Vol 215 (2022)
Статьи
Special uniform Vinberg cones and their applications
Abstract
In this paper, we present basic facts of Vinberg’s theory of homogeneous convex cones, primarily the special Vinberg cones associated with Clifford modules, and their generalization. Applications of the cone theory to differential geometry, physics (including supergravity), information geometry, convex programming, and differential equations are briefly discussed.
3-17
Lie algebras of projective motions of five-dimensional pseudo-riemannian spaces. IV. Structure of projective and affine LIE algebras of five-dimensional rigid h-spaces
Abstract
This work is devoted to the problem of studying multidimensional pseudo-Riemannian manifolds that admit Lie algebras of infinitesimal projective (in particular, affine) transformations, wider than Lie algebras of infinitesimal homotheties. Such manifolds have numerous geometric and physical applications. This paper is the fourth part of the work. The first part: Itogi Nauki i Tekhniki. Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. — 2022. — 212. — P. 10–29. The second part: Itogi Nauki i Tekhniki. Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. — 2022. — 213. — P. 10–37. The third part: Itogi Nauki i Tekhniki. Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. — 2022. — 214. — P. 3–20. The last part will be published in the next issue.
18-31
32-39
Geometric approach to the problem of optimal scalar control of two nonsynchronous oscillators
Abstract
The problem of optimal scalar control of a system of two independent harmonic oscillators is considered. For the solution, methods of geometric control theory are used. The vertical subsystem of the Hamiltonian system is examined. Optimal solutions are found in control classes with various number of switchings. Analytical results are illustrated by simulation.
40-51
Doubling of cyclic algebras
Abstract
In this paper, we construct algebras generalizing the ring of complex quaternions and algebras of hypercomplex Clifford numbers. These algebras are obtained from the algebras of cyclic numbers by a modified doubling procedure. Also, we prove basic properties of these algebras, which are similar to the properties of quadratic hypercomplex numbers.
52-57
Asymptotical enumeration of some labeled geodetic graphs
Abstract
We asymptotically enumerate labeled geodetic k-cyclic cacti and obtain asymptotics for the numbers of labeled connected geodetic unicyclic, bicyclic, and tricyclic n-vertex graphs. We prove that under the uniform probability distribution, the probabilities that a random labeled connected unicyclic, bicyclic, or tricyclic graph is a geodetic graph are asymptotically equal to 1/2, 3/20, and 1/30, respectively. In addition, we prove that almost all labeled connected geodetic tricyclic graphs are cacti.
58-67
Spaces with polylinear forms
Abstract
We consider spaces with multilinear forms whose degree is greater than two. The motion groups of such spaces are subgroups of the general linear group whose transformations preserve the given multilinear form. The search for such groups becomes simpler if the multilinear form is defined on the linear space of some algebra and possesses the multiplicative property with respect to multiplication in this algebra. We prove that such a form exists in any associative algebra.
68-72
Beltrami theorem in Minkowski space
Abstract
E. Beltrami proved a theorem on the relationship of curvatures for families of surfaces of revolution in the three-dimensional Euclidean space, which implies that if some surface of revolution orthogonally intersects all surfaces obtained from a surface of constant curvature by translations along the rotation axis, then the curvature of the surface is also constant and differs from the curvature of the surface only in sign. In this paper, we obtain analogs of this theorem for surfaces of revolution in the three-dimensional Minkowski space.
73-80
Integrable homogeneous dynamical systems with dissipation on the tangent bundles of smooth finite-dimensional manifolds. II. Equations of motion on the tangent bundle of an n-dimensional manifold in a potential force field
Abstract
This paper is the second part of the work on the integrability of general classes of homogeneous dynamical systems with variable dissipation on the tangent bundles of n-dimensional manifolds. The first part of the paper is: Integrable homogeneous dynamical systems with dissipation on the tangent bundles of smooth finite-dimensional manifolds. I. Equations of geodesics on the tangent bundle of a smooth n-dimensional manifold// Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 214 (2022), pp. 82–106.
81-94
Polynomial automorphisms, quantization, and Jacobian conjecture related problems. III. Automorphisms, augmentation topology, and approximation
Abstract
This paper is the third part of a review of results concerning the quantization approach to the some classical aspects of noncommutative algebras. The first part is: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 213 (2022), pp. 110–144. The second part is: Itogi Nauki Tekhn. Sovr. Mat. Prilozh.Temat.Obzory,214 (2022), pp. 107–126. Continuation will be published in future issues.
95-128