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 $r\to +\infty$.

Properties of subsymmetric polynomials, analytic functions and some their generalizations on Banach spaces with subsymmetric bases are considered. We prove that if a polynomial on a complex infinite-dimensional Banach space $X$ has a subsymmetric set of zeros, then it is subsymmetric. From here we deduce that the algebra $\mathcal{P}_{\mathfrak{S}}(X)$ of subsymmetric polynomials on $X$ is factorial. We consider conditions when a subsymmetric function on a Banach space can be approximated by subsymmetric analytic functions or polynomials. In addition we construct some weighted backward shift-like mappings on the metric space of point evaluation functionals on $\mathcal{P}_{\mathfrak{S}}(X)$ and prove their topological transitivity.

This study defines certain properties of 2-variable $q$-truncated Tricomi functions $h_{n,q}(x,y)$, such as integral forms, generating functions and series definitions. Additionally, we present the associated 2-variable $q$-Laguerre polynomials, which we utilize to obtain higher order of 2-variable $q$-truncated Tricomi functions and examine the characteristics they possess.

Applying the theory of tensor products of Banach spaces, we study the Banach spaces of normalized Bloch maps from $\mathbb{D}$ (the complex unit open disc) into $X^*$ (the dual of a complex Banach space $X$) that can be represented canonically as the dual of the completion of the tensor product $\mathrm{lin}(\Gamma(\mathbb{D}))\otimes_\alpha X$, where $\mathrm{lin}(\Gamma(\mathbb{D}))$ is the space of $X$-valued Bloch molecules on $\mathbb{D}$ and $\alpha$ is a Bloch cross-norm on $\mathrm{lin}(\Gamma(\mathbb{D}))\otimes X$. We show that the normalized spaces of Bloch maps, $p$-summing Bloch maps and Bloch maps that factor through a Hilbert space admit such a representation. On the converse problem, we characterize when a Banach normalized Bloch space $B(\mathbb{D},X^*)$ is isometrically isomorphic to $(\mathrm{lin}(\Gamma(\mathbb{D}))\widehat{\otimes}_\alpha X)^*$ for some Bloch cross-norm $\alpha$, in terms of the compactness of its unit ball with respect to the weak* Bloch topology.

In this article, we investigate the normalized Laplacian and Randić spectrum of the cozero-divisor graph of a finite commutative ring $\mathfrak{R}$ with identity $1\neq 0$. Let $Z'(\mathfrak{R})$ be the set of non-unit and non-zero elements of ring $\mathfrak{R}$. The cozero-divisor graph of $\mathfrak{R}$, denoted by $\Gamma'(\mathfrak{R})$, is a simple undirected graph having vertex set $Z'(\mathfrak{R})$ and two distinct vertices $u$ and $v$ are joined by an edge if and only if $u\notin v\mathfrak{R}$ and $v\notin u\mathfrak{R}$, where $\alpha \mathfrak{R}$ is the ideal generated by the element $\alpha$ in $\mathfrak{R}$. Specifically, we describe the normalized Laplacian spectrum and Randić spectrum of the graph $\Gamma'(\mathbb{Z}_n)$ for various values of $n$.

This paper investigates the existence of mild solutions for an initial value problem involving fractional-order differential inclusions with nonlocal boundary conditions, specifically infinite-point or Riemann-Stieltjes integral conditions. We establish sufficient conditions for the uniqueness of the solution and examine its continuous dependence on the given data. To demonstrate the practical applicability of our findings, we conclude with two illustrative examples.

The Householder's method is a root-find algorithm which is a natural extension of both the Newton's method and the Halley's method. The current paper focuses on approximating the square root of a positive real number based on these methods. The resulting algorithms can be expressed using Chebyshev polynomials. An extension to the $n$th root is also proposed.

For functions that take values in an isotropic semilinear metric space we prove two sharp Kolmogorov-type inequalities. In the first one we obtain an estimate for the uniform norm of the derivative (in the Rådström sense) of a function using the uniform norm of the function and the $H^\omega$-norm of the function's derivative; here $\omega$ is an arbitrary modulus of continuity. The second one gives an estimate of the uniform norm of a generalized fractional derivative of a function via its uniform norm and its $H^\omega$-norm.

Let $TM$ be the tangent bundle over an $F$-Kählerian manifold endowed with Berger type deformed Sasaki metric $g_{BS}$. In this paper, we obtain the Levi-Civita connection of this metric and study geodesics on the tangent bundle $TM$ and $F$-unit tangent bundle $T_{1,F}M.$ Secondly, we characterize the geodesic curvatures on $T_{1,F}M$. Finally, we present some conditions for a vector field $\xi :M\rightarrow TM$ to be harmonic and study the harmonicity of the canonical projection $\pi :TM\rightarrow M$. In addition, we search the harmonicity of the Berger type deformed Sasaki metric $g_{BS}$ and the Sasaki metric $g_{S}$ with respect to each other.

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 $(D,\dashv, \vdash)$.