Asian-European Journal of Mathematics

The M-P inverse has many important applications in the aspects of theoretic research of operator and statistics. One of the core problems in M-P inverse is to find the M-P inverse of operator product. In this paper, by using the technique of matrix form of bounded linear operators, we study the existence and give an expression of the M-P inverse of an operator product. In particular, by applying this expression, we introduce a formula for the M-P inverse of a sum of two operators.

The aim of this paper is to establish some metrical coincidence and common fixed point theorems with an arbitrary relation under an implicit contractive condition which is general enough to cover a multitude of well-known contraction conditions in one go besides yielding several new ones. We also provide an example to demonstrate the generality of our results over several well-known corresponding results of the existing literature. Finally, we utilize our results to prove an existence theorem for ensuring the solution of an integral equation.

Sanskruti index of a graph $G(V(G),E(G))$ with vertex-set $V(G)$ and edge-set $E(G)$ is $S(G)={\sum }_{uv\in E(G)}{(\frac{S_u{S}_v}{S_u+S_v-2})}^3$, where $S_u={\sum }_{v\in N_G(u)}{d}_v$. In this paper, the extremal graphs in the class of unicyclic graphs are characterized with respect to the Sanskruti index, where the considered class of unicyclic graphs contains five different large families of graphs.

In 1904, Schur proved his famous result which says that if the central factor group of a given group $G$ is finite, then so is its derived subgroup. In 1994, Hegarty showed that if the absolute central factor group, $G/L(G)$, is finite, then so is its autocommutator subgroup, $K(G)$.

In this paper, we introduce the concept of left and right $\alpha$-commutator, $Z^{\alpha }(G)$, and $K^{\alpha }(G)$, where $\alpha$ is an automorphism of the group $G$. Then under some condition, we prove that the finiteness of $G/Z^{\alpha }(G)$ implies that $K^{\alpha }(G)$ is also finite. We also construct an upper bound for the order of $K^{\alpha }(G)$ in terms of the order of $G/Z^{\alpha }(G)$.

In this paper, we give new characterizations of modular Riesz bases in Hilbert $C^{\ast }$-modules. We prove that modular Riesz bases share many properties with Riesz bases in Hilbert spaces. Moreover, we show that there are also important differences; for example, there exist exact frames that are not modular Riesz bases.

We introduce the concepts of a hyper chromatic, a critically chromatic and a purely chromatic lattice. We obtain some characterizations for purely chromatic and hyper chromatic dismantlable lattices. We prove relationships between the chromatic number of two lattices and their linear sum, vertical sum and adjunct.

The rings considered in this paper are finite commutative rings with identity, which are not fields. For any ring $R$ which is not a field and which is not necessarily finite, we denote the set of all zero-divisors of $R$ by $Z(R)$ and $Z(R)/\{0\}$ by $Z(R)\ast$. Let $\mathrm{\Gamma }(R)$ denote the zero-divisor graph of $R$ and for a finite ring $R$, let $\mathrm{\Delta }(\mathrm{\Gamma }(R))$ denote the maximum degree of $\mathrm{\Gamma }(R)$. We denote $\{a\in Z{(R)}^{\ast }|{deg}_{\mathrm{\Gamma }(R)}(a)=\mathrm{\Delta }(\mathrm{\Gamma }(R))\}$ by $M\mathrm{\text{deg}}(\mathrm{\Gamma }(R))$. The aim of this paper is to study some properties of $M\mathrm{\text{deg}}(\mathrm{\Gamma }(R))$.

The inverse sum indeg index of a graph $G$ is defined as $\mathrm{\text{ISI}}(G)={\sum }_{uv\in E(G)}\frac{d(u)d(v)}{d(u)+d(v)}$, where $d(u)$ is the degree of the vertex $u$. In a recent paper, Pattabiraman [Inverse sum indeg index of graphs, AKCE Int. J. Graphs Combinat. 15(2) (2018) 155–167] gave some lower and upper bounds on $\mathrm{\text{ISI}}$ index of all connected graphs in terms of Harmonic index, second Zagreb index and hyper Zagreb index. But some results were erroneous. In this note, we have corrected these results.

We introduce ∗-primeness and ∗-semiprimeness for a certain class of semirings called MA-Semirings. We prove some results on derivations satisfying certain conditions under which ∗-prime and ∗-semiprime semirings become commutative.

A relation between Connes amenability of dual Banach algebras and the existence of ($\sigma wc$)-virtual diagonals was investigated by Volker Runde. In this paper, we first introduce and study the notion of dual Fréchet algebras. We then study the weak${}^{\ast }$-weakly continuous elements of Fréchet bimodules, and the notion of normality for dual spaces of Fréchet bimodules. We obtain some important results about these notions. Finally, we introduce and study amenability for dual Fréchet algebras and investigate the relation between this notion and the existence of ($\sigma wc$)-virtual diagonals.

