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In this note we give simple explicit examples of free subgroups of rank 2 in the group of infinite upper unitriangular matrices over integers. The proofs that the given subgroup is free are elementary. Using canonical projections mod p (p — any odd prime) we obtain also free subgroups in the group of finite state automata on alphabet of size p, extending results of Aleshin [1], Brunner–Sidki [2] and Olijnyk–Sushchansky [8, 9] in this direction. As application, we give the simple proofs of two classical results on free groups.

A binary word wxY y is called 2-symmetric for a given group G if wgY h whY g for all gY h in G. In this note we describe 2-symmetric words for the relatively free (nilpotent of class 2)-by-abelian groups and for the relatively free centreby-metabelian groups.1. Introduction. Symmetric words for a group G are closely related to the fixed points of the automorphisms permuting generators in their corresponding relatively free groups. The problem of characterizing the 2-symmetric words for a given group G was initiated by Pøonka ([6], [7]) who, among other things, gave a complete description of the 2-symmetric words for nilpotent groups of class % 3. Further descriptions of 2-symmetric words are now known for the free metabelian groups and free soluble groups of derived length 3 (Macedon  ska and Solitar [4]) and free metabelian nilpotent groups of arbitrary nilpotency class c (Hoøubowski [2]). Further work for free nilpotent groups is under investigation by the second author ([3]). In this note we extend the work in [4] and characterize the 2-symmetric words for free centre-by-metabelian groups (Theorem 3) and for free (nilpotent of class 2)-by-abelian groups (Theorem 2).

The third type of neural network called spiking is developed due to a more accurate representation of neuronal activity in living organisms. Spiking neural networks have many different parameters that can be difficult to adjust manually to the current classification problem. The analysis and selection of coefficients’ values in the network can be analyzed as an optimization problem. A practical method for automatic selection of them can decrease the time needed to develop such a model. In this paper, we propose the use of a heuristic approach to analyze and select coefficients with the idea of collaborative working. The proposed idea is based on parallel analyzing of different coefficients and choosing the best of them or average ones. This type of optimization problem allows the selection of all variables, which can significantly affect the convergence of the accuracy. Our proposal was tested using network simulators and popular databases to indicate the possibilities of the described approach. Five different heuristic algorithms were tested and the best results were reached by Cuckoo Search Algorithm, Grasshopper Optimization Algorithm, and Polar Bears Algorithm.

We investigate the commutators of elements of the group UT(∞, R) of infinite unitriangular matrices over an associative ring R with 1 and a commutative group R * of invertible elements. We prove that every unitriangular matrix of a specified form is a commutator of two other unitriangular matrices. As a direct consequence we give a complete characterization of the lower central series of the group UT(∞, R) including the width of its terms with respect to basic commutators and Engel words. With an additional restriction on the ring R, we show that the derived subgroup of T(∞, R) coincides with the group UT(∞, R). The obtained results generalize the results obtained for triangular groups over a field.

In this note we prove that the group G of infinite dimensional upper unitriangular matrices over a finite field contains an explicit countable subgroup 'full' of free subgroups. We deduce from this fact that, in a suitable sense, almost all A;-generator subgroups of G are free groups of rank k.

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