D. N. Azarov, A. A. Kryazheva On the subgroup separability of splitting extensions of groups

Let G be a split extension of a group A by a group B. Suppose that for every natural number n the number of all subgroups of the group A of index n is finite. And let \Omega be a class of groups satisfying the following conditions:
   (i)  if X \in \Omega, then any homomorphic image of X belongs to \Omega,
   (ii) if X \in \Omega, then any subgroup of finite index of X belongs to \Omega.
It is proved that the following two conditions are equivalent.
   1. All \Omega-subgroups in G are finitely separable.
   2. All \Omega-subgroups in A and B are finitely separable, and, moreover, all subgroups of the group A that are the intersections of A with the \Omega-subgroups of G are finitely separable.
This result is a generalization of the following well-known theorem by R. Allenby and R. Gregorac. If the group A is finitely generated, all subgroups of A are finitely separable and all finitely generated subgroups of B are finitely separable, then all finitely generated subgroups of G are finitely separable.

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S. E. Vaganov Polynomial based motion model for optical flow refinement

An algorithm for constructing and refining an optical flow over several video frames by using polynomial models of inter-frame points motion is proposed. A comparative analysis of the accuracy of optical flows constructed in two and three frames is carried out. The paper shows that the use of quadratic models of inter-frame movements of points allows to find a more accurate approximation of the optical flow compared with the two-frame methods.

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B. Ya. Solon, E. V. Eremina E-degree structure of enumerations of natural number sets

The article clarifies the notion of enumeration of the natural number set using one of the fundamental concepts of the computability theory — the concept of enumeration reducibility. With the formalization of the notion of algorithm in terms of Turing machines and the formalization of enumeration, the article gives an equivalent definition of enumeration reducibility.

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S. I. Khashin A comparison of the effectiveness of the activation functions of the neural network

A few simple variants of the neural network architecture with two and three input parameters and with various activation functions are considered. For each case, 40000 attempts are made to train a neural network with various random initial values. The effectiveness of different choices of activation functions is compared and an estimate of the total number of local extremums of the objective function is given.

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J. A. Khashina Representation of an n-quadratic function as a sum of squares

An n-quadratic function represented as a sum of squares of n-linear functions can be represented as such a sum consisting of no more than 2^n–1 terms and a nonnegative constants.

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