Asian-European Journal of Mathematics

Let ${{\mathscr{H}}}_0$ denote the set of all rings $R$ such that $\text{Nil}(R)$ is a divided prime ideal with $\text{Nil}(R)=\text{Z}(R)$. We study the concept of maximal non-$\lambda$-rings in class ${{\mathscr{H}}}_0$ and generalize the results of maximal non-$\lambda$-domains.

In this paper, we define a particular class of semiclassical Fourier integral operators with operator symbol ($h$-FIOs for short). We study the $L^2$-boundedness and $L^2$-compactness of $h$-FIOs. Mainly, we prove that class of $h$-FIOs denotes Hilbert–Schmidt operators if the order function belongs to $L^2({ℝ}^n)$.

Let $G$ be a graph with $n$ vertices and let $d_i$ be the degree of its $i$th vertex ($d_i$ is the degree of $v_i$), then the Zagreb matrix of $G$ is the square matrix of order $n$ whose $(i,j)$-entry is equal to $d_i+d_j$ if the $i$th and $j$th vertex of $G$ are adjacent, and zero otherwise. We give some bounds for the Zagreb spectral radius in terms of the maximum degree and minimum degree of $G$, the Randić index $R_{-1/2}$, and the first Zagreb index $M_1$. We also obtain some bounds for the Zagreb Estrada index.

In this study, graph burning, a deterministic discrete time graph process that can be interpreted as a model for spreading influence in social networks, is considered. The minimum number of steps in a graph-burning process is called the burning number of that graph. Intrinsically, the burning number of the Jahangir graph was examined, and then the path forests whose vertices are already burned were studied. As a result, the burning numbers of cycles with a single chord and theta graphs were obtained.

Let $h:A\to A$ and $𝜀$ be a partial order on $A$. We deal with properties of oriented graphs which correspond to the algebra $(A,h)$ in the case that $h$ is monotone with respect to $𝜀$. We derive that every mono-unary algebra except connected one with a cycle of odd length has the property that there exists a nontrivial partial order such that $(A,h)$ is monotone with respect to it. All mono-unary algebras such that there exists a linear order such that $h$ is monotone with respect to this order will be described; if the number of components of $(A,h)$ is infinite, then the number of such orders is equal to the cardinality of the power set of $A$.

We associate a simple undirected graph to a ∗-ring $R$ whose vertex set is $V({\mathrm{\Gamma }}_s^{\ast }(R))=\{0\ne a\in R$$|$$r_R(aR)\ne \{0\}\}$ and two distinct vertices $a$ and $b$ are adjacent if and only if $aR{b}^{\ast }=0$. This graph is called the strong zero-divisor graph of a ∗-ring $R$ and denoted by ${\mathrm{\Gamma }}_s^{\ast }(R)$. In general ${\mathrm{\Gamma }}_s^{\ast }(R)$ is not connected. It is proved that ${\mathrm{\Gamma }}_s^{\ast }(R)$ is connected and $\mathrm{\text{diam}}({\mathrm{\Gamma }}_s^{\ast }(R))\le 3$, when $R$ is a principally quasi-Baer (p.q.-Baer) ∗-ring. We characterize the $gr({\mathrm{\Gamma }}_s^{\ast }(R))$ using the lattice of central projections of a p.q.-Baer ∗-ring $R$. We proved a sort of Beck’s conjecture for the strong zero-divisor graph ${\mathrm{\Gamma }}_s^{\ast }(R)$ of a p.q.-Baer ∗-ring $R$. It is proved that for a p.q.-Baer ∗-ring $R$, ${\mathrm{\Gamma }}_s^{\ast }(R)$ is uniquely complemented.

In this paper, we introduce and study a new subclass of starlike functions with respect to $(2ȷ,\ell )$-symmetric conjugate points using the principle of subordination. Several relationship with the well-known classes have been established. We have focussed on conic regions when it pertained to applications of our main results. Inclusion results, subordination property and coefficient inequality of the defined class are the main results of this paper. The applications of the results are presented here as corollaries, most of which are extensions of well-known results.

