Linear Geometry describes geometric properties that depend on the fundamental notion of a line. In this paper we survey basic notions and results related to completions and free projectivizations of liners. The paper is the second survey in the series of surveys that describe the contents of the monograph ``Linear Geometry and Algebra'' [1]. The first survey [2] concentrated at properties of liners that depend on flat hulls (flats, ranks, regularity, parallelity). In this paper we survey basic notions and results related to completions and free projectivizations of liners. The material covers Chapters 7, 8, 9 of the book [1].
The paper is devoted to the study of the boundary behavior of mappings with finite distortion, more precisely, open discrete mappings with moduli conditions similar to Poletsky inequality in the inverse direction. We study the case when some majorant in the above inequality is integrable over spheres centered at each finite point. It is established that the indicated mappings have a continuous extension to an isolated boundary point of the boundary of a domain whenever the mapping $f$ omits at least one a point. The proof of the main results is step-by-step and is based on the following logic, in line with which we prove that 1) no two different boundary points of the corresponding mapped domain can belong to the limit set of the mapping $f$ at the point $x_0,$ 2) the cluster set of a mapping at a given point is, in principle, always a singleton, provided that this mapping omits at least one finite point. The proofs of the these statements are made by the contradiction. This contradiction, in turn, is ensured by the property of approaching continua in the preimage under the mapping on the one hand, and by the upper bound on the mapped families of paths (taking into account the definition of the mapping through this upper bound) on the other hand. In order to construct the above approaching continua, we use some geometric constructions that take into account the nature (definition) of the class of mappings under study. In particular, to prove that the cluster set under the mapping does not contain more than one boundary point, we use segments joining elements of some approximate sequences with its limit points, and the corresponding converging continua which are the pre-images of these segments under the mapping. Note that, the result obtained in the article was previously obtained by us first for homeomorphisms, then for open discrete closed mappings, and then for open discrete mappings but with an integrable majorant in the inverse Poletsky inequality. We also previously established a similar result, but in the case where the mapping omits at least two points.
This paper investigates the growth properties of analytic functions defined in the unit polydisc of $\mathbb{C}^n$ having bounded $L$-index in a direction. Special attention is devoted to the behavior of such functions under composition and to the relationships between their growth characteristics and the corresponding properties of their components. We establish new estimates for the maximum modulus and propose sufficient conditions under which the composition preserves given growth classes. The obtained results extend several known one-dimensional theorems to the multidimensional setting and refine existing bounds in the theory of entire and analytic functions. Furthermore, we analyze the interplay between the geometry of the polydisc and the growth indicators, revealing specific features that arise in higher dimensions. At the end, we consider one equation from the electromagnetism's theory written by directional derivative and study its analytic solutions.
We give a negative answer to a question in group theory proposed by Vitalyi Sushchanskyy which is an analogue of Koethe conjecture in ring theory.
There are presented sufficient conditions for the Taylor-Dirichlet series of the form $$ F(x)=\sum_{n=0}^{+\infty}a_ne^{x\lambda_n+\tau(x)\beta_n}, $$ and $\lambda=(\lambda_n)$, $\beta=(\beta_n)$ be positive sequences, $\tau\colon\mathbb{R}_+\to\mathbb{R}_+$ be a differentiable function such that $\tau'(x)\geq 0$ $(x\geq x_0)$ providing validity of Wiman-type inequality outside some set $E$ of finite Lebesgue measure. We prove the following statement: If for the sequences $\lambda=(\lambda_n)$, $\beta=(\beta_n)$ and a twice differentiable function $\tau\colon\mathbb{R}_+\to\mathbb{R}_+$ such that $\tau'(x)\geq 0$ $(x\geq x_0)$, $\tau''(x)\geq 0$ $(x\geq x_0)$ there exists a function $\psi\in\mathcal{L}_1$ that the condition $$ \sup\limits_{x>0}\varlimsup\limits_{u\to+\infty}\frac{\ln \nu\big\{n\geq 0\colon u-\sqrt{\psi(u)}<\lambda_n+\tau'(x)\beta_n\leq u+\sqrt{\psi(u)}\big\}}{\ln u}<+\infty $$ holds, where $\nu(D)=\#\{n\geq 0\colon \lambda_n+\tau'(x)\beta_n\in D\}$, then there exists a set $E$ of finite Lebesgue measure such that for arbitrary $\delta>0$ and $x\in \mathbb{R}_+\setminus E$ we have $$ F(x)\leq \mu(x,F)(\ln\mu(x,F))^{\alpha_0+\delta}. $$
