Let $h$ be a positive increasing on $[0;+\infty)$ function such that $h\Big(x+\frac{1}{h(x)}\Big)=O(h(x))$ as $x\to +\infty$. For measurable by Lebesgue set $E\subset[0;+\infty)$ of finite Lebesgue measure $\mathop{\rm meas}E=\int_{E}dx$, we define the asymptotic $h$-density of $E$ on $+\infty$ by \[{D}_{h}(E)= \varlimsup_{R \rightarrow +\infty} h(R)\cdot \mathop{\rm meas }(E \cap [R;+\infty)).\] Consider the class $\mathcal{H}^{p}$ of an entire functions in $\mathbb{C}^{p}$, that are bounded in an arbitrary domain $\Pi_{R}=\{z=(z_1,\ldots ,z_p)\in\mathbb{C}^p\colon \text{Re} z_j<R_j\}$, $R=(R_{1},\ldots,R_{p})\in\mathbb{R}^{p}_{+}$ as well as in $G(r,A)=G+ rA$ for every fixed $A\in\mathbb{R}^p$ and for each $r>0$, where $G$ is a complete polylinear domain. For a function $F\in \mathcal{H}^p$ and $A\in\mathbb {R}^p$, let $F'_A(w)$ denotes the derivative of $F$ in the direction of $A$ at the point $w\in\mathbb{C}$. Let $F^{(k)}_A(w)=(F^{(k-1)}_A(w))'_A$ denotes the $k$th derivative in the direction of $A$ at the point $w\in\mathbb{C}$. We also denote \[S_F(r,A):=\sup\big\{|F(z)|\colon z\in G(r,A)\big\}=\sup\big\{|F(z)|\colon z\in \partial G(r,A)\big\},\]\[L_F(r,A)=(\ln S_F(r,a))'_+.\] We prove the following statement. Let $F\in \mathcal{H}^p$ and $A\in\mathbb {R}^p$ be such that $L_F(r,A)\uparrow+\infty$ as $r\to+\infty$. Suppose that $\Phi$ is a positive increasing on $[0;+\infty)$ function satisfying $u(r)\ge \Phi(r)$ for all $r\ge r_0$, and $h(r)=o(\Phi(r))$ as $r\to +\infty$. Then there exists a set $E\subset\mathbb {R}_+$ of zero asymptotical $h$-density, i.e. $D_h(E)=0$, such that for every $k\in\mathbb{N}$ we have \[F^{(k)}_A(w)=(1+o(1))\, L^k_F(r,A)\, F(w) \;\;\text{as}\;\; r\to +\infty,\;\; r\in\mathbb {R}_+\setminus E,\] for all points $w\in\partial G(r,A)$ satisfying the inequality $|F(w)|\geq S_F(r,A)/(1+\varepsilon(r))$, where $\varepsilon(r)$ is a given arbitrary function such that $\varepsilon(r)\to + 0$ as $r\to +\infty$.

For an entire transcendental function $f$ and for a sequence $(\lambda_n)$ of positive numbers increasing to $+\infty$, a series $A(z)=\sum\limits_{n=1}^{\infty}a_nf(\lambda_n z)$ is said to be regularly convergent in $\{z: |z|<R[A]<+\infty\}$, if $\mathfrak{M}(r,A)=\sum\limits_{n=1}^{\infty} |a_n|M_f(r\lambda_n)<+\infty$ for all $r\in [0, R[A])$, where $R[A]$ is the radius of a regular convergence of $A(z)$ and $M_f(r)=\max\{|f(z)|:|z|=r\}$. We have found conditions on $(\lambda_n)$ and $f$, under which $\ln\mathfrak{M}(r,A)\sim \ln\,\mu(r,A)$ as $r\to R[A]$, where $\mu(r,A)= \max\{|a_n|M_f(r\lambda_n):n\ge 1\}$ is the maximal term of the series. At the end of the paper, an unresolved problem is stated.

A family $\mathcal{U}$ of non-empty subsets of a set $D$ is called an upfamily if for each set $U\in\mathcal{U}$ any set $F\supset U$ belongs to $\mathcal{U}$. An upfamily $\mathcal L$ of subsets of $D$ is said to be linked if $A\cap B\ne\emptyset$ for all $A,B\in\mathcal L$. A linked upfamily $\mathcal M$ of subsets of $D$ is maximal linked if $\mathcal M$ coincides with each linked upfamily $\mathcal L$ on $D$ that contains $\mathcal M$. The superextension $\lambda(D)$ of $D$ consists of all maximal linked upfamilies on $D$. Any associative binary operation $* : D\times D \to D$ can be extended to an associative binary operation $$*:\lambda(D)\times \lambda(D)\to \lambda(D), \quad \mathcal M*\mathcal L=\Big\langle\bigcup_{a\in M}a*L_a:M\in\mathcal M,\;\{L_a\}_{a\in M}\subset\mathcal L\Big\rangle.$$ In the paper, we investigate the structure of the doppelsemigroup $(\lambda(D),\dashv,\vdash)$ of maximal linked upfamilies on a doppelsemigroup $(D,\dashv,\vdash)$. In particular, we study right and left zeros and identities, commutativity, the center, ideals of the superextension $(\lambda(D),\dashv,\vdash)$ of a doppelsemigroup $(D,\dashv, \vdash)$. We introduce the superextension functor $\lambda$ in the category $\mathbf {DSG}$, whose objects are doppelsemigroups and morphisms are doppelsemigroup homomorphisms, and show that this functor preserves strong doppelsemigroups, doppelsemigroups with left (right) zero, doppelsemigroups with left (right) identity, left (right) zeros doppelsemigroups. Also we prove that the automorphism group of the superextension of a doppelsemigroup $(D,\dashv, \vdash)$ contain a subgroup, isomorphic to the automorphism group of $(D,\dashv, \vdash)$.

