Volume 68 , issue 2 ( 2016 ) | back |

Decompositions of normality and interrelation among its variants | 77$-$86 |

**Abstract**

Interrelation among some existing variants of normality is discussed and characterizations of these variants are obtained. It is verified that Urysohn's type lemma and Tietze's type extension hold for some of these variants of normality. Decomposition of normality in terms of near normality and some factorizations of normality in presence of some lower separation axioms are given.

**Keywords:** $\kappa$-normal; $\theta$-normal; f$\theta$-normal; w$\theta$-normal; wf$\theta$-normal; $\Delta$-normal; w$\Delta$-normal; wf$\Delta$-normal;
$\alpha$-normal; $\beta$-normal; $\pi$-normal; $\Delta$-regular

**MSC:** 54D15

Coefficient inequality for certain subclass of $p$-valent analytic functions whose reciprocal derivative has a positive real part | 87$-$92 |

**Abstract**

The objective of this paper is to introduce certain new subclass of $p$-valent analytic functions in the open unit disc $E=\{z:|z|<1\}$ and obtain sharp upper bound for the second Hankel determinant of functions belonging to this class, using Toeplitz determinants.

**Keywords:** $p$-valent analytic function; function whose derivative has a positive real part; second Hankel determinant; positive real function; Toeplitz determinants

**MSC:** 30C45, 30C50

On $\mathcal{I_{\tau}^{\mathcal{K}}}$-convergence of nets in locally solid Riesz spaces | 93$-$99 |

**Abstract**

In this short note we continue our investigation of nets in locally solid Riesz spaces from [P. Das, E. Savas, {On $\mathcal I$-convergence of nets in locally solid Riesz spaces}, Filomat, 27 (1) (2013), 84--89] and introduce the idea of $\Cal{I}_{\tau}^{\Cal{K}}$-convergence of nets which is more general than $\Cal{I}_{\tau}^*$-convergence and obtain some of its basic properties.

**Keywords:** Ideal; filter; nets; $\mathcal{I_{\tau}}$-convergence; $\mathcal{I_{\tau}^{\mathcal{K}}}$-convergence; $\mathcal{I_{\tau}^{\mathcal{K}}}$-boundedness;
$\mathcal{I}_{\tau}^{\mathcal{K}}$-Cauchy; locally solid Riesz space

**MSC:** 40G15, 40A35

Coarse topologies on the real line | 100$-$118 |

**Abstract**

Let $c=|{\mathbb R}|$ denote the cardinality of the continuum and let $\eta$ denote the Euclidean topology on ${\mathbb R}$. Let ${\mathcal L}$ denote the family of all Hausdorff topologies $\tau$ on ${\mathbb R}$ with $\tau\subset\eta$. Let ${\mathcal L}_1$ resp.~${\mathcal L}_2$ resp.~${\mathcal L}_3$ denote the family of all $\tau\in{\mathcal L}$ where $({\mathbb R},\tau)$ is {\it completely normal} resp.~{\it second countable} resp.~{\it not regular}. Trivially, ${\mathcal L}_1\cap{\mathcal L}_3=\emptyset$ and $|{\mathcal L}_i|\leq|{\mathcal L}|\leq 2^c$ and $|{\mathcal L}_2|\leq c$. For $\tau\in{\mathcal L}$ the space $({\mathbb R},\tau)$ is metrizable if and only if $\tau\in{\mathcal L}_1\cap{\mathcal L}_2$. We show that, up to homeomorphism, both ${\mathcal L}_1$ and ${\mathcal L}_3$ contain precisely $2^c$ topologies and ${\mathcal L}_2$ contains precisely $c$ completely metrizable topologies. For $2^c$ non-homeomorphic topologies $\tau\in{\mathcal L}_1$ the space $({\mathbb R},\tau)$ is {\it Baire}, but there are also $2^c$ non-homeomorphic topologies $\tau\in{\mathcal L}_1$ and $c$ non-homeomorphic topologies $\tau\in{\mathcal L}_1\cap{\mathcal L}_2$ where $({\mathbb R},\tau)$ is of {\it first category}. Furthermore, we investigate the {\it complete lattice} ${\mathcal L}_0$ of all topologies $\tau\in{\mathcal L}$ such that $\tau$ and $\eta$ coincide on ${\mathbb R}\setminus\{0\}$. In the lattice ${\mathcal L}_0$ we find $2^c$ (non-homeomorphic) immediate predecessors of the maximum $\eta$, whereas the minimum of ${\mathcal L}_0$ is a compact topology without immediate successors in ${\mathcal L}_0$. We construct chains of homeomorphic topologies in ${\mathcal L}_0\cap{\mathcal L}_1\cap{\mathcal L}_2$ and in ${\mathcal L}_0\cap{\mathcal L}_2\cap{\mathcal L}_3$ and in ${\mathcal L}_0\cap({\mathcal L}_1\setminus{\mathcal L}_2)$ and in ${\mathcal L}_0\cap({\mathcal L}_3\setminus{\mathcal L}_2)$ such that the length of each chain is $c$ (and hence maximal). We also track down a chain in ${\mathcal L}_0$ of length $2^\lambda$ where $\lambda$ is the smallest cardinal number $\kappa$ with $2^\kappa>c$.

**Keywords:** nonmetrizable Baire spaces; metrizable spaces of first category

**MSC:** 54A10

Radical transversal screen semi-slant lightlike submanifolds of indefinite Kaehler manifolds | 119$-$129 |

**Abstract**

In this paper, we introduce the notion of radical transversal screen semi-slant lightlike submanifolds of indefinite Kaehler manifolds giving characterization theorem with some non-trivial examples of such submanifolds. Integrability conditions of distributions $D_1$, $D_2$ and $RadTM$ on radical transversal screen semi-slant lightlike submanifolds of indefinite Kaehler manifolds have been obtained. Further, we obtain necessary and sufficient conditions for foliations determined by above distributions to be totally geodesic.

**Keywords:** Semi-Riemannian manifold; degenerate metric; radical distribution; screen distribution; screen transversal vector bundle; lightlike transversal vector bundle;
Gauss and Weingarten formulae

**MSC:** 53C15, 53C40, 53C50

Semi parametric estimation of extremal index for ARMAX process with infinite variance | 130$-$139 |

**Abstract**

We consider estimating the extremal index of a maximum autoregressive process of order one under the assumption that the distribution of the innovations has a regularly varying tail at infinity. We establish the asymptotic normality of the new estimator using the extreme quantile approach, and its performance is illustrated in a simulation study. Moreover, we compare, in terms of bias and mean squared error, our estimator with the estimator of Ferro and Segers [Inference for clusters of extreme values, J. Royal Stat. Soc., Ser. B, {65} (2003), 545--556] and Olmo [A new family of consistent and asymptotically-normal estimators for the extremal index, {Econometrics}, 3 (2015), 633--653].

**Keywords:** extreme value theory; max autoregressive processes; tail index estimation

**MSC:** 60G70, 62G32

Common fixed points in $b$-metric spaces endowed with a graph | 140$-$154 |

**Abstract**

We discuss the existence and uniqueness of points of coincidence and common fixed points for a pair of self-mappings defined on a $b$-metric space endowed with a graph. Our results improve and supplement several recent results of metric fixed point theory.

**Keywords:** $b$-metric; reflexive digraph; point of coincidence; common fixed point

**MSC:** 47H10, 54H25