of diverse numerical quantities associated with the optimal strategies for both players, as the number of objects tends to infinity, has been studied.

The authors consider a game of the optimal choice where one of the players seeks to decrease the probability of selection of the best object by the other player by imposing some prohibitions and restrictions on browsing of certain elements. Optimal players' strategies that form the Nash equilibrium are found. The asymptotic behavior of the strategies in case where the number of objects tends to infinity is studied.

On the basis of a system of four qubits, the influence of white and colored noises in the states of initially prepared entangled qubit pairs on the final state obtained as a result of the entanglement swapping has been considered. The corresponding density matrices are obtained, and the redistribution of fractions for the pure state and white and colored noises is analyzed. Conditions for the entanglement preservation and destruction in the course of the transition from the initial to the final state are determined. A comparison between the von Neumann entropy for the initial and final states of qubits is carried out.

The robustness of Bell's [in the Clauser-Horne-Shimony-Holt (CHSH) form] inequality violation for an entangled state under the simultaneous presence of colored and white noises in the system is studied. The two-photon polarization state is modeled by a two-parameter density matrix. By choosing the parameters, one can set a relative fraction of pure entangled Bell's state, as well as the fractions of white and colored noises. The analysis of the dependence of Bell's operator on the parameters is made. Computational results are compared with experimental data [9] and with those computed within the one-parameter density matrix [8] which is a special case of the model considered in this work.

For the systems of three nucleons, (2p, n) and (2n, p), in the doublet state in spin, it is established that, respectively, from six and four spatial components of wave functions in the standard form with the use of the isospin formalism, only two components are independent. The formulas for the construction of the full antisymmetric wave functions in terms of two independent components are obtained. The systems of two equations for independent spatial components in the doublet state of nuclei 3 He and 3 H and one equation for the quartet state are formulated. It is shown that the physical characteristics, which are calculated in the representation with the use of the isospin formalism and in the representation without isospin, coincide. The perspective of the new approach for the execution of precise studies of few-nucleon systems is discussed.

A technique is proposed for evaluating the probability of aircraft collision by the importance sampling technique. Upper limits for the size of an auxiliary sample are found, which ensure the prescribed relative accuracy of probability evaluation. Theoretical conclusions are confirmed by numerical experiments.