Volume 72 , issue 2 ( 2020 ) | back |

$\mathbf{S}$-PSEUDOSPECTRA AND $\mathbf{S}$-ESSENTIAL PSEUDOSPECTRA | 95$-$105 |

**Abstract**

In the present paper, we introduce and study the $\mathcal{S}$-pseudospectra and the essential $\mathcal{S}$-pseudospectra of linear relations. We start by giving the definition and we investigate the characterization and some properties of these $\mathcal{S}$-pseudospectra.

**Keywords:** Linear relation; $\mathcal{S}$-pseudospectra; $\mathcal{S}$-essential pseudospectra.

**MSC:** 47A06

HEMI-SLANT $\xi^\bot$-LORENTZIAN SUBMERSIONS FROM $(LCS)_n$-MANIFOLDS | 106$-$116 |

**Abstract**

The present paper introduce a study of hemi-slant $\xi^\bot$-Lorentzian submersion from $(LCS)_n$-manifolds with an example. We obtain some results and investigate the geometry of foliations. Necessary and sufficient conditions for such submersion to be totally geodesic have been obtained. Finally, we study such submersions with totally umbilical fibers.

**Keywords:** $(LCS)_n$-manifold; Lorentzian submersion; hemi-slant $\xi^\bot$-Lorentzian submersion.

**MSC:** 53C15, 53C43, 53C50

HYPERBOLIC SETS FOR THE FLOWS ON PSEUDO-RIEMANNIAN MANIFOLDS | 117$-$123 |

**Abstract**

In this paper we introduce and consider the hyperbolic sets for the flows on pseudo-Riemannian manifolds. If $\Lambda $ is a hyperbolic set for a flow $\Phi $, then we show that at each point of $\Lambda $ we have a unique decomposition for its tangent space up to a distribution on the ambient pseudo-Riemannian manifold. We prove that we have such decomposition for many points arbitrarily close to a given member of $\Lambda $.

**Keywords:** Hyperbolic set; pseudo-Riemannian manifold; flow; distribution.

**MSC:** 37D20, 37D30

THE RICCI-BOURGUIGNON FLOW ON HEISENBERG AND QUATERNION LIE GROUPS | 124$-$137 |

**Abstract**

In this paper, we study the Ricci-Bourguignon flow on higher dimensional classical Heisenberg nilpotent Lie groups and construct a solution of this flow on Heisenberg and quaternion nilpotent Lie groups. In the end, we investigate the deformation of spectrum and length spectrum on compact nilmanifolds obtained of Heisenberg and quaternion nilpotent Lie groups.

**Keywords:** Ricci-Bouguignon flow; Heisenberg type; Lie group.

**MSC:** 53C44, 22E25

ON CONFORMAL TRANSFORMATION OF $\boldsymbol m$-th ROOT FINSLER METRIC | 138$-$145 |

**Abstract**

The purpose of the present paper is to study the conformal transformation of $m$-th root Finsler metric. The spray coefficients, Riemann curvature and Ricci curvature of conformally transformed $m$-th root metrics are shown to be certain rational functions of direction. Further, under certain conditions it is shown that a conformally transformed $m$-th root metric is locally dually flat if and only if the transformation is a homothety. Moreover the conditions for the transformed metrics to be Einstein and isotropic mean Berwald curvature are also found.

**Keywords:** Finsler space; $m$-th root metric; conformal transformation; locally dually flat metric; Einstein metric; Ricci curvature; isotropic mean
Berwald curvature.

**MSC:** 53B40, 53C60

EXTREMAL $F$-INDEX OF A GRAPH WITH $k$ CUT EDGES | 146$-$153 |

**Abstract**

The so called forgotten index or $F$-index is defined as the sum of cubes of vertex degrees of a molecular graph. In this paper, we have obtained the upper and lower bounds of $F$-index for the graphs with $k$ cut edges and also we have characterized the extremal graphs.

**Keywords:** $F$-index; cut edges; transformation.

**MSC:** 05C35, 05C07, 05C40

STABILITY OF ADDITIVE-QUADRATIC $\boldsymbol\rho$-FUNCTIONAL EQUATIONS IN NON-ARCHIMEDEAN INTUITIONISTIC FUZZY BANACH SPACES | 154$-$164 |

**Abstract**

In this paper, a Hyers-Ulam-Rassias stability result for additive-quadratic $\rho$-functional equations is established. The framework of the study is non-Archimedean intuitionistic fuzzy Banach spaces. These spaces are generalizations of fuzzy Banach spaces. Several studies of functional analysis have been extended to this space.

**Keywords:** Hyers-Ulam stability; non-Archimedean intuitionistic fuzzy normed space; generalized metric space.

**MSC:** 03E72, 97I70, 39B82, 32P05

COUNTING SPACES OF EXCESSIVE WEIGHTS | 165$-$186 |

**Abstract**

Let $\kappa,\lambda$ be infinite cardinal numbers with $\kappa<\lambda\leq 2^\kappa$. We show that there exist precisely $2^\lambda$ T$_0$-spaces of size $\kappa$ and weight $\lambda$ up to homeomorphism. Among these non-homeomorphic spaces we track down (i) $2^{\lambda}$ zero-dimensional, scattered, para\-compact, perfectly normal spaces (which are also extremally disconnected in case that $\lambda=2^\kappa$); (ii) $2^{\lambda}$ connected and locally connected Hausdorff spaces; (iii) $2^{\lambda}$ pathwise connected and locally pathwise connected, paracompact, perfectly normal spaces provided that $\kappa\geq 2^{\aleph_0}$; (iv) $2^{\lambda}$ connected, nowhere locally connected, totally pathwise disconnected, paracompact, perfectly normal spaces provided that $\kappa\geq 2^{\aleph_0}$; (v) $2^\lambda$ scattered, compact T$_1$-spaces; (vi) $2^\lambda$ connected, locally connected, compact T$_1$-spaces; (vii) $2^\lambda$ pathwise connected {ıt and} scattered, compact T$_0$-spaces; (viii) $2^\lambda$ scattered, paracompact $P_\alpha$-spaces whenever $\kappa^{<\alpha}=\kappa$ and $\lambda^{<\alpha}=\lambda$ and $2^\lambda>2^\kappa$.

**Keywords:** Scattered resp. connected; paracompact spaces.

**MSC:** 54A525, 54D20, 54D05