Asian-European Journal of Mathematics

In this paper, we study the $L^2$-boundedness and $L^2$-compactness of a class of $h$-Fourier integral operators ($h$-FIOs) with operator symbol. Mainly, we prove that this class is $L^2$-bounded (respectively, $L^2$-compact) if the order function is bounded (respectively, tends to $0$).

The work presented in this paper is to combine the Sumudu transform method with a variational iteration method for solving linear and nonlinear partial differential equations with local fractional derivative. We apply the proposed method to obtain approximate analytical solutions of Klein–Gordon equations with local fractional derivative. The results obtained for the local fractional Klein–Gordon equation in the three cases presented, are in the form of non-differentiable functions. Through this work it can be said that this method is an alternative analytical method for linear and nonlinear local fractional partial differential equations.

In this paper, we establish the structure of the unit group of the group algebra $FG$ where $G$ is an abelian group of order at most $16$ and $F$ is a finite field of characteristic $p$ with $q=p^n$ elements.

In this paper, we introduce a new operator in order to derive some new symmetric properties of $k$-Fibonacci and $k$-Lucas numbers and Fibonacci polynomials. By making use of the new operator defined in this paper, we give some new generating functions for $k$-Fibonacci and Pell numbers and Fibonacci polynomials.

In this paper, we consider the existence of one solution and three solutions for the boundary value system with Sturm–Liouville boundary conditions

For nonzero polynomials $f$ and $g$ over a field $K$, let $L(f/g)$ be the depth (length) of the continued fraction expansion for $f/g$. An upper bound on $L(fX)$, for nonzero polynomial $f$ and rational function $X$ is obtained. Applying this result, an upper bound on the depth of a linear fractional transformation is also established.

The intention of this paper is to prove various stability results of reciprocal-septic and reciprocal-octic functional equations in non-Archimedean fields and nonzero real numbers relevant to Hyers, Rassias, and Găvruţa stability. Appropriate counter-examples are supplied to invalidate the results in the cases of singularities.

In this paper, we study some covering properties in topological spaces defined by preopen sets. We introduce and investigate the properties of the pre-Menger property, the almost pre-Menger property and their star versions. It is shown that the pre-Menger and the semi-Menger [Covering properties defined by semi-open sets, J. Nonlinear Sci. Appl. 9 (2016) 4388–4398] are independent properties.

New subclasses of analytic functions are defined by using the function $f_{\delta }(z)$ given by the integral transform $f_{\delta }(z)={\int }_0^z{(\frac{1+r}{1-r})}^{\delta }\frac{1}{1-r^2}dr$. Bounds for the Fekete–Szegö coefficient functional $|a_3-\mu a_2^2|$ for these classes of functions are derived.

Let $G$ be a finite groups. The power graph of $G$ is a graph with vertex set $G$ and two distinct element $x,y$ are adjacent if one of them is power another. We give a short elementary proof for $\mathrm{\Gamma }({\mathbb{Z}}_n)$ has a maximum number edges in all groups of order $n$.

In this paper, we establish some linear finite difference inequalities and obtain an explicit bound on the unknown function. These inequalities can be used as handy tools to study the qualitative as well as the quantitative properties of solutions of some finite difference equations and its variants.

The Wiener index $W(G)$ of a connected graph $G$ is the sum of distances between all pairs of vertices of $G$. A connected graph $G$ is said to be a cactus if each of its blocks is either a cycle or an edge. Let ${{𝒢}}_{n,t}$ be the set of all $n$-vertex cacti containing exactly $t$ cycles. Liu and Lu (2007) determined the unique graph in ${{𝒢}}_{n,t}$ with the minimum Wiener index. Gutman, Li and Wei (2017) determined the unique graph in ${{𝒢}}_{n,t}$ with maximum Wiener index. In this paper, we present the second-minimum Wiener index of graphs in ${{𝒢}}_{n,t}$ and identify the corresponding extremal graphs, which solve partially the problem proposed by Gutman et al. [Cacti with $n$-vertices and $t$ cycles having extremal Wiener index, Discrete Appl. Math. 232 (2017) 189–200] in 2017.

