Asian-European Journal of Mathematics

Let $L$ be a Lie algebra and $I\ne 0$ be a proper ideal of $L$. Then $(L,I)$ is called a Camina pair if $I\subseteq [x,L]:=\{[x,y]|y\in L\}$ for all $x\in L\setminus I$. Also, $L$ is called a Camina Lie algebra if $(L,L^2)$ is a Camina pair. In this paper, we give some properties of Camina Lie algebras. Moreover, we show that a nilpotent Camina Lie algebra of finite dimension over an algebraically closed field is nilpotent with nilindex at most $3$.

In the chemical graph theory, graph invariants are usually referred to as topological indices. The second Zagreb index (denoted by $M_2$) is one of the most studied topological indices. For $n\ge 5$, let ${𝕋𝔼𝕋}_n$ be the collection of all non-isomorphic connected graphs with $n$ vertices and $n+3$ edges (such graphs are known as tetracyclic graphs). Recently, Habibi et al. [Extremal tetracyclic graphs with respect to the first and second Zagreb indices, Trans. on Combin. 5(4) (2016) 35–55.] characterized the graph having maximum $M_2$ value among all members of the collection ${𝕋𝔼𝕋}_n$. In this short note, an alternative but relatively simple approach is used for characterizing the aforementioned graph.

It is known that hypermonoid is closely related to GHS-acts. In this paper, we initiate the study of injective object in the category ${𝒢}{\mathscr{H}}{𝒜}$ct-S and explore its properties. Besides other results, we prove that the category ${𝒢}{\mathscr{H}}{𝒜}$ct-S has enough injective objects and every GHS-act has a minimal injective extension.

The distance signless Laplacian matrix $D^Q(G)$ of a connected graph $G$ is defined as $D^Q(G)=\mathrm{\text{Tr}}(G)+D(G)$, where $D(G)$ is the distance matrix of $G$ and $\mathrm{\text{Tr}}(G)$ is the diagonal matrix whose main entries are the vertex transmissions of $G$, and the spectral radius of a connected graph $G$ is the largest eigenvalue of $D^Q(G)$. In this paper, first we obtain the $D^Q(G)$-eigenvalues of the join of certain regular graphs. Next, we give some new bounds on the distance signless Laplacian spectral radius of a graph $G$ in terms of graph parameters and characterize the extremal graphs. Utilizing these results we present some upper and lower bounds on the distance signless Laplacian energy of a graph $G$.

The main aim of this paper is to define the localization of a tetravalent modal algebra $A$ with respect to a topology ${ℱ}$ on $A$. In Sec. 5, we prove that the tetravalent modal algebra of fractions relative to a ∧-closed system (defined in Definition 3.1) is a tetravalent modal algebra of localization.

In this paper, we introduce the concept of algebra cone generalized $b$-metric space over Banach algebra by replacing the constant $s\ge 1$ with $r(s)\ge 1$ where $r(s)$ is the spectral radius of $s$, with this modification we shall prove Banach and Kannan fixed point results for contractive generalized Lipschitz mappings in such a space. Moreover, we present one example in support of our results.

In this paper, we introduce the concepts of semisimple semirings, Jacobson radical and socle of semirings based on their co-ideals and record some results involving them.

Let $L$ be a Lie algebra and $\mathrm{\text{Der}}(L)$ and $\mathrm{\text{IDer}}(L)$ be the set of all derivations and inner derivations of $L$, respectively. A derivation $\alpha$ of a Lie algebra $L$ is pointwise inner if $\alpha (x)\in Ima{d}_x$ for all $x\in L$. The set of all pointwise inner derivations of Lie algebra $L$ denoted by ${\mathrm{\text{Der}}}_c(L)$ form a subalgebra of $\mathrm{\text{Der}}(L)$ containing $\mathrm{\text{IDer}}(L)$. In this paper, we prove that, if $L$ is nilpotent of class $k$ (solvable of length $k$), then ${\mathrm{\text{Der}}}_c(L)$ is nilpotent of class $k-1$ (solvable of length $k$ or $k-l$). We also prove that if ${\mathrm{\text{der}}}_c(\frac{L}{Z_j(L)})$ is nilpotent of class $k$, then ${\mathrm{\text{der}}}_c(L)$ is nilpotent of class at most $k+j$, in which ${\mathrm{\text{der}}}_c(L)=\frac{{\mathrm{\text{Der}}}_c(L)}{\mathrm{\text{IDer}}(L)}$ and $Z_j(L)$ is the $j$th term in the upper central series of $L$.

