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 $$ F(x)\leq \mu(x,F)(\ln\mu(x,F))^{\alpha_0+\delta}. $$
Let $\mathfrak{E}$ be a unital prime $\ast$-algebra. For any $ \mathscr{U}, \mathscr{V} \in \mathfrak{E}$, the product defined by $ \mathscr{U} \diamond \mathscr{V}=\mathscr{U}^{\ast} \mathscr{V}-\mathscr{V}^{\ast} \mathscr{U}$ is known as bi-skew Lie product of $\mathscr{U}$ and $\mathscr{V}$. This paper establishes that if a family $\Delta=\{\zeta_n\}_{ n \in \mathbb{N}}$ of mappings $\zeta_n : \mathfrak{E} \rightarrow \mathfrak{E}$ (not necessarily linear) on $\mathfrak{E}$ with $\zeta_{0} = id_{\mathfrak{E}}$ (the identity map on $\mathfrak{E}$), satisfies the relation $\zeta_n(\mathscr{U} \diamond \mathscr{V} \diamond \mathscr{W}) = \sum\limits_{p+q+r=n} \zeta_p(\mathscr{U}) \diamond \zeta_q(\mathscr{V}) \diamond \zeta_r(\mathscr{W})$ for all $\mathscr{U}, \mathscr{V}, \mathscr{W} \in \mathfrak{E}$ and for each $n \in \mathbb{N},$ then $\Delta$ is an additive $\ast$-higher derivation provided $\zeta_n\left(\frac{\beta\mathscr{I}}{2}\right)$ is self-adjoint for $\beta \in \{1, i\}.$
The paper devoted to investigation strong solutions of the initial-boundary-value problem for some equations of the Kirchhoff type with avariable exponent of nonlinearity. As we know the equations of the Kirchhoff type of the second order with variable exponents of the nonlinearity are not studied yet. Problems for nonlinear partial differential equations with variable exponents of the nonlinearity are investigated in the generalized Lebesgue and Sobolev spaces. In present paper we investigate strong solutions of the initial-boundary value problem for these equations. This article continues the research which starts in the paper Adv. Math. Sci. App., 23 (2013), 509--528, where we investigated the equation with strong damping. We found the sufficient conditions of the existence and uniqueness of the strong solution to given problem. The proof is based on the Faedo-Galerkin method, the derivation of a priori energy estimates, and the use the embedding results for Lebesgue spaces with variable exponent of nonlinearity. The obtained results can be applied to further studies of global solvability and long-time behavior of solutions for our problem.
For an analytic in the unit disk ${\mathbb D}$ solution of the form $f(z)=F(1/(1-z))$, where $F$ is an entire transcendental function, of the differential equation $(1-z)^nw''+a(1-z)^mw'+bw=0$ (with $n>m\ge 0,\, a\in {\mathbb R},\,b\in {\mathbb R}$) the boundedness of $l$-index of $f$ in ${\mathbb D}$ and $l$-index of $F$ in ${\mathbb C}$ are investigated.
We investigate discrete modified projection-type methods for the numerical approximation of nonlinear Hammerstein integral equations with sufficiently smooth kernels. Such equations arise in various applications and require efficient numerical techniques for their accurate resolution. The proposed approach is based on Legendre polynomial bases, which provide a suitable framework for constructing approximate solutions in appropriate function spaces. By combining these bases with sufficiently accurate numerical quadrature rules, we derive discrete formulations of modified Galerkin-type and modified collocation-type methods. These methods are designed to improve the accuracy of classical projection techniques while maintaining computational efficiency. A comprehensive convergence analysis is performed for both approximate and iterated approximate solutions. Under suitable regularity assumptions on the kernel and the exact solution, we establish superconvergence results, showing that the proposed methods achieve higher-order accuracy compared to standard approaches. Moreover, we provide a rigorous error analysis that highlights the role of discretization and quadrature in the overall approximation process. The obtained theoretical estimates demonstrate that the use of Legendre-based discretization leads to significant improvements in convergence behavior. These results will contribute to the development of efficient numerical approaches for solving nonlinear Hammerstein-type integral equations.
For fixed natural numbers $r$ and $s$, where $2\leq s \leq r$, we consider a representation of num\-bers from the interval $[0;\frac{r}{s-1}]$ obtained by encoding numbers by means of the alphabet $A=\{0,1,...,r\}$ via the expansion $$x=\sum\limits_{n=1}^{\infty}s^{-n}\alpha_n=\Delta^{r_s}_{\alpha_1\alpha_2...\alpha_n...}.$$ The algorithm for expanding a number into such a series is justified in the paper. The geometry of this representation is studied, including the geometric meaning of digits, properties of cylinder sets --- particularly the specificity of their overlaps --- and metric relations, as well as the connection between the representation and partial sums of the corresponding series. The paper also presents results on the study of a function $f$ defined by $$f\Big(x=\sum\limits_{n=1}^{\infty}\frac{\alpha_n}{(r+1)^n}\Big)=\Delta^{r_s}_{\alpha_1\alpha_2...\alpha_n...}, \alpha_n\in A.$$ It is proved that the function $f$ is continuous at every point that has a unique representation in the classical numeration system with base $r+1$, and discontinuous at points having two representations. The function has unbounded variation and a self-affine graph. For $r<2s-1$, the function possesses singleton, finite, countable, and continuum level sets, including fractal ones; for $r>2s-2$, every level set is a continuum, and moreover it is fractal or anomalously fractal.
