The category of Chu spaces is constructed from a given symmetric monoidal closed category 𝐾 and a fixed object in it. If the object is not fixed, we obtain the category 𝐶ℎ𝑢(𝐾), whose objects are Chu spaces and whose morphisms are defined in a more general way. E.E. Skurikhin defined and studied the category of 𝒯 -Chu spaces associated with an arbitrary functor 𝒯 from the product of categories 𝑁 𝑜𝑝 and 𝑀 to the category of sets. In this paper, we prove that if the functor 𝒯 is closed, then the category 𝑀 is isomorphic to a reflective subcategory of the category of 𝒯 -Chu spaces. In the case where the categories 𝑁 and 𝑀 are complete, we present constructions of limits and colimits of arbitrary functors with values in the category of 𝒯 -Chu spaces. We also prove that if 𝐾 is a symmetric monoidal closed category, then 𝐶ℎ𝑢(𝐾) admits the structure of a closed right-monoidal category. As a consequence of the results on 𝒯 -Chu spaces, it follows that the category 𝐶ℎ𝑢(𝐾), as well as the categories of Chu spaces over it, are complete and cocomplete whenever 𝐾 is.

Closed classes under superposition are examined in 𝑘-valued logic. E. Post established that the lattice (on inclusion) of all closed classes in two-valued logic is countable. Besides, each closed class has a finite basis in two-valued logic. Yu. I. Yanov and A. A. Muchnick proved that the lattice of all closed classes in 𝑘-valued logic is continuous at each 𝑘 > 3. Besides, there are closed classes without a basis and closes classes of a countable basis in 𝑘-valued logic at 𝑘 > 3. Because of the continuity on the lattice of all closed classes at 𝑘 > 3, its sub-lattices are examined. In particular, the closed class of all functions, that are represented by polynomials modulo 𝑘, is considered in 𝑘-valued logic. This closed class contains all functions of 𝑘-valued logic, if and only if 𝑘 is a prime number. If 𝑘 is a composite number, then this closed class is not even pre-complete. In works of A. N. Cherepov, A. B. Remizov, A. A. Krokhin, K. L. Safin, E. V. Sukhanov, D. G. Meschaninov and of others the structure of sub-lattices and of over-lattices is examined for the closed class of all polynomial functions at composites 𝑘. In this work at each composite number 𝑘 the continuity of the sub-lattice is established for the closed class of all polynomial functions in 𝑘-valued logic.

A non-local search method for extremum in nonlinear optimal control problems is developed, based on the use of the property of hidden convexity of the set of admissible velocities of controlled dynamic systems. Extended optimal control problems are formed, which in some cases can be characterized by convex reachable sets. Five variants of the convexification method of the initial optimal control problem are proposed, depending on the way of accounting for the constraints on auxiliary controls. The results of computational experiments on a test collection of nonlinear optimal control problems with geometric constraints are presented. Conclusions are formulated based on the obtained experimental experience and the use of the expansion correctness criterion. The proposed approach allows for the convexification of the velocity set and expands the applicability of numerical optimization methods in the study of non-convex optimal control problems.

We continue to explore the multi-agent logic of computational trees relative to the relational Kripke semantics of possible worlds: we investigate the question of logical solvability, the complexity of model construction, feasibility testing, and correctness. For the semantics introduced earlier, we proved a strong finite model property, and obtained polynomial estimates of the dimension of minimal models for an arbitrary formulas. We proved the recursive enumerability of finite frames of logic and proposed an effective algorithm for checking the feasibility of formulas, which makes it possible to conclude the solvability of logic. The obtained polynomial estimates are within the framework of theoretical expectations, which makes it possible to perceive the logic under study as an effective tool for analyzing multi-agent distributed systems and practical modelchecking, and to count on a positive resolution of issues of unification and description of the admissibility for inference rules.

This article concerns the notion of weak circular minimality being a variant of o-minimality for circularly ordered structures. We consider the binary level of these structures forming algebras of binary isolating formulas, which are based on families of labels and compositions of related formulas. These algebras are studied for ℵ0-categorical 1-transitive non-primitive weakly circularly minimal theories of convexity rank greater than 1 with a trivial definable closure having a non-trivial monotonic-to-right function to the definable completion of a structure. On the basis of the study, the authors present a description of these algebras. It is shown that for this case there exist only commutative algebras. A strict 𝑠-deterministicity of such algebras for some natural number 𝑠 is also established.