In this paper, the Belousov theorem on linearity of invertible algebras with the Schauffler $\forall \exists (\forall )$-identity is extended over the other $\forall \exists (\forall )$-identities of associativity. As a consequence, we obtain the equivalency of the considered $\forall \exists (\forall )$-identities of associativity and nontrivial hyperidentities of associativity in systems of groups. For the considered formulas, we prove the Schauffler-type theorems, too.

Best proximity point theorems for self-mappings and multi-functions were investigated with different conditions on spaces and the contractions of mappings. In this paper, we will prove best proximity point theorems for some generalized multi-functions that we call $\psi$-proximal multi-functions, which are of two types, $\psi$-proximal multi-functions of the first and second kind.

It is well known that a Sasakian space form is pseudo-symmetric [M. Belkhelfa, R. Deszcz and L. Verstraelen, Symmetry properties of Sasakian space-forms, Soochow J. Math. 31(4) (2005) 611–616], therefore it is Ricci-pseudo-symmetric. In this paper, we proved that a normal complex contact manifold is Ricci-semi-symmetric if and only if it is an Einstein manifold; moreover, we showed that a complex contact space form $M$ with constant $GH$-sectional curvature $c$ is properly Ricci-pseudo-symmetric $(L_{\rho }\ne 0)$ if and only if $c=-1$; in this case $L_{\rho }=1$. We gave an example of properly Ricci-pseudo-symmetric complex contact space form. On the other hand, we proved the non-existence of proper pseudo-symmetric ($L_R\ne 0$) complex contact space form $M.$

In this paper, we investigate warped product semi-transversal lightlike submanifolds of indefinite Kaehler manifolds. It is shown that there does not exist any warped product semi-transversal lightlike submanifold of the type $N_{\perp }{\times }_f{N}_T$ in an indefinite Kaehler manifold. Moreover, a necessary and sufficient condition for an isometrically immersed semi-transversal lightlike submanifold of an indefinite Kaehler manifold to be a semi-transversal lightlike warped product of the type $N_T{\times }_f{N}_{\perp }$ is obtained, in terms of the shape operator.

Laan in Pullbacks and flatness properties of acts I, [Commun. Algebra 29(2) (2001) 829–850] introduced Condition $(E^{\prime })$, a generalization of Condition $(E)$. Golchin and Mohammadzadeh in [On Condition ($P^{\prime }$), Semigroup Forum 86(2) (2012) 413–430] introduced Condition $(P^{\prime })$, a generalization of Condition $(P)$. In this paper similarly, we introduce a generalization of weak flatness property, called ${(WF)}^{\prime }$, and will classify monoids by this property of their acts. We also characterize ${(WF)}^{\prime }$ coherent monoids in general and monoids coming from some special classes.

The aim of this work is to study the Weinstein $L^2$-multiplier operators on ${ℝ}_{+}^{d+1}$ and we give for them Calderón’s reproducing formulas and best approximation using the theory of Weinstein transform and reproducing kernels.

This paper deals with some formulations of the uncertainty principle associated to generalized Fourier transform ${{ℱ}}_{\alpha }$ related to Flensted–Jensen partial differential operators. The aim result is to prove the analogue of Bonami–Demange–Jaming’s theorem : A version of Beurling–Hörmander’s theorem which gives more precision in the form of nonzero functions verifying modified-Beurling’s condition. As application, we get analogous of Gelfand–Schilov’s theorem, Cowling–Price’s theorem and Hardy’s theorem for ${{ℱ}}_{\alpha }$.

For a given ideal $J$ of an almost distributive lattice $A$, we study the smallest and the largest congruence relation on $A$ having $J$ as a congruence class.

The main purpose of this paper is to investigate the zero-divisors of semigroups with zero and semirings. In particular, we discuss eversible and reversible semigroups and semirings. We also introduce a new ring-like algebraic structure called prenearsemiring and generalize Cohn’s theorem for reversible rings in this context. Finally, we discuss when expectation semirings are reversible or eversible.

Asymptotic analysis of an incompressible Stokes fluid in a dynamic regime in a three-dimensional thin domain ${\mathrm{\Omega }}^{𝜀}$ with mixed boundary conditions and Tresca friction law is studied in this paper. The problem statement and variational formulation of the problem are reformulated in a fixed domain. In which case, the estimates on velocity and pressure are proved. These estimates will be useful in order to give a specific Reynolds equation associated with variational inequalities and prove the uniqueness.