Asian-European Journal of Mathematics

The study of biordered set plays a significant role in describing the structure of a regular semigroup and since the definition of regularity involves only the multiplication in the ring, it is natural that the study of semigroups plays a significant role in the study of regular rings. Here, we extend the biordered set approach to study the structure of the regular semigroup $(R,\cdot )$ of a regular ring $(R,+,\cdot )$ by studying the idempotents $E_R$ of the regular ring and show that the principal biorder ideals of the regular ring $R$ form a complemented modular lattice and certain properties of this lattice are studied.

The main objective of this paper is to study the continuous wavelet transform in terms of canonical convolution and its adjoint. A relation between the canonical convolution operator and inverse linear canonical transform is established. The continuity of continuous wavelet transform on test function space is discussed.

Let $G$ be a simple graph. The Randić index of $G$ is defined as the sum of $1/\sqrt{d_u{d}_v}$ over all edges $uv$ of $G$, where $d_v$ denotes the vertex degree of $v$ in $G$. Dehghan-Zadeh, Ashrafi and Habibi gave Tetracyclic graphs with extremal values of Randić index. We first point out that Theorem 1 is not completely correct and the number of nonisomorphic tetracyclic graphs on seven vertices given in Fig. 4 is incomplete and incorrect and in this short note, we present the correct version of it.

In this paper, we discuss several properties of ideals, filters, annihilators and maximisors in a general Almost Distributive Lattices (ADLs) and in particular, Almost Boolean Algebras (ABAs). Also, we characterize extreme ideals and filters. Further, several equivalent conditions are obtained in terms of ideals and filters for an ADL to become an ABA.

In this paper, we study the possible transfer of the property of being an $AB$-ring to trivial ring extensions and amalgamated algebras along an ideal. Also, we extend the notion of an almost GCD-domain to the context of arbitrary rings, and we study the possible transfer of this notion to trivial ring extensions and amalgamated algebras along an ideal. Our aim is to provide examples of new classes of commutative rings satisfying the above-mentioned properties.

It is proved that every finite-dimensional algebra is embeddable in a simple finite-dimensional algebra (a suitable isotope of a matrix algebra). An isotope of the 2nd order matrix algebra over an infinite extension of the ground field may contain a trivial ideal.

Every one-sided isotope of a simple unital alternative or Jordan algebra is a simple algebra. Besides, any isotope of a central simple non-Lie Maltsev algebra of characteristic other than 2 and 3 is a simple algebra. But an isotope of a simple Jordan algebra of the symmetric bilinear form on the infinite dimensional space may contain a trivial ideal.

Uniform representation of semigroups is introduced. It is proved that any uniform representation of an ample semigroup can be expressed as the direct sum of some representations obtained via homogenous representations on primitive adequate semigroups. Also, we give the structure of homogenous representations of primitive adequate semigroups. In addition, we consider indecomposable uniform representations of ample semigroups and their constructions.

A total Roman dominating function on a graph $G$ is a labeling $f:V(G)\to \{0,1,2\}$ such that every vertex with label 0 has a neighbor with label 2 and the subgraph of $G$ induced by the set of all vertices of positive weight has no isolated vertex. A set $\{f_1,f_2,\dots ,f_d\}$ of total Roman dominating functions on $G$ with the property that ${\sum }_{i=1}^d{f}_i(v)\le 2$ for each $v\in V(G)$, is called a total Roman dominating family (of functions) on $G$. The maximum number of functions in a total Roman dominating family on $G$ is the total Roman domatic number of $G$, denoted by $d_{\mathrm{\text{tR}}}(G)$. In this paper, we investigate the properties of total Roman domatic number in graphs. In particular, we present some sharp bounds for $d_{\mathrm{\text{tR}}}(G)$ and we determine the total Roman domatic number of some special graphs.

In this paper, we give a numerical approximation to the Caputo–Fabrizio time fractional diffusion equation. The implicit finite differences method is applied to solve a time-fractional diffusion equation with this new fractional derivative. We present the stability and convergence analysis of the proposed numerical scheme. Some numerical problems will be presented to show the accuracy and effectiveness of the method.

In this paper, we propose a generalized $p$-Laplacian Liénard type differential equation with a variable delay. By applying the Mawhin continuation theorem, we established a set of sufficient conditions on the existence of at least one periodic solution with period $T$. It is significant that the growth degree with respect to the variables $x,y$ imposed on $f(t,x,y)$ is allowed to be greater than $p-2$ and the growth degree with respect to the variables $x,z$ imposed on $g(t,x,z)$ is allowed to be greater than $p-1,$ so the result not only improves but also generalizes. Some examples are provided to illustrate the results.

The eccentricity of a vertex $v$ in a graph $G$ is the maximum distance between $v$ and any other vertex of $G.$ A vertex with maximum eccentricity is called a peripheral vertex. In this paper, we study the distance signless Laplacian matrix of a connected graph $G$ with respect to peripheral vertices and define the peripheral distance signless Laplacian matrix of a graph $G$, denoted by $D_p^Q(G)$. We then give some bounds on various eigenvalues of $D_p^Q(G).$ Moreover, we define energy in terms of $D_p^Q(G)$ and give some bounds on the energy.

The main result is to show one application of rank-1 preservers by classifying additive adjugate-commuting preservers between difference square matrix spaces, which is another proof.

Let $(M,g)$ be a closed Riemannian manifold, under some assumptions on $f,P_g^k$ and $h,$ by applying a method used in [Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. Henri Poincaré 9 (1992) 281–304], we show the existence and multiplicity of solutions of the semi-linear elliptic equation: $P_g^k(u)=f(x)|u{|}^{2_k^{♯}-2}u+h(x)$. When $(M,g)$ is an Einsteinian manifold of positive scalar curvature, under additional conditions, we obtain the existence of positive solutions.

Let $X$ be a digraph of order $n$, and ${\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n$ be spectrum of the Hermitian adjacency matrix. The main purpose of this paper is to introduce the Hermitian energy and Hermitian Estrada index of a digraph, both based on the eigenvalues of the Hermitian matrix. Moreover, we establish upper and lower bounds for these new digraph invariants, and relations between them.

In this paper, we consider oblique dual frames and particularly, we obtain some of their characterizations. Among other things, special attention is devoted to the study of the effect of perturbations of frame sequences on their oblique duals. In particular, as a surprising result, we show that a frame sequence is uniquely determined by the set of its oblique dual.

In this paper, using the second-order formulas, we obtained characterizations of invertible algebras principally isotopic to a group or an abelian group.

Let $G$ be a simple connected graph with $n$ vertices, $m$ edges and let ${\mu }_1\ge {\mu }_2\ge \cdots \ge {\mu }_{n-1}>{\mu }_n=0$ be its Laplacian eigenvalues. The Laplacian resolvent energy of graph $G$ is defined by $\mathrm{\text{RL}}(G)={\sum }_{i=1}^n\frac{1}{n+1-{\mu }_i}$. In this paper, we give some new lower bounds for the invariant $\mathrm{\text{RL}}(G)$.

The evolution of some geometric quantities on a compact Riemannian manifold $M^n$ whose metric is Yamabe soliton is discussed. Using these quantities, lower bound on the soliton constant is obtained. We discuss about commutator of soliton vector fields. Also, the condition of soliton vector field to be a geodesic vector field is obtained.