The main object of our study is the Dirichlet kernel. The properties of this trigonometric polynomial — the sum of cosines of multiple arcs — are of undoubted interest in the theory of trigonometric series. For example, the results on the asymptotic behavior of the Lebesgue constants, which are the integral norms of the Dirichlet kernels, are well known. These results are constantly being developed and generalized as applied to various systems of functions in both one-dimensional and multidimensional situations. In this paper, we find the leading term of the asymptotics for the value of the global minimum of the Dirichlet kernel as its number tends to infinity. The leading term is the product of the said number by a negative constant, which coincides with the value of the global minimum of the sinc-function (cardinal sine). The proof uses the connection between Dirichlet kernels and Chebyshev polynomials of the second kind. As can be seen from the authors’ previous works, the result undergoes quantitative changes in the transition to lacunary sums of cosines. Our interest in such constructions is caused by the problem posed several years ago by L. E. Rossovskii and A. A. Tovsultanov on calculating the spectral radius for a special one-parameter family of functional operators. The question reduces to studying the behavior of “long” products of sines with lacunae in the arguments. It is shown that the revealed asymptotic property of Dirichlet kernels turns out to be useful in a similar “non-lacunary” problem.

Within the framework of the theory of microstructural deformation, a new approach is proposed for constructing a solution to the bending equation of a long rectangular nanoplate that is under the influence of a transverse load. The proposed approach is based on the collocation method using a system of orthogonal Chebyshev polynomials of the first kind. The bending function is represented as a partial sum of a series of these polynomials. The roots of Chebyshev polynomials of the first kind are chosen as the collocation points. By sequentially multiplying the left and right sides of the resulting matrix equation by the inverse matrix to the matrix with the values of Chebyshev polynomials at the collocation points and by the generalized inverse matrix to the degenerate matrix of differentiation of these polynomials, the equation of the bending surface, taking into account boundary conditions, is reduced to a system of linear algebraic equations with respect to unknown coefficients in the representation of the solution. In this case, the elements of each of these matrices are presented explicitly. An estimate of the error of the constructed solution based on an infinite norm is obtained. The results of the conducted computational experiments are presented, which demonstrate the effectiveness of the proposed approach.

Given a finite group 𝐿, let 𝑁(𝐿) denote the set of its conjugacy class sizes. Let 𝑋 and 𝑌 be sets of natural numbers, 𝐺 be a finite group such that 𝑁(𝐺) = 𝑋 × 𝑌 . In the article [16] the question is formulated: for which sets 𝑋 and 𝑌 is it true that 𝐺 ≃ 𝐴 × 𝐵, where 𝑁(𝐴) = 𝑋 and 𝑁(𝐵) = 𝑌 ? More than 30 years ago, J. Thompson formulated a conjecture that any finite simple group is uniquely determined by its set of sizes of conjugacy classes in the class of finite groups with trivial center. In 2019, the validity of this conjecture was proven. In 2020, it was noted that in addition to simple groups, some direct products of simple groups are also determined by this set. We prove that if 𝑁(𝐺) = 𝑁(𝐴𝑙𝑡𝑝 × 𝐴𝑙𝑡5), where 𝑝 is a prime greater than 1361 and the group 𝐺 has a trivial center, then 𝐺 ≃ 𝐴𝑙𝑡5 × 𝐴𝑙𝑡𝑝.

In this paper, Shunkov groups are studied in the context of the well-known question of B. Amberg and L. S. Kazarin about the structure of groups containing direct products of a finite number of dihedral groups. To do this, two additional finiteness conditions are imposed on the Shunkov group: it is required that the Shunkov group be periodic and also be saturated with direct products of a finite number of finite dihedral groups.

The article is devoted to the scientific and pedagogical activity of Doctor of Physical and Mathematical Sciences, Professor of the Department of Function Theory at Siberian Federal University Evgeniy Konstantinovich Leinartas (1954–2024).