Doklady Mathematics最新文献

For functions from the Sobolev space (overset{circ}{W}{} _{infty }^{n}[0;1]) and an arbitrary point (a in (0;1)), the best estimates are obtained in the inequality ({text{|}}f(a){text{|}} leqslant {{A}_{{n,0,infty }}}(a), cdot ,{text{||}}{{f}^{{(n)}}}{text{|}}{{{text{|}}}_{{{{L}_{infty }}[0;1]}}}). The connection of these estimates with the best approximations of splines of a special type by polynomials in ({{L}_{1}}[0;1]) and with the Peano kernel is established. Exact constants of the embedding of the space (overset{circ}{W}{}_{infty }^{n}[0;1]) in ({{L}_{infty }}[0;1]) are found.

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 October 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.

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 asiom. 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. We also show that this product is decidable.

It is known that a finite set of convex figures on the plane with disjoint interiors has at least one outermost figure, i.e., one that can be continuously moved “to infinity” (outside a large circle containing the other figures), while the other figures are left stationary and their interiors are not crossed during the movement. It has been discovered that, in three-dimensional space, there exists a phenomenon of self-trapping structures. A self-trapping structure is a finite (or infinite) set of convex bodies with non-intersecting interiors, such that if all but one body are fixed, that body cannot be “carried to infinity.” Since ancient times, existing structures have been based on the consideration of layers made of cubes, tetrahedra, and octahedra, as well as their variations. In this work, we consider a fundamentally new phenomenon of two-dimensional self-trapping structures: a set of two-dimensional polygons in three-dimensional space, where each polygonal tile cannot be carried to infinity. Thin tiles are used to assemble self-trapping decahedra, from which second-order structures are then formed. In particular, a construction of a column composed of decahedra is presented, which is stable when we fix two outermost decahedra, rather than the entire boundary of the layer, as in previously investigated structures.

The study of distribution functions (with respect to areas, perimeters) for partitioning a plane (space) by a random field of straight lines (hyperplanes) and for obtaining Voronoi diagrams is a classical problem in statistical geometry. Moments for such distributions have been investigated since 1972 [1]. We give a complete solution of these problems for the plane, as well as for Voronoi diagrams. The following problems are solved: 1. A random set of straight lines is given on the plane, all shifts are equiprobable, and the distribution law has the form (F(varphi ).) What is the area (perimeter) distribution of the parts of the partition? 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 A is the closest of the marked set. The idea is to interpret a random polygon as the evolution of a segment on a moving one and construct kinetic equations. It is sufficient to take into account a limited number of parameters: the covered area (perimeter), the length of the segment, and the angles at its ends. We show how to reduce these equations to the Riccati equation using the Laplace transform.

Given a heavy rigid body with one fixed point, we investigate the problem of orbital stability of its periodic motions. Based on the analysis of the linearized system of equations of perturbed motion, the orbital instability of the pendulum rotations is proved. In the case of pendulum oscillations, a transcendental situation occurs, when the question of stability cannot be solved using terms of an arbitrarily high order in the expansion of the Hamiltonian of the equations of perturbed motion. It is proved that the pendulum oscillations are orbitally unstable for most values of the parameters.

We obtain broad sufficient conditions for reconstructing the coefficients of a Kolmogorov operator by means of a solution to the Cauchy problem for the corresponding Fokker–Planck–Kolmogorov equation.

We consider variational inequalities with invertible operators ({{mathcal{A}}_{s}}{text{:}}~,W_{0}^{{1,p}}left( {{Omega }} right) to {{W}^{{ - 1,p'}}}left( {{Omega }} right),) (s in mathbb{N},) in divergence form and with constraint set (V = { {v} in W_{0}^{{1,p}}left( {{Omega }} right){text{: }}varphi leqslant {v} leqslant psi ~) a.e. in ({{Omega }}} ,) where ({{Omega }}) is a nonempty bounded open set in ({{mathbb{R}}^{n}}) (left( {n geqslant 2} right)), p > 1, and (varphi ,psi {{:;Omega }} to bar {mathbb{R}}) are measurable functions. Under the assumptions that the operators ({{mathcal{A}}_{s}}) G-converge to an invertible operator (mathcal{A}{text{: }}W_{0}^{{1,p}}left( {{Omega }} right) to {{W}^{{ - 1,p'}}}left( {{Omega }} right)), ({text{int}}left{ {varphi = psi } right} ne varnothing ,) ({text{meas}}left( {partial left{ {varphi = psi } right} cap {{Omega }}} right)) = 0, and there exist functions (bar {varphi },bar {psi } in W_{0}^{{1,p}}left( {{Omega }} right)) such that (varphi leqslant overline {varphi ~} leqslant bar {psi } leqslant psi ) a.e. in ({{Omega }}) and ({text{meas}}left( {left{ {varphi ne psi } right}{{backslash }}left{ {bar {varphi } ne bar {psi }} right}} right) = 0,) we establish that the solutions u_s of the variational inequalities converge weakly in (W_{0}^{{1,p}}left( {{Omega }} right)) to the solution u of a similar variational inequality with the operator (mathcal{A}) and the constraint set V. The fundamental difference of the considered case from the previously studied one in which ({text{meas}}left{ {varphi = psi } right} = 0) is that, in general, the functionals ({{mathcal{A}}_{s}}{{u}_{s}}) do not converge to (mathcal{A}u) even weakly in ({{W}^{{ - 1,p'}}}left( {{Omega }} right)) and the energy integrals (langle {{mathcal{A}}_{s}}{{u}_{s}},{{u}_{s}}rangle ) do not converge to (langle mathcal{A}u,urangle ).

This paper is dedicated to studying the algorithmic properties of unars with an injective function. We prove that the theory of every such unar admits quantifier elimination if the language is extended by a countable set of predicate symbols. Necessary and sufficient conditions are established for the quantifier elimination to be effective, and a criterion for decidability of theories of such unars is formulated. Using this criterion, we build a unar such that its theory is decidable, but the theory of the unar of its subsets is undecidable.

In 1993, Kahn and Kalai famously constructed a sequence of finite sets in d-dimensional Euclidean spaces that cannot be partitioned into less than ({{(1.203 ldots + o(1))}^{{sqrt d }}}) parts of smaller diameter. Their method works not only for the Euclidean, but for all ({{ell }_{p}})-spaces as well. In this short note, we observe that the larger the value of p, the stronger this construction becomes.