Asian-European Journal of Mathematics

Let $R$ be a commutative ring with unity $1\ne 0$. The annihilating-ideal graph of $R$, denoted by $𝔸𝔾(R)$, is defined to be the graph with vertex set $A^{\ast }(R)$ — the set of non-zero annihilating ideals of $R$ and two distinct vertices $𝔞$ and $𝔟$ adjacent if and only if $𝔞𝔟=(0)$. Some connections between annihilating-ideal graphs and zero divisor graphs are given. We characterize the prime ideals (or equivalently maximal ideals) of ${\mathbb{Z}}_n$ in terms of their degrees as vertices of $𝔸𝔾(R)$. We also obtain the metric dimension of annihilating-ideal graphs of commutative rings.

We extend our approach of asymptotic parametrix construction for Hamiltonian operators from conical to edge-type singularities which is applicable to coalescence points of two particles of the helium atom and related two electron systems including the hydrogen molecule. Up to second-order, we have calculated the symbols of an asymptotic parametrix of the nonrelativistic Hamiltonian of the helium atom within the Born–Oppenheimer approximation and provide explicit formulas for the corresponding Green operators which encode the asymptotic behavior of the eigenfunctions near an edge.

Let $T\in B({\mathscr{H}})$ be a $(p,k)$-quasiposinormal operator on a complex Hilbert space ${\mathscr{H}}$. In this paper, we give basic properties for $T$ and we show that a $(p,k)$-quasiposinormal operator $T$ is polaroid. We also prove that all Weyl type theorems (generalized or not) hold and are equivalent for $f(T)$, where $f$ is an analytic function defined on a neighborhood of $\sigma (T)$.

In this paper, almost closed subspaces and almost closed linear operators are described in a Hilbert space. We show Neubauer’s lemma and we give necessary and sufficient conditions for an almost closed operator to be with closed range and we exhibit sufficient conditions under which it is either closed or closable.

The connection between polylogarithmic functions and Euler sums is well-known. In this paper, we explore the representation and many connections between integrals of products of polylogarithmic functions with alternating signs in the argument and Euler sums. We shall consider mainly, polylogarithmic functions with negative and positive arguments, thereby producing new results and extending the work of Freitas. Many examples of integrals of products of polylogarithmic functions in terms of Riemann zeta values and Dirichlet values will be given.

For a positive integer $n$, if $\sigma (n)$ denotes the sum of the positive divisors of $n$, then $n$ is called a deficient perfect number if $\sigma (n)=2n-d$ for some positive divisor $d$ of $n$. In this paper, we prove some results about odd deficient perfect numbers with four distinct prime factors.

The iota energy of an $n$-vertex digraph $D$ is defined by $E_c(D)={\sum }_{k=1}^n|\mathrm{\text{Im}}(z_k)|$, where $z_1,\dots ,z_n$ are eigenvalues of $D$ and $\mathrm{\text{Im}}(z_k)$ is the imaginary part of eigenvalue $z_k.$ Recently, this concept of iota energy of digraphs is extended to sidigraphs. The iota energy of a sidigraph is defined in an analogous way. In this paper, we consider a class of digraphs with two linear subdigraphs of equal length and find digraphs in this class with extremal iota energy. Furthermore, we take a similar class of sidigraphs and find minimal and maximal iota energy among the sidigraphs in this class.

We are interested in the behavior of Billingsley dimensions under projections onto a lower dimensional linear subspace. The results in this paper establish the connections with various dimensions of subsets $E$ of ${ℝ}^n$ and their projections, and generalize many known results about Hausdorff and packing dimensions of projections of $E$. In particular, we improve, through these results, one of the main theorems of Selmi et al. in [Multifractal variation for projections of measures, Chaos Solitons Fractals  91 (2016) 414–420] and treat an unsolved case in their work.

The total domination game is played on a graph $G$ by two players, named Dominator and Staller. They alternately select vertices of $G$; each chosen vertex totally dominates its neighbors. In this game, each chosen vertex must totally dominates at least one new vertex not totally dominated before. The game ends when all vertices in $G$ are totally dominated. Dominator’s goal is to finish the game as soon as possible, and Staller’s goal is to prolong it as much as possible. The game total domination number is the number of chosen vertices when both players play optimally, denoted by ${\gamma }_{\mathrm{\text{tg}}}(G)$ when Dominator starts the game and denoted by ${\gamma }_{\mathrm{\text{tg}}}^{\prime }(G)$ when Staller starts the game. In this paper, we show that for any graph $G$ and a vertex $v$, where $G-v$ has no isolated vertex, we have ${\gamma }_{\mathrm{\text{tg}}}(G)-{\gamma }_{\mathrm{\text{tg}}}(G-v)\le 2$ and ${\gamma }_{\mathrm{\text{tg}}}^{\prime }(G)-{\gamma }_{\mathrm{\text{tg}}}^{\prime }(G-v)\le 2$. Moreover, all such differences can be realized by some connected graphs.

