The results of forecasting the first wave of the spread of COVID-19 coronavirus infection based on the simplified Baroyan–Rvachev model are presented.
The properties of the topology $\tau_{inf}$, which is the infimum of the set of all topologies generated by the Hausdorff metrics on the hyperspace $\exp X$ of a metrizable topological space $X$ are studied. As one of the main results necessary and sufficient conditions for the metrizability (with Hausdorff metric) of $\tau_{inf}$ are obtained. We also show that $\exp X$ with the topology $\tau_{inf}$ is first-countable space if and only if a space $X$ is locally compact and second-countable. Besides we investigate relations between $\tau_{inf}$ and other topologies on the $\exp X$: Vietoris topology, Fell topology and locally finite topology.
Throughout the article, all groups are finite. We say that a subgroup $A$ of $G$ is $\pi$-quasinormal in $G$, if $A$ is $1 \pi$-subnormal and modular in $G$. We prove that if the group $G$ is $\pi _{0}$-solvable, where $\pi _{0}=\pi (D) $ and $D$ is the $\pi $-special residual of $G$, and $\pi$-quasi-normality is a transitive relation in $G$, then $D$ is an abelian Hall subgroup of odd order in $G$.
A subgroup $H$ of a finite group $G$ is called a weakly $\mathbb{P}$-subnormal subgroup if $H$ is generated by two subgroups, one of which is subnormal in $G$, and the other one can be connected to $G$ by a subgroup chain with prime indexes. We establish the properties of weakly $\mathbb{P}$-subnormal subgroups and one makes possible to extend the known results on finite groups with sets of $\mathbb{P}$-subnormal subgroups to finite groups with weakly $\mathbb{P}$-subnormal subgroups. In particular, we prove that a finite group with weakly $\mathbb{P}$-subnormal normalizers of Sylow subgroups is supersolvable and a group with weakly $\mathbb{P}$-subnormal $B$-subgroups is metanilpotent.
Let $\mathfrak{X}$ be a non-empty class of finite groups. A complete lattice $\theta$ of formations is said $\mathfrak{X}$-separable if for every term $\eta(x_1, \ldots , x_n)$ of signature $\{\cap, \vee_{\theta}\}$, $\theta$-formations $\mathfrak{F}_1, \ldots , \mathfrak{F}_n$, and every group $A\in \mathfrak{X}\cap \eta(\mathfrak{F}_1, \ldots , \mathfrak{F}_n)$ are exists $\mathfrak{X}$-groups $A_1\in\mathfrak{F}_1, \ldots , A_n\in\mathfrak{F}_n$ such that $A\in\eta(\theta\textup{form}(A_1), \ldots , \theta\textup{form}(A_n))$. In particular, if $\mathfrak{X}=\mathfrak{G}$ is the class of all finite groups then the lattice $\theta$ of formations is said $\mathfrak{G}$-separable or, briefly, separable. It is proved that the lattice $c^{\tau}_{\omega_\infty}$ of all $\tau$-closed totally $\omega$-composition formations is $\mathfrak{G}$-separable.
Under certain restrictions, Weyl submodules with small highest weights in the restrictions of irreducible representations of simple algebraic groups to subsystem subgroups of type $A_1$ over a field of positive characteristic are found.
In this article we study minimal $\sigma$-local non-$\mathfrak H$-formations of finite groups (or, in other words, $\mathfrak H_\sigma$-critical formations), i. e. such $\sigma$-local formations not included in the class of groups $\mathfrak H$, all of whose proper $\sigma$-local subformations are contained in $\mathfrak H$. A description of minimal $\sigma$-local non$\mathfrak H$-formations for an arbitrary $\sigma$-local formation $\mathfrak H$ of classical type is obtained (а $\sigma$-local formation is called a $\sigma$-local formation of classical type if it has a $\sigma$-local definition such that all its non-Abelian values are $\sigma$-local). The main result of the work in the class of $\sigma$-local formations solves the problem of L. A. Shemetkov (1980) on the description of critical formations for given classes of finite groups. As corollaries, descriptions of $\mathfrak H_\sigma$-critical formations are given for a number of specific classes of finite groups, such as the classes of all $\sigma$-nilpotent, meta-$\sigma$-nilpotent groups, as well as the class all groups with $\sigma$-nilpotent commutator subgroup.
The subgroups $A$ and $B$ are said to be $\mathrm{cc}$-permutable, if $A$ is permutable with $B^x$ for some ${x\in \langle A,B\rangle}$. A subgroup $A$ of a finite group $G$ is called conditionally seminormal subgroup in $G$, if there exists a subgroup $T$ of $G$ such that $G=AT$ and $A$ is $\mathrm{cc}$-permutable with all subgroups of $T$. In this paper, we proved the supersolubility of a group $G = AB$, where $A$ and $B$ are supersoluble conditionally seminormal subgroups in $G$, in the following cases: the derived subgroup $G^\prime$ is nilpotent; ${(|A|,|B|)=1}$; $G$ is metanilpotent and ${(|G:A|,|G:B|)=1}$; $G$ is metanilpotent and ${(|A/A^{\frak N}|,|B/B^{\frak N}|)=1}$. Besides, we obtained the supersolubility of a group in which maximal subgroups, Sylow subgroups, maximal subgroups of every Sylow subgroup, minimal subgroups, 2‑maximal subgroups are conditionally seminormal subgroups.
The article is the third one in a series of papers, where for a set $\pi$ consisting of odd primes, finite π‑solvable irreducible complex linear groups of degree $2|H|+1$ are investigated, for which Hall π‑subgroups are $TI$-subgroups and are not normal in groups. The purpose of the series is to prove solvability and to determine the conditions for factorization of such groups. The proof of the theorem is continued. Further properties of the minimal counterexample to the theorem are established.
