Asian-European Journal of Mathematics

In ordered semigroups, the notion of $\alpha$-ideals is a generalized concept of $(m,n)$-ideals, where $m,n\in {\mathbb{N}}_0$. We apply the notion of $\alpha$-ideals to characterize all regularities in ordered semigroups.

In this paper, we investigate several forms of transitivity and sensitivity in the dynamical system of semigroup actions. Firstly, we derive some equivalent conditions for a dynamical system to be sensitive. In particular, it is proved that an M-system with an equicontinuous point is minimal as well as equicontinuous, every syndetically transitive system with equicontinuous point is minimal as well as equicontinuous. Also, we derive equivalent conditions for a system to be almost equicontinuous in terms of sensitivity and equicontinuity. Besides, some results on point transitive and densely point transitive system on uniform spaces are obtained.

In this paper, we establish several strong and weak convergence results for a class of mappings which is essentially wider than the class of Suzuki mappings using $K$ iteration process in the context of uniformly convex Banach spaces. We also provide an example for support of our results. The obtained results are the extension and improvement of some recent results in the fixed point approximations theory.

Any positive integer $n\ge 2$ can be written uniquely as $n=mr$ where $m$ is square-full, $r$ is square-free and $gcd(m,r)=1.$ We denote the square-full part and the square-free part of $n$ by $\mathrm{\text{pow}}(n)$ and $\mathrm{\text{sq}}(n),$ respectively. In this paper, we use an elementary method to study the distribution of square-free and square-full part of an integer. Moreover, we also study the distribution of primitive roots modulo a prime $p$ that are square-full number.

In this paper, we define the $q$th Vandermonde determinant and determine the coefficient bounds for the second- and third-order Vandermonde determinant for a Sakaguchi type function ${{𝒮}}^{\ast }({𝕃}_s)$ and ${𝒞}({𝕃}_s)$ which map open unit disc onto the region bounded by Limaçon

The purpose of this paper is to present new additive results for $n$-quasi-isometries operators on Hilbert spaces. In particular Precisely, we focus on the study of the descent of a $n$-quasi-isometry operator. Some spectral properties for this class of operators and decomposition theorems are also given. Part of the results proved in this paper improve and generalize some results known for isometries and quasi-isometries.

Let $𝕍$ be a vector space over a field ${ℱ}$ with basis ${ℬ}=\{{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n\}$. Let $W$ be an $m$-dimensional subspace of $𝕍$ with basis $\{{\alpha }_{w_1},{\alpha }_{w_2},\dots ,{\alpha }_{w_m}\}$, where ${\alpha }_{w_i}$ is some ${\alpha }_j\in {ℬ}$. We introduce the subspace-based nonzero component graph, denoted by ${\mathrm{\Gamma }}_W({𝕍}_{\alpha })=(V,E)$, of the finite dimensional vector space $𝕍$ with respect to $W$ and ${ℬ}$ as follows: $V=𝕍\setminus W$ and for $u,v\in V$, there is an edge between $u$ and $v$ if and only if either $W_u\subset W_v$ or $W_v\subset W_w$, where $u=u_1{\alpha }_1+u_2{\alpha }_2+\cdots +u_k{\alpha }_k$, $1\le k\le n$ and $W_u=\{{\alpha }_{w_1},{\alpha }_{w_2},\dots ,{\alpha }_{w_m}\}\cup \{{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_k\}$. We investigate the connectivity, diameter and completeness of ${\mathrm{\Gamma }}_W({𝕍}_{\alpha })$. Further, we find its domination number and independence number. If $W$ is a subspace of a vector space, we show that the graph is complete if and only if $W$ is a hyperspace. Finally, we determine the degree of each vertex in case the base field is finite.

Let $R$ be a commutative ring with nonzero identity and $Z{(R)}^{\ast }$ be the set of all nonzero zero-divisors of $R$. The zero-divisor graph of $R$, denoted by $\mathrm{\Gamma }(R)$, is a simple graph whose vertex set consists of all elements of $Z{(R)}^{\ast }$, and two distinct vertices $x$ and $y$ are adjacent if and only if $xy=0$. In this paper, we investigate the $A_{\alpha }$-spectral of the zero-divisor graph of the ring ${\mathbb{Z}}_n$. Specially, we study the $A_{\alpha }$-spectral of the zero-divisor graph of the ring ${\mathbb{Z}}_n$ for $n=pq$, $p^{2}q$, $pqr$, where $p$, $q$ and $r$ are distinct prime numbers and $p<q<r$. Moreover, we study the $A_{\alpha }$-spectral of the zero-divisor graph of the ring ${\mathbb{Z}}_{p^t}$, where $p$ is a prime number and $t\ge 2$.

In this paper, using the $q$-Dunkl harmonic analysis introduced in Bettaibi and Bettaieb [q-Analog of the Dunkl transform on the real line, Tamsui Oxf. J. Math. Sci. 25(2) (2007) 117–205] and motivated by Wong’s approach [M. W. Wong, Wavelet Transforms and Localization Operators, Vol. 136 (Springer Science & Business Media, Berlin, 2002)], we first define and study the $q$-wavelets, the continuous $q$-wavelet transforms. Next, we introduce the two-wavelet localization operators, the Schatten–von Neumann properties of these localization operators are established, and for trace class localization operators, the traces and the trace class norm inequalities are presented.