Let ${𝔽}_q$ be finite field with $q$ elements, where $q$ be power of an odd prime $p.$ In this paper, we determine generator polynomials of all cyclic and negacyclic codes of length $8{\ell }^m{p}^n$ over ${𝔽}_q,$ where $p,\ell$ are distinct odd primes and $m,n$ are positive integers. We also determine all self-dual, self-orthogonal, complementary-dual cyclic and negacyclic codes of length $8{\ell }^m{p}^n$ over ${𝔽}_q.$

In this paper, we study skew cyclic codes over the ring $F_q+u{F}_q+v{F}_q$, where $u^2=u$, $v^2=v$, $uv=vu=0$, $q=p^m$ and $p$ is a prime. We define a Gray map from $F_q+u{F}_q+v{F}_q$ to $F_q^3$ and investigate the structural properties of skew cyclic codes over $F_q+u{F}_q+v{F}_q$ using decomposition method. It is shown that the Gray images of skew cyclic codes of length $n$ over $F_q+u{F}_q+v{F}_q$ are the skew $3$-quasi cyclic codes of length $3n$ over $F_q$. Finally, the idempotent generators of skew cyclic codes over $F_q+u{F}_q+v{F}_q$ have also been discussed.

In this paper, we introduce and study some strongly almost convergent double sequence spaces by Riesz mean associated with four-dimensional bounded regular matrix and a Musielak–Orlicz function over $n$-normed spaces. We make an effort to study some topological and algebraic properties of these sequence spaces. We also study some inclusion relations between the spaces. Finally, we establish some relation between weighted lacunary statistical sequence spaces and Riesz lacunary almost statistical convergent sequence spaces over $n$-normed spaces.

In this paper, power options pricing is driven via time-fractional PDE when the dynamic of underlying asset price follows a regime switching model in which the risky underlying asset depends on a continuous-time hidden Markov chain process. An exact solution for power options pricing is driven under our considered model.

Let $G=(V(G),E(G))$ be a simple graph. A hop dominating set $S\subseteq V(G)$ is called a connected hop dominating set of $G$ if the induced subgraph $〈S〉$ of $S$ is connected. The smallest cardinality of a connected hop dominating set of $G$, denoted by ${\gamma }_{\mathrm{\text{ch}}}(G)$, is called the connected hop domination number of $G$. In this paper, we characterize the connected hop dominating sets in the join, corona and lexicographic product of graphs and determine the corresponding connected hop domination number of these graphs. The study of these concepts is motivated with a social network application.

In this paper, a collocation method by using Clenshaw–Curtis points is proposed to solve the Fredholm integral equations (FIEs) with highly oscillatory kernels. The collocation method is being applied to graded and uniform meshes. Due to the highly oscillatory kernels of integral equations, the discretized collocation equation will lead to the computation of the oscillatory integrals which will be computed by using the efficient Filon-type method. Finally, the effectiveness and accuracy of the proposed method are confirmed by numerical examples.

A ring $R$ is called $n$-coherent if every $(n-1)$-presented ideal of $R$ is $n$-presented. In this paper, the $n$-coherence of the perfect closure $R^{\infty }$ is investigated. We show that if $R^{\infty }$ is $n$-coherent, then it is normal. Also, it is shown that the perfect closure of a Noetherian reduced ring is $n$-coherent if and only if the perfect closure of its Henselization is $n$-coherent. Among other results, we study the $n$-coherence of $R^{\infty }$ when $R$ is a retract algebra.

In 1970, R. Schmidt gave a structural classification for CA-groups. In this paper, we consider a condition upon the intersection of element centralizers which turns out to be equivalent to the definition of a CA-group. We then weaken which centralizers we chose to intersect and structurally classify this new family of groups. Furthermore, we apply a similar weakening to the class of F-groups introduced by Itô in 1953 and classified by Rebmann in 1971.

A supercharacter theory for a finite group $G$ is a set of superclasses each of which is a union of conjugacy classes together with a set of sums of irreducible characters called supercharacters that together satisfy certain compatibility conditions. The aim of this paper is to give a description of some finite simple groups with exactly three or four supercharacter theories.