We consider packings of congruent circles on a square flat torus, i.e., periodic (w.r.t. a square lattice) planar circle packings, with the maximal circle radius. This problem is interesting due to a practical reason the problem of "super resolution of images." We have found optimal arrangements for N = 6, 7 and 8 circles. Surprisingly, for the case N = 7 there are three different optimal arrangements. Our proof is based on a computer enumeration of toroidal irreducible contact graphs.

Mapping every simplex in the Delaunay mosaic of a discrete point set to the radius of the smallest empty circumsphere gives a generalized discrete Morse function. Choosing the points from a Poisson point process in R n , we study the expected number of simplices in the Delaunay mosaic as well as the expected number of critical simplices and non-singular intervals in the corresponding generalized discrete gradient. Observing connections with other probabilistic models, we obtain precise expressions for the expected numbers in low dimensions. In particular, we get the expected numbers of simplices in the Poisson-Delaunay mosaic in dimensions n ≤ 4.

The algorithm to train artificial neural networks for intelligent decision support systems has been constructed. A distinctive feature of the proposed algorithm is that it conducts training not only for synaptic weights of an artificial neural network, but also for the type and parameters of membership function. In case of inability to ensure the assigned quality of functioning of artificial neural networks due to training of parameters of artificial neural network, the architecture of artificial neural networks is trained. The choice of the architecture, type and parameters of membership function occurs taking into consideration the computation resources of the facility and taking into consideration the type and the amount of information entering the input of an artificial neural network. In addition, when using the proposed algorithm, there is no accumulation of an error of artificial neural networks training as a result of processing the information entering the input of artificial neural networks.Development of the proposed algorithm was predetermined by the need to train artificial neural networks for intelligent decision support systems in order to process more information given the unambiguity of decisions being made. The research results revealed that the specified training algorithm provides on average 16–23 % higher the efficiency of training artificial neural networks training that is on average by 16–23 % higher and does not accumulate errors in the course of training. The specified algorithm will make it possible to conduct training of artificial neural networks; to determine effective measures to enhance the efficiency of functioning of artificial neural networks. The developed algorithm will also enable the improvement of the efficiency of functioning of artificial neural networks due to training the parameters and the architecture of artificial neural networks. The proposed algorithm reduces the use of computational resources of decision support systems. The application of the developed algorithm makes it possible to work out the measures aimed at improving the effectiveness of training artificial neural networks and to increase the efficiency of information processing

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