In this paper, we give generalizations followed by refinements of the integral version of Niezgoda’s inequality in several different ways by using weights, functions with nondecreasing increments and isotonic linear functionals.

A family of finite sets is said to be strongly coverable if it has a subfamily with the computable set of canonical indices, such that every set of the coverable family is a subset of some set of the covering subfamily. We prove that a set $A$ is low if and only if every finite strongly coverable $A$-computable family has a positive $A$-computable numbering. We also show that in the previous criterion the strong coverability of a finite family cannot be skipped.

In this paper, we introduce the concept of generalized left derivations in incline algebras and give some basic properties, which will be helpful to extend some results of algebra to generalized left derivations in incline algebras. At the end, we prove under some certain conditions that a Jordan generalized left derivation in incline algebra $M$ is a generalized left derivation in $M$.

Let $k$ be a positive integer. A restrained $k$-rainbow dominating function (RkRD-function) of a graph $G=(V,E)$ is a function $f$ from the vertex set $V$ to the set of all subsets of the set $\{1,2,\dots ,k\}$ satisfying: (i) for any vertex $v\in V$ with $f(v)=\varnothing$ the condition ${\bigcup }_{u\in N(v)}f(u)=\{1,2,\dots ,k\}$ is fulfilled, where $N(v)$ is the set of all vertices adjacent to $v;$ (ii) the subgraph induced by the vertices assigned $\varnothing$ under $f$ has no isolated vertices. The weight of an RkRD-function $f$ is the value $\omega (f)={\sum }_{v\in V}|f(v)|$, and the restrained $k$-rainbow domination number of $G$, denoted by ${\gamma }_{rrk}(G)$, is the minimum weight of an RkRD-function of $G$. In this paper, we continue the study of the restrained $k$-rainbow reinforcement number $r_{rrk}(G)$ of a graph $G$ defined as the cardinality of a smallest set of edges that we must add to $G$ to decrease ${\gamma }_{rrk}(G).$ We shall first show that the decision problem associated with ${\gamma }_{rrk}(G)$ is NP-hard for arbitrary graphs. Then several properties as well as some sharp bounds of the restrained $k$-rainbow reinforcement number are investigated.

The concept of integral as an inverse to a derivation was already introduced for rings and recently also for lattices. Since semirings generalize both rings and bounded distributive lattices, it is natural to investigate integration in semirings. This is our aim in the present paper. We show properties of such integrals from the point of view of semiring operations. Examples of semirings with derivation where integrals are introduced are presented in the paper. These illuminate rather specific properties of such integrals. We show when the set of all integrals on a given semiring forms a semiring again.

A generalized Chen-type inequality and corresponding equality consequences for sequential warped products are proved in this paper. These results, which are based on such warped products having factors holomorphic, totally real and pointwise slant, extend Chen-type inequality for various warped products on nearly Kähler manifolds. Moreover, we also examine the related special cases.

In this study, we establish the binomial transform of the generalized third-order Jacobsthal sequence. We also describe the binomial transform of four special cases of third-order Jacobsthal sequence such as the binomial transform of the third-order Jacobsthal, third-order Jacobsthal–Lucas, modified third-order Jacobsthal–Lucas, third-order Jacobsthal–Perrin sequences. Moreover, we examine their features in more detail.

We present a new family of Gauss $(k,t)$-Horadam numbers and obtain Binet formula of this family. We give the relationship between this family and the known Gauss $t$-Horadam number. Then we prove the Cassini and Catalan identities for this family. Furthermore, we investigate the sums, the recurrence relations and generating functions of this family.

There are various mathematical models that have been designed for forecasting the future behavior of coronavirus spreading, which helps to rapidly control the process while there is no treatment and vaccines. The main aim of this study is to describe COVID-19 dynamics in Turkey by using a Susceptible–Exposed–Infected–Recovered–Deceased (SEIRD) model. For this purpose, a new SEIRD model of nCOVID-19 and its fractional-order version are designed. The basic reproduction number is calculated with the generation operator method. All possible equilibria of the dynamic model are investigated in terms of the basic reproduction number. Further, stability conditions are obtained through the Routh–Hurwitz and Lyapunov stability theories. Finally, some numerical simulations of the dynamic system and its fractional version are given based on the data from the number of nCOVID-19 cases in Turkey. These results provide to implicate the theoretical findings corresponding to the model.

In this paper, we will prove a uniqueness theorem in the case of meromorphic functions on the annuli share $q(q\ge 5)$ distinct elements with different multiple values.

In this paper, a mathematical model of the eye corneal shape is introduced which is a nonlinear boundary value problem. This boundary value problem is converted to a nonlinear integral equation and by means of the collocation method, the problem is reduced into a system of the nonlinear algebraic equations. The approximate solutions are obtained by solving this nonlinear system and numerical results are provided to show that introduced method is efficient. Also, comparisons with other works are made to confirm the reliability of the method.