Asian-European Journal of Mathematics

Let $P(S)$ be the set of all nonempty subsets of a ternary semigroup $S$. Then $P(S)$ is a ternary semigroup with respect to the ternary multiplication defined by $ABC=\{abc:a\in A,b\in B,c\in C\}$ for all $A,B,C\in P(S)$. If $S_1$ and $S_2$ are isomorphic ternary semigroups, then the corresponding power ternary semigroups $P(S_1)$ and $P(S_2)$ are obviously isomorphic. It is quite natural to ask whether the converse is true, i.e. is it true that for any ternary semigroups $S_1$ and $S_2$, $P(S_1)\cong P(S_2)$ implies that $S_1\cong S_2$? If the class ${𝒜}$ of algebra has this property, we say that ${𝒜}$ is a globally determined class. In this paper, we provide some class of globally determined ternary semigroups. We show that all ternary semigroups are not globally determined but some special classes of ternary semigroups are globally determined. We show that the class of finite left (right) zero ternary semigroups, ternary groups and principal ideal (P. I.) ternary semigroups are globally determined but the class of involution ternary semigroups is not globally determined.

The harmonic index of a graph $G$, denoted by $H(G)$, is defined as the sum of weights $2/[d(u)+d(v)]$ over all edges $uv$ of $G$, where $d(u)$ denotes the degree of a vertex $u$. Hu and Zhou [WSEAS Trans. Math. 12 (2013) 716–726] proved that for any unicyclic graph $G$ of order $n\ge 3$, $H(G)\le \frac{n}{2}$ with equality if and only if $G=C_n$. Recently, Zhong and Cui [Filomat 29 (2015) 673–686] generalized the above bound and proved that for any unicyclic graph $G$ of order $n\ge 4$ other than $C_n$, $H(G)\le \frac{n}{2}-\frac{2}{15}$. In this paper, we generalize the aforemention results and show that for any connected unicyclic graph $G$ of order $n\ge 3$ with maximum degree $\mathrm{\Delta }$,

This paper deals with the multiple values and algebraic dependences problem of meromorphic mappings sharing moving hyperplanes in projective space. We give some algebraic dependences theorems for meromorphic mappings sharing moving hyperplanes without counting multiplicity, where all zeros with multiplicities more than a certain number are omitted. Basing on these results, some unicity theorems regardless of multiplicity for meromorphic mappings in several complex variables are given. These results are extensions and strong improvements of some recent results.

In this paper, we attempt to collect the knowledge on generators in the category Pos-$S$ and to apply this to proceed on the questions of homological classification of ordered monoids, that is results of the type: all generators in the category Pos-$S$, have a flatness property if and only if $S$ has a certain property.

In this paper, we will discuss and study the structure of quasi-coretractable modules. Some characterization theorems of quasi-coretractable modules in terms of Kasch rings, CS modules, mono-coretractable modules, projective modules, epi-retractable modules and retractable modules will be given. Some equivalent conditions related with the quasi-coretractable and the mono-coretractable module will also be considered and investigated. Some recent results given by Singh on coretractable modules will be extended and enriched.

Let $R$ be an associative ring with unity, $\alpha$ be an endomorphism of $R$ and $\delta$ an $\alpha$-derivation of $R$. We introduce the notion of $\alpha$-nilpotent p.p.-rings, and prove that the $\alpha$-nilpotent p.p.-condition extends to various ring extensions. Among other results, we show that, when $R$ is a nil-$\alpha$-compatible and $2$-primal ring with $\mathrm{\text{Nil}}(R)$ nilpotent, then $\mathrm{\text{Nil}}(R[x;\alpha ,\delta ])=\mathrm{\text{Nil}}(R)[x;\alpha ,\delta ]$; and when $R$ is a nil Armendriz ring of skew power series type with $\mathrm{\text{Nil}}(R)$ nilpotent, then $\mathrm{\text{Nil}}(R[[x;\alpha ]])=\mathrm{\text{Nil}}(R)[[x;\alpha ]],$ where $\mathrm{\text{Nil}}(R)$ is the set of nilpotent elements of $R$. These results extend existing results to a more general setting.

Let $\psi :{𝒜}\to {𝒜}$ be a surjective map between some operator algebras such that $\psi (A){\circ }_2\psi (B)=A{\circ }_{2}B$ for all $A,B\in {𝒜}$, where $A{\circ }_{2}B$ defined by $A{\circ }_{2}B=(A\circ B)\circ B$ and $A\circ B$ is Jordan product, i.e. $A\circ B=AB+BA$. In this paper, we determine the concrete form of map $\psi$ on some operator algebras. Such operator algebras include standard operator algebras, properly infinite von Neumann algebras and nest algebras. Particularly, if ${𝒜}$ is a factor von Neumann algebra that satisfies $\psi (A){\circ }_2\psi (P)=A{\circ }_{2}P$ for all $A\in {𝒜}$ and idempotents $P\in {𝒜},$ then there exists nonzero scalar $\lambda$ with ${\lambda }^3=1$ such that $\psi (A)=\lambda A$ for all $A\in {𝒜}.$

In this paper, we introduce the concept of strongly torsion property of semigroups. Then we prove that the strongly torsion property of a semigroup is a necessary and sufficient condition for a finitely generated semigroup to be finite, and hence, the strongly torsion property is equivalent to the torsion property together with the permutation property for the class of finitely generated semigroups. Finally, we give an application of our main result to language theory. A question proposed in [H. Prodinger, Congruences defined by languages and filters, Inf. Control 44 (1980) 36–46] is consequently answered.