The paper examines the construction and analysis of a new class of mixed exponential statistical structures that combine the properties of stochastic models and linear positive operators. The aim of the study is to introduce and analyze a generalized family of mixed exponential statistical structures and their corresponding linear positive operators, which include known operators as particular cases. We define two auxiliary statistical structures $\mathbf{B}$ and $\mathbf{H}$ through differential relations between their elements, and construct the main Phillips-type structure. Recurrent relations for the central moments are obtained, their properties are established, and the convergence and approximation accuracy of the constructed operators are investigated. The proposed approach allows mixed exponential structures to be viewed as a generalization of known statistical systems, providing a unified analytical and stochastic description. The results demonstrate that mixed exponential statistical structures can be used to develop new classes of positive operators with controllable preservation and approximation properties. The proposed methodology forms a basis for further research in constructing multidimensional statistical structures, analyzing operators in weighted spaces, and studying their asymptotic characteristics.
For an entire transcendental function $f$ and a sequence $(\lambda_n)$ of positive numbers increasing to $+\infty$ let $A(z)=\sum_{n=1}^{\infty}a_nf(\lambda_n z)$ be a series in the system ${f(\lambda_nz)}$ regularly convergent in $\{z:|z|
The paper is dedicated to the study of the problem of estimating the intensity in the frequency domain of point wave sources moving in three-dimensional space. It is assumed that the wave propagation speed is constant, and the point sources move along known trajectories. The frequency characteristics of the moving sources are considered unknown, and to estimate them, values of the wave field with errors are given at certain discrete moments in time or over a given time interval at known points in space. It is also assumed that the frequency characteristics and errors belong to known sets of corresponding spaces. In the case of specifying the wave field at discrete moments in time, guaranteed estimates of the intensity of point wave sources are obtained as solutions to certain matrix linear equations, while in the case of continuous time, guaranteed estimates of the intensity of point wave sources are obtained as solutions to a system of integral equations. The obtained theoretical results are confirmed by test examples.
The article describes the tensor products of approximation spaces associated with regular elliptic operators on tensor products of Lebesgue spaces $L_2(\partial\Omega)$, where $\partial \Omega$ considers as smooth manifold that describes in the usual way by local system of local coordinates. We use the quasi-normed approximation spaces and subspaces of exponential type functions associated with such operators.} A connection between the tensor products of approximation spaces and interpolation spaces obtained by the real method of interpolation is showed. We prove the direct and inverse approximation theorems for Bernstein–Jackson type inequalities as well as we give the explicit dependence of constants on parameters of approximation spaces. Such constants are expressed via some normalization factor. Application to spectral approximations on tensor products of interpolation spaces associated with regular elliptic operators on compact manifolds is shown. In the article also consider the spectral approximations (Theorem 2), since the subspaces of entire functions of exponential type of regular elliptic operators on compact manifolds coincide with their spectral subspaces (Lemma 3).
Problems for partial differential equations are arise in various branches of mathematics, mechanics, engineering, economics, ecology, and other sciences. Equations with degeneracy in spatial variables by an operator, in particular the singular Bessel operator, describe some diffusion processes, thermal mass transfer phenomena, radial oscillations are found in crystallography and other applied problems. The conditions for the unique solvability of a nonlocal multipoint problem in time for $2b$-parabolic equations with degeneration are investigated. The coefficients of the parabolic equations admit power singularities and degeneration of arbitrary order in any variables on a certain set of points. Estimates of the solutions of the problem in Hölder spaces with power weight are established. The order of the power weight is determined by the magnitudes of the power singularities and degenerations of the coefficients of the $2b$-parabolic equations.
The motivation for writing this article was the discovery of photographs of Sala Weinlös and Irena Krampner. For mathematicians, the first one is especially valuable, as it is the only known photograph of the only woman who earned a doctorate in mathematics in Lviv before World War II. The archival findings presented here complement the few existing publications about Sala Weinlös and introduce another female student of the University of Lviv also highly successful in mathematics, Irena Krampner, who completed a doctoral dissertation in philosophy. In the 1920s--1930s, professors Hugo Steinhaus and Kazimierz Twardowski played a~significant role in their education and shaping their subsequent careers.
Let \( f \) be a meromorphic function in the complex plane \( \mathbb{C} \). The uniqueness problems concerning \( f(z) \) and its shifted counterpart \( f(z+c) \) under various shared-value and growth conditions have been extensively studied over the past two decades. Many researchers have explored the uniqueness of the derivatives of these functions when they share values, considering both cases: ignoring multiplicities (IM) and counting multiplicities (CM). Recently, attention has been directed toward the uniqueness of \( f^{(j)}(z) \) and \( f^{(k)}(z+c) \) in the context of three shared values—specifically, one shared value in the IM sense and two shared values in the partial CM~sense. The objective of this study is to refine and extend the existing sharing conditions. In this article, we investigate the uniqueness between \( f^{(j)}(z) \) and \( f^{(k)}(z+c) \) when sharing two values in the CM sense. We provide examples to demonstrate that the proposed conditions are optimal. Additionally, we examine how deficiency conditions affect value sharing and their influence on the unicity results. Our final result extends our investigation by proving the uniqueness between \( f^{(j)}(z) \) and \( f^{(k)}(z+c) \) for one CM shared value, subject to suitable deficiency conditions. We also show that the same result holds when these derivatives share a value in the IM sense, along with a stronger deficiency condition.
