Volume 71 , issue 3 ( 2019 ) | back |

A NOTE ON TWO OF VUKMAN'S CONJECTURES | 190$-$195 |

**Abstract**

In this paper we prove, under certain condition, when $R$ is a semiprime ring with suitable characteristic restrictions, that every nonzero $(m,n)$-Jordan triple centralizer (resp., $(m,n)$-Jordan triple derivation) is a two-sided centralizer (resp., a derivation which maps $R$ into $Z(R)$). This give partial affirmative answers to two conjectures of Vukman.

**Keywords:** von Neumann regular ring; semiprime ring; Jordan triple derivation; $(m,n)$-Jordan triple centralizer.

**MSC:** 16E50, 16W25, 16N60, 16W99

ON LIE-YAMAGUTI COLOR ALGEBRAS | 196$-$206 |

**Abstract**

Lie-Yamaguti color algebras are defined and some examples are provided. It is shown that any Leibniz color algebra has a natural Lie-Yamaguti structure. For a given Lie-Yamaguti color algebra, an enveloping Lie color algebra is constructed and it is proved that any Lie color algebra with reductive decomposition induces a Lie-Yamaguti structure on some of its subspaces.

**Keywords:** Lie color algebra; Akivis color algebra; Leibniz color algebra; Lie-Yamaguti color algebra.

**MSC:** 17B55, 17B60, 17B99

A SIMPLE METHOD FOR FINDING THE INVERSE MATRIX OF VANDERMONDE MATRIX | 207$-$213 |

**Abstract**

A simple method for computing the inverse of Vandermonde matrices is presented. The inverse is obtained by finding the cofactor matrix of Vandermonde matrices. Based on this, it is directly possible to evaluate the determinant and inverse for more general Vandermonde matrices.

**Keywords:** Vandermonde matrix; inverse of a matrix; determinant of a matrix.

**MSC:** 11C20, 15A09, 15A15

SOME PROPERTIES OF COMMON HERMITIAN SOLUTIONS OF MATRIX EQUATIONS $A_{1}XA_{1}^{*}=B_{1}$ AND $A_{2}XA_{2}^{*}=B_{2}$ | 214$-$229 |

**Abstract**

In this paper we provide necessary and sufficient conditions for the pair of matrix equations $ A_{1}XA_{1}^{*}=B_{1} $ and $ A_{2}XA_{2}^{*}=B_{2} $ to have a common hermitian solution in the form $ \frac{X_{1}{+}X_{2}}{2} $, where $ X_{1} $ and $ X_{2} $ are hermitian solutions of the equations $ A_{1}XA_{1}^{*}=B_{1} $ and $ A_{2}XA_{2}^{*}=B_{2}$ respectively.

**Keywords:** Moore-Penrose inverse; matrix equation; rank; inertia; hermitian solution; submatrices.

**MSC:** 40A05, 40A25, 45G05

COINCIDENCE POINT AND COMMON FIXED POINT RESULTS FOR A HYBRID PAIR OF MAPPINGS VIA DIGRAPHS | 230$-$243 |

**Abstract**

In this paper, we introduce the concept of $(\alpha, \psi, \xi )-G$-contractive mappings in a metric space endowed with a directed graph $G$. We investigate the existence and uniqueness of points of coincidence and common fixed points for such mappings under some conditions. Our results extend and generalize several well-known comparable results in the literature. Some examples are provided to justify the validity of our results.

**Keywords:** Digraph; weakly compatible mappings; point of coincidence; common fixed point.

**MSC:** 54H25, 47H10

REMARKS ON ALMOST $\eta$-SOLITONS | 244$-$249 |

**Abstract**

A general definition of almost soliton is considered and the particular cases of almost $\eta$-Ricci, -Einstein and -Yamabe solitons are stretched. Conditions of existence of conjugate solitons are also provided and examples from paracontact geometry are constructed. In the gradient case, two inequalities are deduced and a Bochner-type formula is obtained.

**Keywords:** Almost solitons; linear connections.

**MSC:** 35C08, 35Q51, 53B05

$n$-ARY 2-ABSORBING AND 2-ABSORBING PRIMARY HYPERIDEALS IN KRASNER $(m,n)$-HYPERRINGS | 250$-$262 |

**Abstract**

Let $R$ be a commutative Krasner $(m,n)$-hyperring with the scalar identity $1_R$. In this paper, we introduce and study the concept of $n$-ary 2-absorbing and 2-absorbing primary hyperideals of $R$. These concepts are a generalisation of $n$-ary prime and primary hyperideals.

**Keywords:** $n$-ary prime hyperideal; $n$-ary 2-absorbing hyperideal; $n$-ary 2-absorbing primary hyperideal; Krasner $(m,n)$-hyperring.

**MSC:** 16Y99

ON THE VERTEX-EDGE WIENER INDICES OF THORN GRAPHS | 263$-$276 |

**Abstract**

The vertex-edge Wiener index is a graph invariant defined as the sum of distances between vertices and edges of a graph. In this paper, we study the relation between the first and second vertex-edge Wiener indices of thorn graph and its parent graph and examine several special cases of the results. Results are applied to compute the first and second vertex-edge Wiener indices of thorn stars, Kragujevac trees, and dendrimers.

**Keywords:** Topological index; thorn graph; bipartite graph; Kragujevac tree; dendrimer.

**MSC:** 05C12, 05C76

AN EXISTENCE RESULT FOR A CLASS OF $p$-BIHARMONIC PROBLEM INVOLVING CRITICAL NONLINEARITY | 277$-$283 |

**Abstract**

This paper is concerned with the following elliptic equation with Hardy potential and critical Sobolev exponent \begin{align*} \Delta(|\Delta u|^{p-2}\Delta u)-\lambda \frac{|u|^{p-2}u}{|x|^{2p}}=\mu h(x)|u|^{q-2}u+|u|^{p^{*}-2}u\quad \text{in }\Omega , \quad u\in W^{2,p}_0(\Omega). \end{align*} By means of the variational approach, we prove that the above problem admits a nontrivial solution.

**Keywords:** Topological index; thorn graph; bipartite graph; Kragujevac tree; dendrimer.

**MSC:** 35J60, 35D05, 35J20, 35J40