In this paper, the conformable Laguerre and associated Laguerre differential equations are solved using the Laplace transform. The solution is found to be in exact agreement with that obtained using the power series method. In addition, some properties and some recursion relations of the Laguerre and associated Laguerre functions are discussed and proved. Then, the conformable Rodriguez’s formula and generating function are proposed.

A new graph invariant called the secure vertex cover pebbling number, which is a combination of two graph invariants, namely, ‘secure vertex cover’ and ‘cover pebbling number’, is introduced in this paper. The secure vertex cover pebbling number of a graph, $G$, is the minimum number $m$ so that every distribution of $m$ pebbles can reach some secure vertex cover of $G$ by a sequence of pebbling moves. In this paper, the complexity of the secure vertex cover problem and secure vertex cover pebbling problem are discussed. Also, we obtain some basic results and the secure vertex cover pebbling number for complete $r$-partite graphs, paths, Friendship graphs, and wheel graphs.

In this paper, we study an $(A,Q)$-functional equation

In this paper, we define $n$-generalized Schützenberger-bicrossed product from the point of Combinatorial Group Theory. We give to be a monoid of this new product, then we obtain a presentation for $n$-generalized Schützenberger-bicrossed product of arbitrary monoids. Additionally, we give the necessary and sufficient conditions for the regularity of this product.

Let $G$ be a finite group and ${\mathrm{\text{A}}}_G$ be the automorphism group of $G$ (i.e. $\mathrm{\text{Aut}}(G)$). We associated a bipartite graph, denoted by ${\mathrm{\Gamma }}_G$, to $G$ and its automorphism group $\mathrm{\text{Aut}}(G)$ as follows: two parts of the vertex set are $G\backslash L(G,Z(G))$ and ${\mathrm{\text{A}}}_G\backslash {\mathrm{\text{Aut}}}_c(G)$, where $L(G,Z(G))$ is the set of elements $g\in G$ such that $g^{-1}\alpha (g)\in Z(G)$ for all $\alpha \in {\mathrm{\text{A}}}_G$ and ${\mathrm{\text{Aut}}}_c(G)$ is the set of automorphisms $\beta \in {\mathrm{\text{A}}}_G$ such that $g^{-1}\beta (g)\in Z(G)$ for all $g\in G$. Two vertices $g\in G\backslash L(G,Z(G))$ and $\alpha \in {\mathrm{\text{A}}}_G\backslash {\mathrm{\text{Aut}}}_c(G)$ are adjacent if and only if $g^{-1}\alpha (g)\notin Z(G)$. In this paper, we investigate some fundamental properties of ${\mathrm{\Gamma }}_G$ such as connectivity, diameter, girth, Hamiltonian, independence and dominating numbers. Moreover, planarity and outer planarity of the graph are studied.

Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $S\subseteq V$ such that every vertex not in $S$ is adjacent to at least one vertex in $S$. The cardinality of a smallest dominating set of $G$, denoted by $\gamma (G)$, is the domination number of $G$. A dominating set $S$ is an isolate dominating set of $G$, if the induced subgraph $G[S]$ has at least one isolated vertex. The isolate domination number, ${\gamma }_0(G)$, is the minimum cardinality of an isolate dominating set of $G$. In this paper, we count the number of isolate dominating sets of some specific graphs.

In this paper, a comprehensive study is made on the continuity, compactness, and spectrum of the lower triangular terraced matrix, introduced by Rhaly [$p$-Cesaro matrices, Houston J. Math. 15(1) (1989) 137–46], acting on the weighted null sequence spaces with bounded, strictly positive weights. Several spectral subdivisions such as point spectrum, residual spectrum, and continuous spectrum are also discussed. In addition, a new class of operator ideal ${\chi }_{c_0(r)}^{(s)}$ associated to the Rhaly operator on weighted $c_0$ space is defined using the concept of $s$-number and it is proved that under certain condition, ${\chi }_{c_0(r)}^{(s)}$ forms a quasi-Banach closed operator ideal.