In this paper, we propose a new three-point iterative scheme for solving nonlinear equations, which achieves seventh-order convergence. The method begins with a standard Newton iteration, followed by two weighted-Newton steps constructed using power series expansions. The present manuscript enhances the order of convergence by integrating divided difference techniques with power series approaches, leading to an efficient and reliable iterative process. The order of convergence has been established rigorously as seven, and the corresponding error equations are derived to validate the theoretical results. A comprehensive convergence analysis is carried out, encompassing both local and semilocal convergence aspects. The local convergence results are obtained under assumptions involving only the first derivative of the operator, and a computable radius of convergence is derived. Moreover, the uniqueness of the solution within this radius is also discussed in detail. For the semilocal analysis, we employ the majorizing sequence technique, which ensures convergence from a wider range of initial approximations. Extensive numerical experiments are performed to demonstrate the validity and accuracy of the proposed method. The calculated results show excellent agreement with the theoretical predictions, confirming the robustness and efficiency of the new algorithm, particularly when compared in terms of the number of iterations and the approximated computational order of convergence.
In the paper, we present new findings concerning Cesaro-type operators. Special attention is given to the Cesaro \((C, \alpha)\) summation operators, which form a widely used family of summation methods in Fourier analysis. We establish sharp inequality for the upper bound of uniform deviations of Cesaro \((C, \alpha)\) summation operators of the second order for the class of continuous periodic functions. Let $n\in \mathbb N$ and $\displaystyle x_k^{(n)} = x_{k-1}^{(n)} + {2\pi}/{(2n+1)},$\ $k\in\{ 0, \pm1, \ldots, \pm(n-1), n\},$ be the points from the interval $[-\pi, \pi]$ such that $\displaystyle -\pi \leq x_{-n}^{(n)} < x_{-n+1}^{(n)} < \ldots < x_{-1}^{(n)} < x_0^{(n)} < x_1^{(n)} < \ldots < x_n^{(n)} \leq \pi;$ the set of points $\{x_k^{(n)}\}$ is uniquely determined by the value of $x_0^{(n)}$. The Ces\`{a}ro $(C, \alpha)$ summation operators are defined by $\displaystyle \sigma_n^{(\alpha)}[f]\left( \left\{x_k^{(n)}\right\}; x\right) = \frac{2}{2 n+1} \sum_{k=-n}^n f\left(x_k^{(n)}\right) K_n^{\alpha}\left(x - x_k^{(n)}\right),$ $K_n^{(\alpha)}(t)=\frac{1}{A_n^\alpha}\sum_{\nu=0}^n A_{n-\nu}^{\alpha-1} D_\nu (t),$ where $ D_\nu(t)=\frac{\sin\left(\left(\nu+{1}/{2}\right)t\right)}{2 \sin\left(t/2\right)}$ is the Dirichlet kernel, and $ A_n^\alpha = \frac{(\alpha + 1) \ldots (\alpha + n)}{n!}$ $(n \in \mathbb N),$\ $A_0^\alpha = 1,$\ are the Cesaro numbers, $\alpha > -1$. Let $\mathbb{T} = [-\pi, \pi]$ and $\mathrm C(\mathbb T)$ be the space of continuouson $\mathbb{T}$ functions with the norm $\|f\|_{\mathrm C} = \max\{|f(t)|\colon t\in \mathbb T\}.$ The main result is contained in the following statement (Theorem 1): Let $f\in \mathrm C(\mathbb T)$. Then the inequality $$ \left\|f - \sigma^{(2)}_n[f]\left(\left\{x_k^{(n)}\right\}\right)\right\|_{\mathrm C} \leq \frac{11}{9\ln2}\cdot \ln(n+1) \cdot\omega\left(f; {2\pi}/{(2n+1)}\right),\ n\in \mathbb N, $$ holds. The constant ${11}/{(9\ln2)}$ in this inequality is sharp.
The article is aimed at investigation of a third initial-boundary value problem for a parabolic pseudo-differential equation that is related to an isotropic $\alpha$-stable stochastic process in multidimensional Euclidean space $\mathbb{R}^d$. The equation is linear with constant coefficients {with respect to} the partial derivative in time of unknown function and its fractional Laplacian of the order $\alpha \in (1,2)$. The boundary condition is formed on a bounded closed surface that is sufficiently smooth. It equates a liner combination of inside and outside limits of the pseudo-derivative of order $\beta\in (0, \alpha-1)$ in the normal direction to the surface, and the value of the unknown function itself. We have established some estimates of the fundamental solutions of the given problem.