In this paper, we show some $q$-difference analogues of the second main theorems for algebraically nondegenerate meromorphic mappings over the field ${\varphi }_q^0$ of zero-order meromorphic functions in ${ℂ}^m$ satisfying $f(qz)=f(z)$ intersecting hypersurfaces, located in subgeneral position in ${\mathbb{P}}^n(ℂ)$, where $q=(q_1,\dots ,q_m)\in {ℂ}^m$ and $q_j$ may be different. As an application, we give some unicity theorems for meromorphic mappings of ${ℂ}^m$ into ${\mathbb{P}}^n(ℂ)$ under the growth condition “order $=0$”, which are analogous to Picard’s theorems.

In this paper, we study the notions of $k$-prime and $k$-semiprime ideals of semirings, $k$-$m$-system and $k$-$p$-system. We produce some properties and characterizations for $k$-prime and $k$-semiprime ideals of semirings in terms of $k$-$m$-system and $k$-$p$-system, respectively. We initiate the notion of prime semiring and semiprime semiring to set up characterizations for these two ideals of a semiring. The concept of power semiring associated to a semigroup is also studied in this paper.

Let $G$ be a group and $L_c(G)$ be the $c$-absolute center of $G$, that is, the set of all elements of $G$ fixed by all class preserving automorphisms of $G$. In this paper, we classify all finite groups $G$, whose $c$-absolute central factors are isomorphic to the direct product of cyclic groups, $C_p\times C_p$ and $Q_8$. Moreover, we consider finite groups which can be written as the union of centralizers of class preserving automorphisms and study the structure of $G/L_c(G)$ for groups, in which the number of distinct centralizers of class preserving automorphisms is equal to 4 or 5.

After investigating the concept of congruences and formation of quotient of $le$-semigroups, some semigroups like isomorphism theorems on $le$-semigroups have been proved.

In this paper, nil extensions of simple regular ordered semigroups, left simple and right regular ordered semigroups, etc. have been characterized. Also, we describe the ordered semigroups which are complete semilattices of nil extensions of left simple and right regular ordered semigroups, left group like ordered semigroups, etc.

Let ${𝒜}$ be the family of analytic and normalized functions $f$ in the open unit disc $\left|z\right|<1$. In this paper, we consider the following classes:

Quasi-automatic semigroups are extensions of a Cayley graph of an automatic group. Of course, a quasi-automatic semigroup generalizes an automatic semigroup. We observe that a semigroup $S$ may be automatic only when $S$ is finitely generated, while a semigroup may be quasi-automatic but it is not necessary finitely generated. Similar to the usual automatic semigroups, a quasi-automatic semigroup is closed under direct and free products. Furthermore, a semigroup $S$ is graph automatic if and only if $S$ with a zero element adjoined is graph automatic, and also a semigroup $S$ is graph automatic if and only if $S$ with an identity element adjoined is graph automatic. However, the class of quasi-automatic semigroups is a much wider class than the class of automatic semigroups. In this paper, we show that every automatic semigroup is quasi-automatic but the converse statement is not true (see Example ??). In addition, we notice that the quasi-automatic semigroups are invariant under the changing of generators, while a semigroup may be automatic with respect to a finite generating set but not the other. Finally, the connection between the quasi-automaticity of two semigroups $S$ and $T$, where $T$ is a subsemigroup with finite Rees index in $S$ will be investigated and considered.

The authors, in 2016, introduced the notion of central autocommutator subgroup of a group $G$ and proved some new results concerning the central kernel subgroup of $G$, which was introduced earlier by Haimo in $1955$. In this paper, we establish some results on central kernel quotient of $G$. Finally, we introduce the concept of central autonilpotent (henceforth $\mathrm{\text{CA}}$-nilpotent) groups and determine its relationship with the nilpotency of the subgroups of central automorphism of $G$.