Nowadays, wavelets have played a crucial role in approximating the solution of a wide range of problems arising in science and engineering. Wavelets have been used in numerous areas of applied mathematics as diverse as signal analysis, statistics, computer-aided geometric, image processing and numerical analysis. The uniform Haar wavelets are most widely used wavelets for solving stochastic integral equations, outperforming Legendre, Daubechies, etc. The uniform Haar wavelets are generated from pairs of piecewise constant functions and can be simply integrated. Furthermore, the uniform Haar functions are orthogonal and form a good transform basis. In this paper, we have proposed an efficient numerical method to solve a class of stochastic linear Volterra integral equations (SLVIEs) based on the uniform Haar wavelets. The aim of this paper is to apply numerical method to obtain approximate solution of SLVIEs. By using this method, the SLVIEs can be reduced to a linear system of algebraic equations. Also, the error analysis of this method is performed. Finally, some numerical examples are given to demonstrate pertinent features of this method. Furthermore, we compare obtained results from this method with the achieved results from relevant studies.

The concept of normal category was introduced by Nambooripad in his theory of cross connections for describing the structure of regular semigroups. These are essentially the categories of principal left ideals and the categories of principal right ideals of regular semigroups with appropriate morphisms. These categories also have an associated partial order on the set of objects. Discrete normal categories are normal categories in which the partial order on the object set is the identity relation. A cross connection relates two normal categories ${𝒞}$ and ${𝒟}$ and produces a regular semigroup $S$ such that one is the category of principal left ideals and the other the category of principal right ideals of the semigroup $S$. This paper considers the question of compatibility of two normal categories in order that they can be connected by a cross connection. It is known that given any normal category ${𝒞}$ there is a normal category ${𝒟}={{𝒩}}^{\ast }{𝒞}$ called the normal dual of ${𝒞}$ such that ${𝒞}$ is compatible with ${𝒟}$. The question of finding all normal categories ${𝒟}$ that are compatible with a given normal category ${𝒞}$ remains unsettled. Such categories ${𝒟}$ are called compatible duals of ${𝒞}$. For the class of discrete normal categories the characterization of all compatible duals has been done here.

Various existing works in the literature have addressed the solvability of different infinite systems of boundary value problems in different types of Banach spaces. In most of these works, the authors have restricted themselves in systems of differential equations with boundary conditions defined at two points. In this work, we have addressed the solvability of an infinite system of third-order differential equations with boundary conditions defined at three points in the sequence spaces $c_0$ and ${\ell }_1$. We have also illustrated the results for both sequence spaces with suitable examples.

An endo-commutative algebra is a nonassociative algebra in which the square mapping preserves multiplication. In this paper, we give a complete classification of endo-commutative curled algebras of dimension 2 over an arbitrary nontrivial field, where a curled algebra satisfies the condition that the square of any element is a scalar multiple of that element. We list all multiplication tables of the algebras up to isomorphism.

For a connected graph $G$ of order $n$ with normalized signless Laplacian eigenvalues ${\gamma }_1^{+}\ge {\gamma }_2^{+}\ge \cdots \ge {\gamma }_n^{+}\ge 0$, the normalized signless Laplacian resolvent energy of $G$ is defined as $\mathrm{\text{ERNS}}(G)={\sum }_{i=1}^n\frac{1}{3-{\gamma }_i^{+}}$. In this paper, we derive new inequalities on $\mathrm{\text{ERNS}}(G)$ as well as relations on $\mathrm{\text{ERNS}}(G)$ with Randić (normalized) incidence energy. We also deduced that some of our results improve the existing results in the literature.

Consider a Hom–Lie superalgebra, denoted by ${𝒯}$. In this paper, we inquire the Lie triple derivation $hTDer({𝒯})$ and generalized Lie triple derivation $hGTDer({𝒯})$ of Hom–Lie superalgebra. Later, we address the triple centroid, triple quasi-centroid and central triple derivations of Hom–Lie superalgebra. Moreover, we attempt to show the relationship of triple centroid, triple quasi-centroid and central triple derivation with $hTDer({𝒯})$ and $hGTDer({𝒯})$. We give complete classification of triple derivations of Hom–Lie superalgebra, where we show that for the simple Hom–Lie superalgebras $hTDer({𝒯})=hDer({𝒯})$.