In this paper, some properties and lemmas in weighted anisotropic Sobolev spaces are proved. As an application, we study the following noncoercive degenerate elliptic equation $$ \begin{array}{lll} Au + H(x,u,\nabla u) = \mu & \quad \mbox{ in } \ \Omega, \\ u = 0 &\quad \mbox{ on }\ \partial \Omega, \end{array}$$ where $\mu$ belongs to $L^{1}(\Omega) + W^{-1,\vec{p'}}(\Omega,\vec{\omega}^{*})$. We establish the existence of solutions for the unilateral problem associated to the strongly nonlinear and noncoercive elliptic equation in anisotropic weighted Sobolev spaces and conclude some regularity results.

{Let $D$ be a circular sector bounded by the rays $\arg z=\pm\pi/4$ and the arc of the unit circle $|z|=1$. If one removes \textquotedblleft one half \textquotedblright of its boundary, the resulting domain becomes a fundamental region for a particular case of a dihedral rotation group. On this domain, a three-term linear functional equation with coefficients holomorphic in $D$ is considered. Its regularization is achieved by means of a Carleman boundary value problem with shift induced by the generating transformations of the group and their inverses. Conditions on the values of the coefficients at the vertices that guarantee such a regularization are established. The Borel transform makes it possible to apply the obtained results to moment problems for entire functions.

Tauberian theorems have found applications in various areas of mathematics and constitute an effective tool for deriving asymptotic formulas of different kinds. There are many problems for which the application of Tauberian theorems leads to remarkably deep results. The nature of the problems solved with the help of Tauberian theorems is very diverse. In solving some of them, it is sufficient to determine only the leading term of the desired asymptotic expansion. In such cases, the classical Tauberian theorems are adequate. In the present paper, we study Tauberian theorems with Riesz and Ces\`aro means for Taylor--Dirichlet series.

The paper studies a family of conformal mappings $f=f(z,t)$, $f:R(t)\times[0,1]\rightarrow\Delta(t)$, $t\in[0,1]$. This family maps rectangles $R(t)$ onto a family of quadrilaterals $\Delta(t)$. The quadrilateral family $\Delta(t)$ is obtained by a parallel translation of one of the sides of some initial quadrilateral $\Delta$. For a fixed $t$, the mapping $f$ maps the rectangle $R(t)$ onto the quadrilateral $\Delta(t)$ in such a way that the vertices of the rectangle $R(t)$ correspond to the vertices of the quadrilateral $\Delta(t)$. A second-order differential equation has been derived for the family of mappings $f$. An ordinary differential equation with a Cauchy initial condition has been obtained for the parameter $p(t)$ (the modulus of the quadrilateral $\Delta(t)$). For certain specific quadrilaterals, the parameter $p(t)$ was found numerically using the derived differential equation.

In an associative ring $\mathcal{R}$ with a unit element denoted by 1, given two elements $a$ and $b\in \mathcal{R}$ satisfying the equations $a^{m+1}=b a^{m} \mbox{ and } b^{m+1}=ab^m\ \mbox{ for some }m\geq 2$. We study common spectral properties for (strong Drazin, generalized strong Drazin, Hirano, generalized Hirano) invertibility of $1-a$ (respectively, $a$) and $1-b$ (respectively, $b$). An application of the obtained results to the $\nu$-convergence in a Banach algebra is given.

This study aims at combining the concepts of $g$-frame and $K$-frame in a Hilbert $C^*$-module $U$ for an operator $K\in {\rm End}_{\mathcal{A}}^*(U)$, where ${\mathcal{A}}$ denotes a complex $C^{*}$-algebra and ${\rm End}_{\mathcal{A}}^*(U)$ denotes the class of all adjointable $\mathcal{A}$-linear maps on $U$. As a result, the concept of a $c$-$K$-$g$-frame for a Hilbert $C^{*}$-module is introduced and then characterizations on a $c$-$K$-$g$-frame are provided. Finally, some results on dual $c$-$K$-$g$-Bessel system in Hilbert $C^{*}$-modules are obtained.

One of the promising directions in number theory is the construction of polynomial primality tests for integers. Such tests can be used in cryptography to improve modern encryption protocols. In the article \textquotedblleft On the Baillie PSW Conjecture\textquotedblright\ we considered a combined primality test based on verifying two conditions: Fermat's little theorem $2^{n-1}\equiv 1(\bmod\ n)$ and Lucas' theorem $F_{n+1}\equiv 0(\bmod\ n)$, where $F_n$ is the $n$-th term of the Fibonacci sequence for numbers $n\equiv\pm 2(\bmod\ 5)$. Odd composite numbers passing this combined test were called FL-pseudoprimes. We established strong restrictions on FL-pseudoprimes and proved that none of them exist in the interval up to $10^{23}$. In the article \textquotedblleft In the present paper, we consider a combination of the Fermat test with a primality test based on elliptic curves over finite fields. This test works successfully for odd integers of a special form, namely, $n\equiv 2(\bmod\ 3)$, for which we formulate a hypothesis analogous to the Baillie-PSW hypothesis.