Metrization of weighted graphs

We find a set of necessary and sufficient conditions under which the weight $w:E\to\mathbb R^+$ on the graph $G=(V,E)$ can be extended to a pseudometric $d:V\times V\to\mathbb R^+$. If these conditions hold and $G$ is a connected graph, then the set $\mathfrak M_w$ of all such extensions is nonvoid and the shortest-path pseudometric $d_w$ is the greatest element of $\mathfrak M_w$ with respect to the partial ordering $d_1 \leqslant d_2$ if and only if $d_1(u,v) \leqslant d_2(u,v)$ for all $u,v\in V$. It is shown that every nonvoid poset $(\mathfrak M_w,\leqslant)$ contains the least element $\rho_{0,w}$ if and only if $G$ is a complete $k$-partite graph with $k\geqslant 2$ and in this case the explicit formula for computation of $\rho_{0,w}$ is obtained.Comment: 7 figure

On quasiplanes in Euclidean spaces

A variational inequality for the images of $k$-dimensional hyperplanes under quasiconformal maps of the $n$-dimensional Euclidean space is proved when $1\le k\le n-2 .$Comment: 12 page

Equilibrium measures for uniformly quasiregular dynamics

We establish the existence and fundamental properties of the equilibrium measure in uniformly quasiregular dynamics. We show that a uniformly quasiregular endomorphism $f$ of degree at least 2 on a closed Riemannian manifold admits an equilibrium measure $\mu_f$, which is balanced and invariant under $f$ and non-atomic, and whose support agrees with the Julia set of $f$. Furthermore we show that $f$ is strongly mixing with respect to the measure $\mu_f$. We also characterize the measure $\mu_f$ using an approximation property by iterated pullbacks of points under $f$ up to a set of exceptional initial points of Hausdorff dimension at most $n-1$. These dynamical mixing and approximation results are reminiscent of the Mattila-Rickman equidistribution theorem for quasiregular mappings. Our methods are based on the existence of an invariant measurable conformal structure due to Iwaniec and Martin and the \cA-harmonic potential theory.Comment: 17 page

An alternative capacity in metric measure spaces

Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 18, No. 2, pp. 196–208, April–June, 2021.Peer reviewe

Equivalence of Viscosity and Weak Solutions for the p(x)p(x)-Laplacian

We consider different notions of solutions to the $p(x)$-Laplace equation -\div(\abs{Du(x)}^{p(x)-2}Du(x))=0 with $1<p(x)<\infty$. We show by proving a comparison principle that viscosity supersolutions and $p(x)$-superharmonic functions of nonlinear potential theory coincide. This implies that weak and viscosity solutions are the same class of functions, and that viscosity solutions to Dirichlet problems are unique. As an application, we prove a Rad\'o type removability theorem

Hardy's inequality for functions vanishing on a part of the boundary

We develop a geometric framework for Hardy's inequality on a bounded domain when the functions do vanish only on a closed portion of the boundary.Comment: 26 pages, 2 figures, includes several improvements in Sections 6-8 allowing to relax the assumptions in the main results. Final version published at http://link.springer.com/article/10.1007%2Fs11118-015-9463-

Integrated satellite-terrestrial connectivity for autonomous ships:Survey and future research directions

An autonomous vessel uses multiple different radio technologies such as satellites, mobile networks and dedicated narrowband systems, to connect to other ships, services, and the remote operations center (ROC). In-ship communication is mainly implemented with wired technologies but also wireless links can be used. In this survey paper, we provide a short overview of autonomous and remote-controlled systems. This paper reviews 5G-related standardization in the maritime domain, covering main use cases and both the role of autonomous ships and that of people onboard. We discuss the concept of a connectivity manager, an intelligent entity that manages complex set of technologies, integrating satellite and terrestrial technologies together, ensuring robust in-ship connections and ship-to-outside connections in any environment. This survey paper describes the architecture and functionalities of connectivity management required for an autonomous ship to be able to operate globally. As a specific case example, we have implemented a research environment consisting of ship simulators with connectivity components. Our simulation results on the effects of delays to collision avoidance confirm the role of reliable connectivity for safety. Finally, we outline future research directions for autonomous ship connectivity research, providing ideas for further work

Korn’s inequalities for generalized external cusps

In this paper we consider a general class of external cusps defined by linking appropriate collections of John domains. For that class, weighted Korn inequalities are proved by means of rather elementary arguments.Fil: Acosta, Gabriel. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Ojea, Ignacio. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santalo". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santalo"; Argentin