In this paper, we determine the degree of trigonometric approximation of $2\pi$-periodic functions and their conjugates, in terms of the moduli of continuity associated with them, by matrix means of corresponding Fourier series. We also discuss some analogous results with remarks and corollaries.

An attempt has been made to understand the complex dynamics of a spatial predator–prey system with Beddington–DeAngelis type functional response in the presence of prey-taxis and subjected to homogenous Neumann boundary condition. To describe the active movement of predators to the regions of high prey density or if the predator is following some sort of odor to find the prey, the prey-taxis phenomenon is included in a general reaction–diffusion equation. We have studied the linear stability analysis of both spatial and non-spatial models. We have performed extensive simulations to identify the conditions to generate spatiotemporal patterns in the presence of prey-taxis. It has been observed that the increasing predator active movement from the bifurcation value, the system shows chaotic behavior whereas increasing value of random movement brings the system back to order from the disordered state.

Let $(R,𝔪)$ be a local ring and ${(-)}^{\vee }$ denote the Matlis duality functor. Assume that $R$ possesses a normalized dualizing complex $D$ and $X$ and $Y$ are two homologically bounded complexes of $R$-modules with finitely generated homology modules. We will show that if G-dimension of $X$ and injective dimension of $Y$ are finite, then $H_{𝔪}^i(X,Y)\cong {Ext}_R^{-i}{(Y,D{\otimes }_R^{L}X)}^{\vee }.$ Also, we prove that if Gorenstein injective dimension of $Y$ and projective dimension of $X$ are finite, then $H_{𝔪}^i(X,Y)\cong {Ext}_R^{-i}{(R{Hom}_R(D,Y),X)}^{\vee }.$ These results provide some generalizations of Suzuki’s Duality Theorem and the Herzog–Zamani Duality Theorem.

The aim of the present article is to investigate the structure of rings $R$ satisfying the condition: for any family $\{S_i|i\in \mathbb{N}\}$ of simple right $R$-modules, every essential extension of ${\oplus }_{i\in \mathbb{N}}E(S_i)$ is a direct sum of lifting modules, where $E(-)$ denotes the injective hull. We show that every essential extension of ${\oplus }_{i\in \mathbb{N}}E(S_i)$ is a direct sum of lifting modules if and only if $R$ is right Noetherian and $E(S)$ is hollow. Assume that $M$ is an injective right $R$-module with essential socle. We also prove that if every essential extension of $M^{(\mathbb{N})}$ is a direct sum of lifting modules, then $M$ is $\mathrm{\Sigma }$-injective. As a consequence of this observation, we show that $R$ is a right V-ring and every essential extension of $S^{(\mathbb{N})}$ is a direct sum of lifting modules for all simple modules $S$ if and only if $R$ is a right $\mathrm{\Sigma }$-V-ring.

Let $R$ be an associative ring with identity, $S$ a monoid and $\omega :S\to \mathrm{\text{End}}(R)$ a monoid homomorphism. When $S$ is a u.p.-monoid and $R$ is a reversible $S$-compatible ring, then we observe that $R$ satisfies a McCoy-type property, in the context of skew monoid ring $R\ast S$. We introduce and study the $(S,\omega )$-McCoy condition on $R$, a generalization of the standard McCoy condition from polynomial rings to skew monoid rings. Several examples of reversible $S$-compatible rings and also various examples of $(S,\omega )$-McCoy rings are provided. As an application of $(S,\omega )$-McCoy rings, we investigate the interplay between the ring-theoretical properties of a general skew monoid ring $R\ast S$ and the graph-theoretical properties of its zero-divisor graph $\overline{\mathrm{\Gamma }}(R\ast S)$.

In this paper, we solve Fekete–Szegö problem for a Ma–Minda type class $M_{{}_{\mathrm{\Sigma }}}^{{}_{a,b,c}}(\lambda ,\nu ,{\psi }_1,{\psi }_2)$ of bi-univalent functions associated with the Hohlov operator which is a specialized case of the widely-investigated Dziok–Srivastava operator which, in turn, is contained in the Srivastava–Wright operator. Results for some of its consequent classes ${{𝒮}{𝒯}}_{\mathrm{\Sigma }}({\psi }_1,{\psi }_2),$ ${{𝒦}}_{\mathrm{\Sigma }}({\psi }_1,{\psi }_2),$ ${{ℳ}}_{\mathrm{\Sigma }}({\psi }_1,{\psi }_2)$ and $M_{{}_{\mathrm{\Sigma }}}^{{}_{a,b,c}}(\lambda ,{\psi }_1)$ are also given. A function $f$ in this class may satisfy one subordinate condition, whereas $f^{-1}$, another. One result for that is also given by considering certain ${\psi }_1,{\psi }_2$. Some useful consequences and relationships with some of the known results are also pointed out.

