Volume 515, Nº 1 (2024)

Orbital invariants of integrable billiards on two-dimensional book tables are studied at constant energy values. These invariants are calculated from rotation functions defined on one-parameter families of Liouville 2-tori. For two-dimensional billiard books, a complete analogue of Liouville’s theorem is proved, action-angle variables are introduced, and rotation functions are defined. A general formula for the rotation functions of such systems is obtained. For a number of examples, the monotonicity of these functions was studied, and edge orbital invariants (rotation vectors) were calculated. It turned out that not all billiards have monotonic rotation functions, as was originally assumed by A. Fomenko’s hypothesis. However, for some series of billiards this hypothesis is true.

We study the asymptotic behavior of the solution to the diffusion equation in a planar domain, perforated by tiny sets of different shapes with a constant perimeter and a uniformly bounded diameter, when the diameter of a basic cell ε goes to 0. This makes the structure of the heterogeneous domain aperiodical. On the boundary of the removed sets (or the exterior to a set of particles, as it arises in chemical engineering), we consider the dynamic unilateral Signorini boundary condition containing a large-growth parameter β(ε). We derive and justify the homogenized model when the problem’s parameters take the “critical values”. In that case, the homogenized is universal (in the sense that it does not depend on the shape of the perforations or particles) and contains a “strange term” given by a non-linear, non-local in time, monotone operator H that is defined as the solution to an obstacle problem for an ODE operator. The solution of the limit problem can take negative values even if, for any ε, in the original problem, the solution is non-negative on the boundary of the perforations or particles.

The present work deals with the characteristic polynomial of Laplacian matrix for circulant graphs. We show that it can be decomposed into a finite product of algebraic function evaluated at the roots of a linear combination of Chebyshev polynomials. As an important consequence of this result we get the periodicity of characteristic polynomials evaluated at the prescribed integer values. Moreover, we can show that the characteristic polynomials of circulant graphs are always perfect squares up to explicitly given linear factors.

The work is devoted to the analysis of a nonlinear integral equation that arises as a result of parametric closure of the third spatial moment in a single-species model of logistic dynamics by U. Dieckmann and R. Law. The case of piecewise constant kernels is analyzed, which is very important for further computer modeling. Sufficient conditions have been found to guarantee the existence of a nontrivial solution to the equilibrium equation. The use of constant kernels made it possible to obtain more accurate results compared to earlier works, in particular, more accurate estimates were obtained for the norm of the solution, as well as for the closure parameter.

In classical texts equations for gravitation and electromagnetic fields are proposed without derivation of the right-hand sides [1–4]. Here we suggest the derivation of the right-hand sides and analyze momentum-energy tensor in the framework of Vlasov–Maxwell–Einstein equations and Milne–McCree model. We propose new models of accelerated expansion of the Universe without Einstein lambda.

The study of distribution functions (by areas, perimeters) for partitioning a plane (space) by a random field of straight lines (hyperplanes) and for Voronoi mosaics is a classical problem of statistical geometry. Starting from 1972 [1] to the present, moments for such distributions have been investigated. We give a complete solution of these problems for the plane, as well as for Voronoi mosaics. We investigate the following tasks:

1. A random set of straight lines is given on the plane, all shifts are equally probable, and the distribution law has the form F(φ). What is the distribution of the parts of the partition by areas (perimeters)?
2. A random set of points is marked on the plane. Each point A is associated with a “region of attraction”, which is a set of points on the plane to which the point A is the closest of the set marked.

The idea is to interpret a random polygon as the evolution of a segment on a moving one and construct kinetic equations. At the same time, it is sufficient to take into account a limited number of parameters: the area covered (perimeter), the length of the segment, the angles at its ends. We will show how to reduce these equations to the Riccati equation using the Laplace transform. (see theorems 1, 1 and 2).

We consider products of modal logics in topological semantics and prove that the topological product of S4.1 and S4 is the fusion of logics S4.1 and S4 plus one extra axiom. This is an example of a topological product of logics that is greater than the fusion but less than the semiproduct of the corresponding logics.

In 1993, Kahn and Kalai famously constructed a sequence of finite sets in d-dimensional Euclidean spaces that cannot be partitioned into less than  parts of smaller diameter. Their method works not only for the Euclidean, but for all l_p-spaces as well. In this short note, we observe that the larger the value of p, the stronger this construction becomes.

Since 2012 NFV (Network Functions Virtualisation) technology has evolved significantly and became widespread. Before the advent of this technology, proprietary network devices had to be used to process traffic. NFV technology allows you to simplify the configuration of network functions and reduce the cost of traffic processing by using software modules running on completely standard datacenter servers (in virtual machines). However, deploying and maintaining virtualised network functions (such as firewall, NAT, spam filter, access speed restriction) in the form of software components, changing the configurations of these components, and manually configuring traffic routing are still complicated operations. The problems described exist due to the huge number of network infrastructure components and differences in the functionality of chosen software, network operating systems and cloud platforms. In particular, the problem is relevant for the biomedical data analysis platform of the world-class Scientific Center of Sechenov University.

In this article, we propose a solution to this problem by creating a framework TOMMANO that allows you to automate the deployment of virtualised network functions on virtual machines in cloud environments. It converts OASIS TOSCA [5][6] declarative templates in notation corresponding to the ETSI MANO [2] for NFV standard into normative TOSCA templates and sets of Ansible scripts. Using these outputs an application containing virtualised network functions can be deployed by the TOSCA orchestrator in any cloud environment it supports. The developed TOMMANO framework received a certificate of state registration of the computer program No. 2023682112 dated 10.23.2023.

In addition, this article provides an example of using this framework for the automatic deployment of network functions. In this solution Cumulus VX is used as the provider operating system of network functions. Clouni is used as an orchestrator. Openstack is used as a cloud provider.

A change has been added to the article by A.A. Onoprienko, “On analogs of Erbran and Harrop's theorems for the joint logic of QHC problems and statements,” published in 2023, vol. 514, pp. 123-128 (https://doi.org/ 10.31857/S2686954323602324, EDN: WMAEPN).