Asian-European Journal of Mathematics

Consider a finite ordered set $X=(X;\le )$ whose order forms a zigzag. For an equivalence $E$ on $X$, let $O_E(X)$ be the set of all order homomorphism preserving $E$. In this paper, we completely describe the regularity of elements and Green’s relations for $O_E(X)$ under a certain condition.

The pre-period of a finite algebra is the maximum pre-period of its endomorphisms. We know that the pre-period of any finite modular lattice is less than or equal to the length of the lattice. A finite modular lattice is said to have the maximum pre-period property (MPP) if its pre-period and its length are equal. In this paper, we study MPP of the direct product of chains.

Let $G$ be a simple, connected graph. The matrix $A_{\alpha }(G)$ is defined as $A_{\alpha }(G)=\alpha D(G)+(1-\alpha )A(G)$, $\alpha \in [0,1]$, where $A(G)$ is the adjacency matrix and $D(G)$ is the degree matrix of $G$. Let $G_1\star G_2$, $G_1\odot G_2$, $G_1\ominus G_2$, $G_1⊡G_2$ and $G_1\boxminus G_2$ denote the neighborhood corona, subdivision vertex corona, subdivision edge corona, central vertex corona and central edge corona of two graphs $G_1$ and $G_2$, respectively. In this paper, we determine the $A_{\alpha }$-characteristic polynomial and $A_{\alpha }$-spectra of non-regular graphs obtained from the above operations. As an application, we construct infinitely many pairs of non-regular $A_{\alpha }$-cospectral graphs. Also, we estimate the $A_{\alpha }$-energy of the graphs obtained by operating a regular graph $G_1$ and a non-regular graph, $K_{p,q}$.

In order to find an analogue of the structure theorem,“a semigroup is a full subdirect product of a semilattice and a group if and only if it is an $E$-inversive sturdy semilattice of cancellative monoids (Theorem 14 of [H. Mitsch, Subdirect products of E-inversive semigroups, J. Austral. Math. Soc. 48 (1990) 66–78])”, in the setting of seminearrings we have characterized the seminearrings which are full subdirect products of a bi-semilattice and a (zero-symmetric) near-ring and which are subdirect products of a distributive lattice and a (zero-symmetric) near-ring.

This paper concerns a new semi analytical method for approximate solution of the first type generalized Abel integral equations. An especial numerical fractional derivative which is L1 method will be used for solving this kind of equations. Also, existence, uniqueness, convergence and stability of the given scheme will be studied through some theorems and lemmas. Moreover, some numerical examples are presented and the results are compared with their exact solutions and some other numerical methods to illustrate the capability of this algorithm for solving these generalized Abel integral equations.

We define and study the windowed Fourier transform, called also the Gabor transform, associated with the spherical mean operator and we prove for this transform the reproducing inversion formulas. We also prove several versions of Heisenberg type inequalities, Donoho–Stark’s uncertainty principles and finally Faris–Price’s uncertainty principle for the previous transform.

As a continuity to the study by Ouchenane et al. [On the Porous-elastic system with thermoelasticity of type III and distributed delay: Well-posedness and stability, J. Funct. Spaces (2021) 1–12], we consider a one-dimensional linear thermoelastic system of porous type with past history and distributed delay term. This model deals with dynamics of engineering structures and non-classical problems of mathematical physics. It includes new results and their proofs with a help of the energy method combined with Lyapunov functional, we discuss the stability of system for the case of equal speeds of wave propagation.

Recently, the spectrum of the q-Cesàro matrix, which is considered an operator on $c_0$, $c$ and ${\ell }^p$, and its various spectral competitions have been determined by Yildirim, Durna–Turkay and El-Shabrawy, respectively. Later, Türkay determined the spectral decompositions of the generalized $\alpha q$-Cesàro operators on $c_0$ and $c$, giving this matrix in its special case. This paper examines the spectra and fine spectra of the generalized $\alpha q$-Cesàro operators on ${\ell }^p$, taking the result of El-Shabrawy as a special case.

Let $Y$ be a nonempty subset of $X$. The invariant subsemigroup of the full transformation semigroup $T(X)$ is denoted by

We extend the concept of nil-extensions and retractive nil-extensions of different classes of semigroups to semirings and find a semiring analog of structure theorems on nil-extensions of different classes of additively regular semirings.

In this paper, we study a class of ${\mathrm{\Delta }}_{\gamma }$-biharmonic parabolic equations with logarithmic nonlinearity. Using potential well method, we first prove the invariance of some sets and vacuum isolating of solutions. Then, by the Galerkin approximation technique and the Sobolev inequality, we obtain the existence of the unique global weak solutions. In addition, we also provide sufficient conditions for the large time decay of global weak solutions and the finite time blow-up of weak solutions.

Many authors gave definitions of a new kind of derivative and integral, moving the roles of subtraction and addition to division and multiplication, and thus established a new calculus, called multiplicative calculus. In this paper, our aim is to bring up this calculus to the attention of researchers and demonstrate its usefulness. We first discussed multiplicative normed spaces by giving some topological properties of the relevant multiplicative normed spaces. Also, we give a generalization of the open mapping theorem and the uniform boundedness principle for nonlinear operators with a number of important examples.

Let $R$ be a commutative ring with unity and $A(R)$ be the set of annihilating-ideals of $R$. The strong annihilating-ideal graph of $R$, denoted by $\mathrm{\text{SAG}}(R)$, is an undirected graph with vertex set $A{(R)}^{\ast }$ and two vertices $I_1$ and $I_2$ are adjacent if and only if $I_1\cap \mathrm{\text{Ann}}(I_2)\ne (0)$ and $I_2\cap \mathrm{\text{Ann}}(I_1)\ne (0)$. In this paper, first we characterize commutative Artinian rings whose strong annihilating-ideal graph is isomorphic to some well-known graphs and then we classify commutative Artinian rings whose strong annihilating-ideal graph is planar, toroidal or projective.