In the paper, we obtained exact order estimates of the best $m$-term trigonometric approximation and the best orthogonal trigonometric approximation of functions from the Nikol'skii-Besov $B^{\boldsymbol{r}}_{p,\theta}(\mathbb{T}^d)$ and Sobolev $W^{\boldsymbol{r}}_{p,\boldsymbol{\alpha}}(\mathbb{T}^d)$ classes in the Lebesque subspaces $B_{q,1}(\mathbb{T}^d)$ for some relations between the parameters $p$ and $q$. The received results yield that estimates of the considered approximation characteristics in the multivariate case, in contrast to the univariate, in the spaces $B_{q,1}(\mathbb{T}^d)$ and $L_q(\mathbb{T}^d)$ differ in order. Besides, for some parameter values the obtained exact order estimates of the best $m$-term trigonometric approximation and the best orthogonal trigonometric approximation in the spaces $B_{q,1}(\mathbb{T}^d)$ still remain unknown for the $L_q(\mathbb{T}^d)$-space.

In this paper, our primary focus is on studying the properties of self-similar solutions to the Cauchy problem. We specifically examine the behavior of these solutions in a double nonlinear time-dependent parabolic equation and their absorption in a non-homogeneous medium. Through the research of the topic matter, our aim is to deliver a more thorough comprehension of the finite speed perturbations propagation in the solution of the Cauchy problem for a nonlinear parabolic equation. This establishment of a property is essential to understand the dynamic nature of these equations. Furthermore, we delve into the self-similar analysis of the solution, which allows us to ascertain the condition of Fujita type global solvability for the Cauchy problem in a double nonlinear degenerate-type parabolic equation within a non-homogeneous medium. This analysis provides valuable insights into the behavior and potential solvability of these equations on a global mean. Additionally, we establish estimates for weak solutions depending on the growing density and the value of numerical parameters. By establishing these estimates, we provide a more comprehensive understanding of the behavior of the solutions in different scenarios.

We investigate the following question: does there exist a compatible extension of a given compatible partial metric $p:A^2\to\mathbb R$ on a closed subset $A$ of a partially metrizable space $X$? We obtain a positive answer to this question in the case when the corresponding quasi-metric $q_p(x,y)=p(x,y)-p(x,x)$ has an extension that generates a weaker topology on $X$ (in particular, if $q_p$ is bounded). Moreover, we give an example which shows that in general the answer to the question is negative.

The biharmonic equation in Cartesian coordinates is considered for the case of the upper half-plane. The solution of such a fourth-order partial differential equation for given boundary conditions is represented in the form of an integral of the product of the function and the delta-shaped kernel, which in this paper plays the role of an approximating aggregate. In the paper, we found an exact equality for the upper bound of the deviation of Hölder class functions from the considered biharmonic Poisson operator in the uniform metric.

Let $G$ be a group with binary operation $\ast$. The non-inverse graph (in short, $i^*$-graph) of $G$, denoted by $\Gamma$, is a simple graph with vertex set consisting of elements of $G$ and two vertices $x, y \in \Gamma$ are adjacent if $x$ and $y$ are not inverses of each other. That is, $x- y$ if and only if $x\ast y \neq i_G \neq y \ast x$, where $i_G$ is the identity element of $G$. In this paper, we extend the study of $i^\ast$-graphs to signed graphs by defining $i^\ast$-signed graphs. We characterize the graphs for which the $i^\ast$-signed graphs and negated $i^\ast$-signed graphs are balanced, sign-compatible, consistent and $k$-clusterable. We also obtain the frustration index of the $i^\ast$-signed graph. Further, we characterize the homogeneous non-inverse signed graphs and study the properties like net-regularity and switching equivalence.

In this study, we define Kantorovich-Stancu type Lototsky-Chlodowsky operators and obtain some approximation properties in weighted space of these operators. Furthermore, we prove Voronovskaja type results for the Kantorovich-Stancu type Lototsky-Chlodowsky operators.

This article investigates the properties and monomiality principle within Bell-based Apostol-Bernoulli-type polynomials. Beginning with the establishment of a generating function, the study proceeds to derive explicit expressions for these polynomials, providing insight into their structural characteristics. Summation formulae are then derived, facilitating efficient computation and manipulation. Implicit formulae are also examined, revealing underlying patterns and relationships. Through the lens of the monomiality principle, connections between various polynomial aspects are elucidated, uncovering hidden symmetries and algebraic properties. Moreover, connection formulae are derived, enabling seamless transitions between different polynomial representations. This analysis contributes to a comprehensive understanding of Bell-based Apostol-Bernoulli-type polynomials, offering valuable insights into their mathematical nature and applications.