Volume 75 , issue 4 ( 2023 ) | back |

SOME RESULTS ON SCALABLE $K$-FRAMES | 225$-$234 |

**Abstract**

We investigate the scalability of $K$-frames and derive a characterization for scalable $K$-frames. We investigate whether or not a particular $K$-frame is scalable, as well as the existence and uniqueness of scalings. Using the concept of trace of an operator, we analyse the possible scalings, if a given $K$-frame is scalable. In $\mathbb{C}^{n}$, we look at the scalability of $K$-frames independently.

**Keywords:** Frames; $K$-frames; scalable $K$-frames.

**MSC:** 42C15, 47A63

APPROXIMATION SPACES VIA IDEALS AND GRILLS | 235$-$246 |

**Abstract**

In this paper, we use the notions of lower set $L_{R}(A)$ and the upper set $U_{R}(A)$ to define the interior operator ${\rm int}_{R}^{A}$ and the closure operator ${\rm cl}_{R}^{A}$ associated with a set $A$ in an approximation space $(X,R)$. These operators generate an approximation topological space different from the generated Nano topological space in $(X,R)$. Ideal approximation spaces $(X,R, \ell)$ based on an ideal $\ell$ joined to the approximation space $(X,R)$ are introduced as well. The approximation continuity and the ideal approximation continuity are defined. The lower separation axioms $T_{i}, i= 0,1,2$ are introduced in the approximation spaces and also in the ideal approximation spaces. Examples are given to explain the definitions. Connectedness in approximation spaces and ideal connectedness are introduced and the differences between them are explained. The interior and the closure operators are deduced using a grill ${\cal G}$ defined on $(X,R)$, yielding the same results.

**Keywords:** Rough set; approximation space; approximation continuity; ideal approximation continuity; separation axioms; approximation connectedness; ideal approximation connectedness.

**MSC:** 54A40, 54A05, 54A10, 03E20

EQUIPRIME FUZZY GRAPH OF A NEARRING WITH RESPECT TO A LEVEL IDEAL | 247$-$264 |

**Abstract**

In this paper, we introduce an equiprime fuzzy graph of a nearring with respect to the level ideal of a fuzzy ideal. We interrelate graph theoretical properties of the graph and ideal theoretical properties of nearring. We show that the properties like vertex cut, connectedness of the graph depend on the properties of the fuzzy ideal. We define ideal symmetry of the graph and find conditions for the graph to be ideal symmetric. If the fuzzy ideal is equiprime then we show that the level set induces a fuzzy clique. We find conditions required for the level set to be the vertex cover of the graph. We find interrelation between equiprime fuzzy graph and fuzzy graph of nearring with respect to level ideal. We study properties of the graph under neaaring homomorphism. We prove that the connectedness of the graph in homomorphic image depends on properties of ideal. We obtain conditions required for homomorphic image of an equiprime fuzzy ideal to be an equiprime fuzzy ideal.

**Keywords:** Nearring; equiprime; ideal; fuzzy graph; homomorphism.

**MSC:** 16Y30, 03E72, 16Y99

ON THE SUBTRACTIVE SUBSEMIMODULE-BASED GRAPH OF SEMIMODULES | 265$-$274 |

**Abstract**

Let $M$ be a semimodule over a commutative semiring $R$ and $K$ be a subtractive subsemimodule of $M$ with $K^{*}=K\setminus \{0\}$. The subtractive subsemimodule-based graph of $M$ is defined as the simple undirected graph $\Omega=\Gamma_{K^{*}}(M)$ with vertex set $V(\Omega)=\{v\in M\setminus K : v+v'\in K^{*} \textrm{ for some }v\neq v'\in M\setminus K\}$, and two distinct vertices $m$ and $n$ are adjacent if and only if $m+n\in K^{*}$. In this paper, we study the interplay between semimodule properties and the properties of the graph. Among other results, we compute the diameter and the girth of $\Gamma_{K^{*}}(M)$.

**Keywords:** Semiring; subtractive subsemimodule; partitioning subsemimodule.

**MSC:** 16Y60, 05C753

MORE ON THE GENERALIZED PASCAL TRIANGLES | 275$-$285 |

**Abstract**

The aim of this article is to obtain a new factorization of a generalized Pascal triangle. This factorization particularly emphasizes that there is a close relation between generalized Pascal triangles and Toeplitz matrices. However, we will show that in general there is no such relation between generalized Pascal triangles and Hankel matrices.

**Keywords:** Generalized Pascal triangle; factorization of a matrix; Töeplitz matrix; Hankel matrix.

**MSC:** 15A15, 15A23, 11C20

MULTIVALUED COUPLED COINCIDENCE POINT RESULTS IN METRIC SPACES | 286$-$295 |

**Abstract**

In this paper, we use an inequality involving a coupled multivalued mapping and a singlevalued mapping to obtain a coupled coincidence point theorem. We discuss special conditions under which coupled common fixed point theorems are obtained. The result combines several ideas prevalent in fixed point theory studies. There are several corollaries and illustrative examples. The Hausdorff-Pompeiu metric between sets is used. The work is in the context of metric spaces and is a part of set-valued analysis with the singlevalued consequences.

**Keywords:** Metric space; Hausdorff-Pompeiu distance; MT - functions; coupled coincidence point; coupled common fixed point.

**MSC:** 54H10, 54H25, 47H10

SOME REMARKS ON MONOTONICALLY STAR COUNTABLE SPACES | 296$-$302 |

**Abstract**

A topological space $X$ is monotonically star countable if for every open cover $\mathcal U$ of $X$ we can assign a subspace $s(\mathcal U)\subseteq X$, called the kernel, such that $s(\mathcal U)$ is a countable subset of $X$, and $st(s(\mathcal U),\mathcal U)=X$, and if $\mathcal V$ refines $\mathcal U$, then $s(\mathcal U)\subseteq s(\mathcal V)$, where $st(s(\mathcal U),\mathcal U)=\bigcup\{U\in\mathcal U:U\cap s(\mathcal U)\neq\emptyset\}.$ In this paper we study the relation between monotonically star countable spaces and related spaces, and we also study topological properties of monotonically star countable spaces.

**Keywords:** Star finite; monotonically star finite; star countable; monotonically star countable; star Lindel{ö}f; monotonically star Lindel{ö}f.

**MSC:** 54D20, 54D30, 54D40