Let $P(z):={\sum }_{j=0}^n{a}_j{z}^j$ be a polynomial of degree at most $n$ with real or complex coefficients, then by a prize winning result of Bernstein

In this paper, we define measurable $S$-acts on monoids and construct a new category, namely Meas Act-$S$. We investigate the behavior of this category with respect to product, coproduct, pushout, pullback, equalizer and coequalizer. Next, we study the injectivity and complete injectivity in category Meas Act-$S$ and show any measurable $S$-act can be embedded into complete injectivity. Moreover, we investigate the Skornjakov’ theorem for measurable S-acts.

In this paper, we consider a class of conformable fractional bilinear systems with multiple inputs. We study the exponential stabilization problem for these systems by using the quadratic Lyapunov functions and some additional hypothesis on the unit sphere. A numerical example is given to illustrate the efficiency of the obtained result.

The reduced power graph of a finite group $\mathrm{\text{G}}$, denoted by ${ℛ}{𝒫}(\mathrm{\text{G}})$, is the graph whose vertices are the elements of the group $\mathrm{\text{G}}$ and two vertices $u,v$ are adjacent if and only if $〈u〉⫋〈v〉$ or $〈v〉⫋〈u〉$. In this paper, we first describe the structure of the reduced power graph of the finite cyclic group ${\mathbb{Z}}_n$. Consequently, we provide a short and alternative proof of one of the results published in (R. Rajkumar and T. Anitha, Laplacian spectrum of reduced power graph of certain finite groups, Linear Multilinear Algebra (2019) 1–18). We characterize the values of $n$ for which ${ℛ}{𝒫}({\mathbb{Z}}_n)$ is a line graph. We then deduce the signless Laplacian spectrum of ${ℛ}{𝒫}({\mathbb{Z}}_n)$ using its structure. We provide lower and upper bounds on the signless Laplacian spectral radius of ${ℛ}{𝒫}({\mathbb{Z}}_n)$. Finally, we conclude the paper by determining the signless Laplacian spectrum of ${ℛ}{𝒫}({\mathrm{\text{D}}}_n)$, where ${\mathrm{\text{D}}}_n$ denotes the dihedral group of order $2n$.

Given a regular multiset $M$ on $[n]=\{1,2,\dots ,n\}$, a partial order $R$ on $M$, and a label map $\pi :[n]\to \mathbb{N}$ defined by $\pi (i)=k_i$ with ${\sum }_{i=1}^n\pi (i)=N$, we define a pomset block metric $d_{(Pm,\pi )}$ on the direct sum ${\mathbb{Z}}_m^{k_1}\oplus {\mathbb{Z}}_m^{k_2}\oplus \cdots \oplus {\mathbb{Z}}_m^{k_n}$ of ${\mathbb{Z}}_m^N$ based on the pomset $\mathbb{P}=(M,R)$. The pomset block metric extends the classical pomset metric introduced by Sudha and Selvaraj and generalizes the poset block metric introduced by Alves et al. over ${\mathbb{Z}}_m$. The space $({\mathbb{Z}}_m^N,d_{(Pm,\pi )})$ is called the pomset block space and we determine the complete weight distribution of it. Further, $I$-perfect pomset block codes for ideals with partial and full counts are described. Then, for block codes with chain pomset, the packing radius and Singleton bound are established. The relation between maximum distance separable (MDS) codes and $I$-perfect codes for any ideal $I$ is investigated. Moreover, the duality theorem for an MDS pomset block code is established when all the blocks have the same size.

The main purpose of this work is to connect the recently introduced iterative approximation scheme known as $KF$-iterative scheme with the class of generalized $(\alpha ,\beta )$-nonexpansive mappings. We consider the setting of a Banach space to prove our weak and strong convergence results. Once again, we prove that the class of $(\alpha ,\beta )$-nonexpansive mapping includes properly the classes of Suzuki nonexpansive, generalized $\alpha$-nonexpansive and as well Reich–Suzuki type nonexpansive mappings by providing a numerical example. Eventually, we prove that $KF$-iterative scheme of this example is essentially more efficient than the other well-known iterative scheme of the literature. The presented work is new and extends some other well-known results in the iterations theory.

In this paper, we firstly provide several new characterizations of quasi-Frobenius rings by using some generalized injectivity of rings with certain chain conditions. For example, $(1)$ a ring $R$ is quasi-Frobenius if and only if $R$ is right $C_{11}$, right minfull with ACC on right annihilators; $(2)$ a ring $R$ is quasi-Frobenius if and only if $R$ is two-sided min-CS with ACC on right annihilators in which $\mathrm{\text{Soc}}{(}_{R}R){\le }_e{R}_R$; $(3)$ a ring $R$ is quasi-Frobenius if and only if $R$ is right Johns left $C_{11}$; $(4)$ a ring $R$ is quasi-Frobenius if and only if $R$ is quasi-dual two-sided $C_{11}$ with ACC on right annihilators. Moreover, it is shown that a ring $R$ is quasi-Frobenius if and only if $R$ is a left $P$-injective left IN-ring with right RMC and $Z(R_R)=Z{(}_{R}R)$. Also, we prove that if $R$ is a right duo, right QF-$3^{+}$ left quasi-duo ring satisfying ACC on right annihilators, then $R$ is quasi-Frobenius. In this paper, several known results on quasi-Frobenius rings are reproved as corollaries.