Domination game is a game on a graph $G$ played by two players called Dominator and Staller. They alternately choose vertices of $G$ such that each move dominates at least one new undominated vertex. The game ends when all vertices are dominated. Dominator’s goal is to finish the game as soon as possible, while Staller’s goal is to prolong it as much as she can. When Dominator moves first, the game is a Dominator-start game; when Staller moves first, it is a Staller-start game. The game domination numbers ${\gamma }_g(G)$ and ${\gamma }_g^{\prime }(G)$ are the sizes of the final dominating sets when both players play optimally for the Dominator-start game and for the Staller-start game, respectively. For a graph $G$ and an edge $e$, we show that $0\le {\gamma }_g(G)-{\gamma }_g(G_e)\le 2$ and $0\le {\gamma }_g^{\prime }(G)-{\gamma }_g^{\prime }(G_e)\le 2$, where $G_e$ is is the graph resulting from subdividing the edge $e$ in $G$. We also demonstrate that each difference satisfying the above bounds are realizable by infinitely many connected graphs.

We deal with existence and multiplicity results for the following nonhomogeneous and homogeneous equations, respectively:

Let $T_n$ and $P_n$ be the monoids of all (full) transformations and of all partial transformations, on a finite chain $X_n=\{1,\dots ,n\}$ under its natural order, respectively. Moreover, let ${\mathrm{\text{OD}}}_{n,r}$ (${\mathrm{\text{POD}}}_{n,r}$) be the subsemigroup of $T_n$ ($P_n$) consists of all monotone transformations (monotone partial transformations) with height less than or equal to $r$ for $1\le r\le n-1$ ($0\le r\le n-1$). In this paper, we develop a new and general approach to find a (minimal) generating set of ${\mathrm{\text{OD}}}_{n,r}$ (${\mathrm{\text{POD}}}_{n,r}$).

In this paper, we characterize strongly right $C_2$-rings in terms of finite-direct-injective modules which is a generalization of direct-injective modules (or $C_2$-modules). Using this result, we give an example of a finite-direct-injective module which is not a direct-injective module. We prove that if every finite-direct-injective right $R$-module is a direct-injective module, then the ring $R$ must be right Noetherian. Also, we characterize semisimple artinian rings, regular right FGC-rings in terms of finite-direct-injective modules.

In this paper, we study two new subclasses ${𝒯}{{𝒮}}_m^{\ast }(\alpha ,\lambda )$ and ${𝒯}{{𝒞}}_m(\alpha ,\lambda )$ of analytic functions which are defined by means of a differential operator. Some results connected to partial sums and neighborhoods and integral means related to these subclasses are obtained.

The purpose of this paper is to define and investigate a new equational class of algebras which we call MS-almost distributive lattices (MS-ADLs) as a common abstraction of De Morgan ADLs and Stone ADLs. It is observed that the class of MS-ADLs properly contains the class of MS-algebras, and most of the properties of MS-algebras are extended to the class of MS-ADLs.

In this paper, we introduce a class of sub-almost distributive lattices in an associate almost distributive lattice through a filter. We obtain several algebraic properties on the class of sub-almost distributive lattices and prove that the above class forms a distributive lattice. We derive a necessary and sufficient condition that the class to become a Boolean algebra.

$\chi$-Posets and $\chi$-lattices are generalizations of posets and lattices introduced by Juhani Nieminen, where $\chi$ denotes a choice function. In this paper, some new characterizations of $\chi$-lattices and $0,1$-posets are obtained.

It is shown that the classes ${\gamma }_{{\mathrm{\Sigma }}_0}=\{R|{\mathrm{\Sigma }}_{0R}=\varphi \}$ and ${\gamma }_{{\mathrm{\Sigma }}_1}=\{R|{\mathrm{\Sigma }}_{1R}=\varphi \}$ of semirings are radical classes, where ${\mathrm{\Sigma }}_{0R}$ is the class of subtractive-simple right $R$-semimodules and ${\mathrm{\Sigma }}_{1R}$ is the class of right $R$-semimodules isomorphic to $R/{\rho }_I$ for some maximal-subtractive right ideal $I$ of $R$. We define the lower Jacobson Bourne radical ${\mathrm{\text{rad}}}_l(R)\subseteq \mathrm{\text{ker}}({\mathrm{\Sigma }}_{0R})$ and upper Jacobson Bourne radical ${\mathrm{\text{rad}}}_u(R)\subseteq \mathrm{\text{ker}}({\mathrm{\Sigma }}_{1R})$ of $R$. For a semiring $R$, ${\mathrm{\text{rad}}}_l(R)\subseteq \mathrm{\text{rad}}(R)$ holds, where $\mathrm{\text{rad}}(R)$ is the Jacobson Bourne radical of $R$. The radical ${\mathrm{\text{rad}}}_l(R)=\mathrm{\text{ker}}({\mathrm{\Sigma }}_{0R})$ and also coincides with $\mathrm{\text{rad}}(R)$, if we restrict the class ${\mathrm{\Sigma }}_{0R}$ to additively cancellative semimodules$.$ The upper radical ${\mathrm{\text{rad}}}_u(R)=\mathrm{\text{ker}}({\mathrm{\Sigma }}_{1R})$ and $\mathrm{\text{rad}}(R)\subseteq$${\mathrm{\text{rad}}}_u(R)$, if $R$ is additively cancellative. Further, ${\mathrm{\text{rad}}}_u(R/{\rho }_{{\mathrm{\text{rad}}}_u(R)})=\{0\}$, if $R$ is a commutative semiring with $1_R.$ The subtractive-primitiveness and subtractive-semiprimitiveness of $R$ are closely related to the upper radical ${\mathrm{\text{rad}}}_u(R).$ Finally, we show that ${\mathrm{\text{rad}}}_u$-semisimplicity of semirings are Morita invariant property with some restrictions.