Volume 67 , issue 3 ( 2015 ) | back |

B-Fredholm spectra and Riesz perturbations | 155$-$165 |

**Abstract**

Let $T$ be a bounded linear Banach space operator and let $Q$ be a quasinilpotent one commuting with $T$. The main purpose of the paper is to show that we do not have $\sigma_{*}(T+Q)=\sigma_{*}(T)$ where $\sigma_{*}\in\{\sigma_{D},\sigma_{LD}\}$, contrary to what has been announced in the proof of Lemma 3.5 from M. Amouch, {Polaroid operators with SVEP and perturbations of property (gw)}, Mediterr. J. Math. {6} (2009), 461--470, where $\sigma_{D}(T)$ is the Drazin spectrum of $T$ and $\sigma_{LD}(T)$ its left Drazin spectrum. However, under the additional hypothesis $\operatorname{iso}\sigma_{ub}(T)=\emptyset$, the mentioned equality holds. Moreover, we study the preservation of various spectra originating from B-Fredholm theory under perturbations by Riesz operators.

**Keywords:** B-Fredholm spectrum; Riesz perturbations.

**MSC:** 47A53, , 47A10, 47A11

Existence of positive solutions for a class of nonlocal elliptic systems with multiple parameters | 166$-$173 |

**Abstract**

In this paper, we study the existence of positive solutions to the following nonlocal elliptic systems $$ \cases - M_1\left(\int_\Omega |\nabla u|^p\,dx\right)\Delta_p u = \alpha_1 a(x)f_1(v) + \beta_1b(x)g_1(u), \quad x \in \Omega,\\ - M_2\left(\int_\Omega |\nabla v|^q\,dx\right)\Delta_q v = \alpha_2 c(x)f_2(u) + \beta_2d(x)g_2(v), \quad x \in \Omega,\\ u = v = 0, \quad x \in \partial\Omega, \endcases $$ where $\Omega$ is a bounded domain in $\Bbb{R}^N$ with smooth boundary $\partial\Omega$, $1

**Keywords:** Nonlocal elliptic systems; positive solutions; sub and supersolutions method.

**MSC:** 35D05, 35J60

Topology generated by cluster systems | 174$-$184 |

**Abstract**

In this paper, we prove that $(X,\tau)$ and the new topology $(X,\tau_{\Cal E})$ have the same semiregularization if ${\Cal E}$ is a $\pi$-network in $X$ with the property ${\Cal H}$. Also, we discuss the properties of ${\Cal E},\tau_{\Cal E}$ and study generalized Volterra spaces and discuss their properties. We show that $\tau_{\Cal E}$ coincides with the $\star$-topology for a particular ${\Cal E}$.

**Keywords:** $\pi$-network; ideal; $\star$-topology; semiregularization; submaximal and Volterra spaces.

**MSC:** 54A05, 54A10, 54F65, 54E99

Faber polynomial coefficient estimates for a subclass of analytic bi-univalent functions defined by S\u{a}l\u{a}gean differential operator | 185$-$193 |

**Abstract**

In this work, considering a subclass of analytic bi-univalent functions defined by S\u{a}l\u{a}gean differential operator, we determine estimates for the general Taylor-Maclaurin coefficients of the functions in this class. For this purpose, we use the Faber polynomial expansions. In certain cases, our estimates improve some of existing coefficient bounds.

**Keywords:** Bi-univalent functions; Taylor-Maclaurin series expansion; Coefficient bounds and coefficient
estimates; Faber polynomials; S\u{a}l\u{a}gean differential operator.

**MSC:** 30C45, 30C80

$IA$-automorphisms of $p$-groups, finite polycyclic groups and other results | 194$-$200 |

**Abstract**

In this paper, the group $IA(G)$ of all $IA$-automorphisms of a group $G$ is studied. We prove some results regarding non-triviality, polycyclicity and commutativity of $IA(G)$ in addition to proving some basic results. We also prove some results analogous to a result by Schur and a weak form of its converse in the context of $IA$-automorphisms.

**Keywords:** Central automorphism; $IA$-automorphism; $p$-groups.

**MSC:** 20D45, 20D10, 20D15

Fuzzy representable modules and fuzzy attached primes | 201$-$211 |

**Abstract**

Let $M$ be a non-zero unitary module over a non-zero commutative ring $R$. A kind of uniqueness theorem for a non-zero fuzzy representable submodule $\mu$ of $M$ will be proved, and then the set of fuzzy attached primes of $\mu$ will be defined. Then among other things, it will be shown that, whenever $R$ is Noetherian, a fuzzy prime ideal $\xi$ is attached to $\mu$ if and only if $\xi$ is the annihilator of a fuzzy quotient of $\mu$. The behavior of fuzzy attached primes with fuzzy quotient and fuzzy localization techniques will be studied.

**Keywords:** Fuzzy coprimary submodules; fuzzy coprimary representation; fuzzy attached primes.

**MSC:** 08A72

A note on $I$-convergence and $I^{\star}$-convergence of sequences and nets in topological spaces | 212$-$221 |

**Abstract**

In this paper, we use the idea of $I$-convergence and $I^{\star}$-convergence of sequences and nets in a topological space to study some important topological properties. Further we derive characterization of compactness in terms of these concepts. We introduce also the idea of $I$-sequentially compactness and derive a few basic properties in a topological space.

**Keywords:** ${I}$-convergence; ${I}^{\star}$-convergence;
${I}$-limit point; ${I}$-cluster point; ${I}$-sequentially compact.

**MSC:** 54A20, 40A35

On $(n-1,n)$-$\phi$-prime ideals in semirings | 222$-$232 |

**Abstract**

Let $S$ be a commutative semiring and $T(S)$ be the set of all ideals of $S$. Let $\phi\:T(S)\to T(S)\cup \{\emptyset\}$ be a function. A proper ideal $I$ of a semiring $S$ is called an $(n-1,n)$-$\phi$-prime ideal of $S$ if $a_{1}a_{2}\cdots a_{n}\in I\setminus \phi(I)$, $a_{1},a_{2},\dots,a_{n}\in S$ implies that $a_{1}a_{2}\cdots a_{i-1}a_{i+1}\cdots a_{n}\in I$ for some $i\in \{1,2,\dots,n\}$. In this paper, we prove several results concerning $(n-1,n)$-$\phi$-prime ideals in a commutative semiring $S$ with non-zero identity connected with those in commutative ring theory.

**Keywords:** Semiring; $(n-1,n)$-$\phi$-prime ideal; $\phi$-subtractive ideal; $Q$-ideal.

**MSC:** 16Y30, 16Y60