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Algebras of distributions of binary isolating and semi-isolating formulae are derived objects for a given theory and reflect binary formula relations between 1-type realizations. These algebras are related to the following natural classification questions: 1) for a given class of theories, determine which algebras correspond to theories from that class, and classify those algebras; 2) classify theories from the class according to the isolating and semi-isolating formulae algebras defined by those theories. The description of a finite algebra of binary isolating formulas unambiguously implies the description of an algebra of binary semi-isolating formulas, which makes it possible to trace the behavior of all binary formula relations of a given theory. The paper describes algebras of binary formulas for tensor products. The Cayley tables are given for the obtained algebras. Based on these tables, theorems are formulated describing all algebras of binary formulae distributions for tensor multiplication theory of regular polygons on an edge. It is shown that they are completely described by two algebras

Recently, many control designers have worked on design methods that meet several design specifications called multi-objective control design. However, the main challenge for the Model Predictive Control design is the high computational load preventing its application to the fast dynamic control of the system in real time. To meet this challenge, this paper proposes a new modified Model Predictive Control design for nonlinear systems with probabilistic uncertainties that guarantees robust stability and performance of the systems, using the Linear Matrix Inequality"LMI". Introducing our robust Model Predictive Control state feedback, the control law will be calculated by step-by-step optimization, and the LMI solutions can be found to stabilize the Linear Parameter-Varying "LPV"system with disturbance rejection ability. Then, a Tensor Product (TP) model transformation is constructed as a powerful tool in the modeling of the complex nonlinear systems.

The article considers construction of generalized crossed product of an arbitrary quaternions skew field and Klein four-group relative to factor system. It is well known that such crossed product is semisimple ring. Under specific conditions it is easy to show that crossed product of simple algebra and its inner automorphism group is central simple algebra. Finding out the conditions under which the crossed product is division algebra we face to difficulties of general case linked to analysis of system of linear equations defined over non-commutative rings. In the terms of anisotropic quadratic forms, there are sufficient conditions under those the crossed product of skew field of quaternions and four-group relative to factor system is division algebra. In addition, it is proved that the crossed product is the tensor product of two quaternions skew fields.