For a topological $T_1$-space we consider a $\Omega$-saturation, which is canonically embedded in the Wallman extension $\omega X$. In a certain sense, this saturation is maximal with respect to inclusion among all saturations of this type. A class of maps $X \overset{f}{\longrightarrow} Y$ which admit a continuous extension $s_{\Delta}X \overset{\tilde{f}}{\longrightarrow} s_{\Delta} Y$, where $s_{\Delta}X$ and $s_{\Delta}Y$ are the $\Omega$-saturations (mentioned above) of the spaces $X$ and $Y$ respectively is found. It is shown that these maps, together with the class of topological $T_1$-spaces, form a category, and the construction of the $\Omega$-saturation considered in the paper defines a covariant functor from the indicated category into the category TOP of topological spaces and continuous maps.
Representation varieties of one Baumslag-Solitar group are investigated. All irreducible components of the representation variety of this group are found, their dimensions are calculated and their rationality is proved.
This paper studies the behavior of $\mathfrak{F}$-accessible subgroups in generalized Frattini extensions.
We discuss methods of functional identification, inverse dynamical systems and stepwise suboptimal optimization for solving inverse problems of reconstruction of coefficients, boundary conditions and transport sources in the nonlinear heat conduction equation.
We consider the linear control periodic system with constant matrix under control. It is supposed that the average matrix coefficient has singular upper left diagonal block. The control problem of asynchronous spectrum is solved.
A non-nilpotent finite group whose all proper subgroups are nilpotent is called a Schmidt group. A subgroup $H$ of a group $G$ is called weakly subnormal in $G$ if $H$ is generated by two subgroups, one of which is subnormal in $G$ and the other is seminormal in $G$. We establish $3$-solvability of a finite group with weakly subnormal $\{2,3\}$-Schmidt subgroups. This implies solvability of a finite group with weakly subnormal $\{2,3\}$-Schmidt subgroups and $5$-closed $\{2,5\}$-Schmidt subgroups. We prove nilpotency of the derived subgroup of a finite group in which all Schmidt subgroups are weakly subnormal.
The article considers conjugate rational trigonometric Fourier series. An integral representation of their partial sums and the Dini test for the convergence of the given series were obtained. The approximation of functions conjugate to $|\sin x|^s$, $s>0$ by partial sums of conjugate rational trigonometric Fourier series is investigated. An integral representation, uniform and point estimates for the above-mentioned approximation were obtained. On the base of the uniform estimate polynomial, a fixed number of geometrically different poles, and general cases were studied.
Classical solutions of problems for a quasilinear hyperbolic equation of the second order in the case of two independent variables with given conditions for the desired function in combination both on characteristic lines and on non-characteristic lines are obtained in the paper. The problems are reduced to a system of equations with a completely continuous operator. Solutions are constructed using the method of successive approximations. In addition, for each problem considered, the uniqueness of the resulting classical solution is shown. Necessary and sufficient matching conditions of given functions are proved in the case of each of the problems considered in the paper, under which classical solutions exist in the presence of a certain smoothness of the given functions.
The present work is devoted to the construction and the rigorous justification of the solution of a boundary value problem of the longitudinal impact on a homogeneous elastic rod of a constant cross-section in the case when one of its ends is rigidly fixed, and the other end has a linear elastic element at the end and was subjected to the impact by some load.
The article is the second in a series of papers where for a set $\pi$ of odd primes $\pi$-solvable finite irreducible complex linear groups of degree $2|H|+1$ whose Hall $\pi$-subgroups are $TI$-subgroups and are not normal in groups. The goal of this series is to prove the solvability and determine the factorization of such groups. The proof of the theorem is continued. Further properties of the minimal counterexample to the theorem are established.
It is proved that there is no algorithm for multiplication of $3\times3$ matrices of multiplicative length $\leq23$ that is invariant under a certain group isomorphic to $S_4\times S_3$. The proof uses description of the orbits of this group on decomposable tensors in the tensor cube $(M_3({\mathbb C}))^{\otimes}$ 3 which was obtained earlier.