In this paper, we present a numerical scheme for solving the Abel equation. The approach is based on the shifted Chebyshev polynomials together with operational method. We reduce the problem to a set of nonlinear algebraic equations using operational matrix method. In addition, convergence analysis of the method is presented. Some numerical examples are given to demonstrate the validity and applicability of the method. The only a small number of Chebyshev polynomials is needed to obtain a satisfactory result.

We take a Galois ring $GR(p^m,g)$ and discuss about the self-dual codes and its properties over the ring. We will also describe the relationship between Clifford-Weil group and Jacobi forms by constructing the invariant polynomial ring with the complete weight enumerator.

In this article we study a class of mixed monotone operators with convexity on ordered Banach spaces and present some new tripled fixed point theorems by means of partial order theory, we get the existence and uniqueness of tripled fixed points without assuming the operator to be compact or continuous, which extend the existing corresponding results. As applications, we utilize the results obtained in this paper to study the existence and uniqueness of positive solutions for a fractional differential equation boundary value problem.

The Wiener number $W(G)$ of a graph $G$ was introduced by Harold Wiener in connection with the modeling of various physic-chemical, biological and pharmacological properties of organic molecules in chemistry. Milan Randić introduced a modification of the Wiener index for trees (acyclic graphs), and it is known as the hyper-Wiener index. Then Klein et al. generalized Randić’s definition for all connected (cyclic) graphs, as a generalization of the Wiener index, denoted by $WW(G)$ and defined as $WW(G)=\frac{1}{2}{\sum }_{u\in V}[d(u|G)+d^2(u|G)]$. In this paper, we establish some upper and lower bounds for $WW(G)$, in terms of other graph-theoretic parameters. Moreover, we compute hyper-Wiener number of some classes of graphs.

An involuted semilattice $(S;\vee ,e{,}^{\ast })$ is a semilattice $(S;\vee ,e)$ with an identity element $e$ and with an involution ${}^{\ast }$ satisfying $x^{\ast \ast }=x$ and ${(x\vee y)}^{\ast }=x^{\ast }\vee y^{\ast }$. We consider involuted semilattices $(S;\vee ,e{,}^{\ast })$ with an identity $e$ such that there is a subsemilattice $(A;\vee )$ without $e$ with the property that any $x\in S$ belongs to exactly one of the following four sets : $A$, $A^{\ast }$, $A\vee A^{\ast }$ or $\{e\}$. In this paper, we introduce an associative binary operation $+$ on $S$ in the following quite natural way: $A+x=\{x\}$, $A^{\ast }+x=\{x^{\ast }\}$, $A\vee A^{\ast }+x=\{x\vee x^{\ast }\}$, and $\{e\}+x=\{e\}$ for $x\in S$ and characterize all endomorphisms of the orthodox semigroup $(S;+)$.

This paper investigates a high even-order nonclassical differential equation with a spectral parameter. We proved that this equation has a countable system of nontrivial solutions if spectral parameter is negative. We consider two cases, one where the spectral parameter is equal to eigenvalues and one where the spectral parameter is not equal to eigenvalues. In both cases, we proved the existence of regular solutions of boundary value problems for this equation. To do this, we combined the Fourier method and the method of a priori estimates. Moreover, we found some conditions for unsolvability of boundary value problems. In addition, for adjoint problems, we proved that there is no complex eigenvalues.

Harmonic maps and exponentially harmonic maps are different. We study equivariant exponentially harmonic maps between warped product manifolds. We then investigate equivariant exponentially harmonic maps from manifolds with metrics of $(p,q)$-signatures into manifolds with metrics of $(p^{\prime },q^{\prime })$-signatures, obtain the reduction theorem, and provide a few examples.

In this paper, we study the regularity and the projective dimension of the Stanley–Reisner ring of a $k$-decomposable simplicial complex and explain these invariants with a recursive formula. To this aim, the graded Betti numbers of decomposable monomial ideals which is the dual concept for $k$-decomposable simplicial complexes are studied and an inductive formula for the Betti numbers is given. As a corollary, for a shellable simplicial complex $\mathrm{\Delta }$, a formula for the regularity of the Stanley–Reisner ring of $\mathrm{\Delta }$ is presented. Finally, for a chordal clutter ${\mathscr{H}}$, an upper bound for $\mathrm{\text{reg}}(I({\mathscr{H}}))$ is given in terms of the regularities of edge ideals of some chordal clutters which are minors of ${\mathscr{H}}$.

Firstly, we show that the variety of all left(right) regular bands and the variety of all normal bands are closed in the variety of all left(right) semiregular bands and the variety of all medial semigroups, respectively. Then, we show that the class of all regular medial semigroups satisfying certain condition is absolutely closed.