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| Year 2020 |
| Issue 1 |
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E. D. Loginova, D. I. Moldavanskii
On the conjugacy finitely separability of cyclic subgroups A subgroup H of a group G is said to be conjugacy finitely separable if, for every element g \in G that is not conjugate to any element from H, there exists a homomorphism \varphi of the group G onto some finite group such that the element g\varphi is not conjugate in the group G\varphi to any element from the subgroup H\varphi. It is known that, in any free group, all finitely generated subgroups are conjugacy finitely separable, but the direct product of two free groups of rank 2 contains a finitely generated subgroup that is not conjugacy finitely separable. It is proved, however, that all cyclic subgroups of the direct product of two free groups are conjugacy finitely separable. It is also proved that the free product of any family of groups with the property of conjugacy finitely separability of all cyclic subgroups is a group, in which all cyclic subgroups are conjugacy finitely separable.
E. V. Sokolov
On the separability of subgroups of bounded solvable groups This paper is an extended version of the report made by the author at the meeting of the algebraic seminar of Ivanovo State University dedicated to the 110th birthday of academician Anatoly Ivanovich Mal'cev. Some results of A. I. Mal'cev on the finite separability of subgroups and their analogues for the case of separability by the class of finite \pi-groups are described. The application of these results for studying the residual properties of free constructions of groups is indicated.
B. Ya. Solon
Weakly total and cototal functions and partial degrees The article deals with new notions of weak totality and cototality of functions and enumeration degrees of such functions. For the first time, these notions applied to sets were considered by the author in 2005. Recently, a group of American mathematicians referred to the topic of cototal sets and enumeration degrees in view of the fact that the most fundamental concept in various applications is the property of cototality.
S. I. Khashin
A comparison of the activation functions of the neural network We consider a 3-layer neural network with two inputs for the regression problem: 10 neurons in the first and the second layers. The 3rd, output, layer always contains 1 neuron. For all neurons in each inner layer, the same activation function is selected. We consider 9 different activator functions in each layer, 81 options in total. As an example, three different training matrices are considered (4096 training vectors in the first, 255025 in the second, and 70000 in the third). The paper compares the effectiveness of various selections of activator functions. |
| Issue 2 |
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D. N. Azarov, B. Ya. Solon
To the 80th anniversary of Professor D. I. Moldavanskii A brief overview of the scientific and pedagogical activities of Professor D. I. Moldavanskii is given.
A. F. Vyalov
On the set of feasible solutions of the linear programming problem At research the not completely degenerate problem of linear programming in the n-dimensional Euclidean space, it is proved in the paper that the (n-m)-dimensional set of feasible solutions of the problem represents either a convex polyhedron, or a blunted cone in an (n-m)-dimensional plane. Possible outcomes of the problem solution are proved using the projection of an arbitrary set of points to a straight line in the n-dimensional Euclidean space. The necessary and sufficient condition is formulated at which the set of feasible solutions of the linear programming problem is empty.
D. I. Moldavanskii
About isolators of finitely generated subgroups of free groups It is well known that in any free group the isolator of a finitely generated subgroup is a finitely generated subgroup. A very simple proof of this statement is proposed.
B. Ya. Solon
Service computing systems and enumeration operators In this article, the concept of a service system is formulated. For such systems, the concept of a realizable relation is defined and a thesis similar to the Turing thesis is formulated. Using this thesis, it is proved that a suitable formalization of the concept of service allows us to consider systems with an outside service as enumeration operators.
S. I. Khashin
Forecasting the brightness of the next point using a neural network When compressing images, an important step in any algorithm is predicting the value of each byte (of the image data before filtering) based on the corresponding byte of the pixel to the left, the pixel above, and the pixel above and to the left or some combination thereof. For example, in the png algorithm for predicting the brightness of the next point through the previous one, one of five methods is used: "None", "Sub", "Up", "Average", "Paeth". In this paper, we use neural network methods to predict the brightness of the next point. We propose the most universal way to describe the architecture of neural networks and the format of training data. The results of a large number of numerical experiments are considered. The error of neural networks is compared with the error obtained by other methods. |
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