Volume 75 , issue 2 ( 2023 ) | back |

SOME FUNCTION SPACES AND THEIR APPLICATIONS TO ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS | 71$-$86 |

**Abstract**

In this paper we prove Fefferman's inequalities associated to potentials belonging to a generalized Morrey space or a Stummel class. We also show that the logarithm of a non-negative weak solution to a second order elliptic partial differential equation with potential in a generalized Morrey space or a Stummel class, under some assumptions, belongs to the bounded mean oscillation class. As a consequence, this elliptic partial differential equation has the strong unique continuation property. An example of an elliptic partial differential equation with potential in a Morrey space or a Stummel class which does not satisfy the strong unique continuation is presented.

**Keywords:** Morrey spaces; Stummel classes; Fefferman's inequality; srong unique continuation property.

**MSC:** 26D10, 46E30, 35J15

PROPERTIES OF ZERO-DIVISOR GRAPH OF THE RING $\mathbf{F}_{p^l} \times \mathbf{F}_{q^m} \times \mathbf{F}_{r^n}$ | 87$-$96 |

**Abstract**

In this paper, we study some basic properties of the zero-divisor graph of ring $F_{p^l} \times F_{q^m} \times F_{r^n}$, where $F_{p^l}$, $F_{q^m}$ and $F_{r^n}$ are fields of order $p^l$, $q^m$ and $r^n$, respectively, $p, q$ and $r$ are primes (not necessarily distinct) and $l, m, n \geq 1$ are positive numbers. Also, we discuss some topological indices of the graph $\Gamma(F_{p^l} \times F_{q^m} \times F_{r^n})$.

**Keywords:** Zero-divisor graph; direct product of rings; graph parameters..

**MSC:** 05C10, 05C12, 05C25

CHARACTERIZATION OF MATRICES OF OPERATORS ON THE $L_{\phi}$-VALUED CESÀRO SEQUENCE SPACES | 97$-$105 |

**Abstract**

In this paper, we characterized the matrices of operators that transform the generalized Cesàro sequence space to the convergence sequence space in Banach spaces. Our results generalize the characterization of the sequence space in $L_p$, $1

**Keywords:** Matrices of operators; Cesàro sequence spaces; Orlicz function.

**MSC:** 40A30, 40J05

SOME PINCHING RESULTS FOR STATISTICAL SUBMANIFOLDS IN COSYMPLECTIC STATISTICAL MANIFOLDS | 106$-$117 |

**Abstract**

In this article, we discuss the curvature properties of statistical submanifolds in cosymplectic statistical manifolds with constant curvature. We also establish some pinching results for such submanifolds and hypersurfaces in cosymplectic statistical manifolds having constant curvature. As an application of the main result we also obtain an obstruction condition for such immersion.

**Keywords:** Chen-Ricci inequality; dual connections; statistical manifolds; cosymplectic statistical manifolds.

**MSC:** 53B05, 53B20, 53C40

NEW CHARACTERIZATIONS OF FUZZY TOPOLOGY | 118$-$133 |

**Abstract**

Following the generalization of Moore-Smith convergence of nets to fuzzy topological spaces which was given in Pu Pao-Ming, Liu Ying-Ming, \emph{Fuzzy topology I. Neighborhood structure of a fuzzy point and Moore-Smith convergence}, J. Math. Anal. Appl., \textbf{76} (1980), 571--599, a characterization theorem between fuzzy topologies and fuzzy convergence classes was introduced in Ying-Ming Liu, \emph{On fuzzy convergence classes}, Fuzzy Sets and Systems, \textbf{30} (1989), 47--51. Our goal in this paper is to provide modified versions of this characterization. Specifically, we will introduce the concept of fuzzy semi-convergence class to give an alternative characterization of fuzzy topology, in relation to the ordinary convergence of fuzzy nets, and then we will introduce the concept of fuzzy ideal convergence class to obtain analogous results, in relation to the ideal convergence of fuzzy nets.

**Keywords:** Fuzzy set; fuzzy topology; fuzzy convergence class; fuzzy ideal convergence class.

**MSC:** 54A20

THE $\sigma$-POINT-FINITE $cn$-NETWORKS ($ck$-NETWORKS) OF PIXLEY-ROY HYPERSPACES | 134$-$137 |

**Abstract**

In this paper, we study the relation between a space $X$ satisfying certain generalized metric properties and the Pixley-Roy hyperspace $\mathcal F[X]$ over $X$ satisfying the same properties. We prove that if $X$ has a $\sigma$-point-finite $cn$-network (resp., $ck$-network), then $\mathcal F[X]$ also has a $\sigma$-point-finite $cn$-network (resp., $ck$-network).

**Keywords:** Pixley-Roy; hyperspace; $cn$-network; $ck$-network; $\sigma$-point-finite.

**MSC:** 54B20, 54D20

A NEW TYPE OF WEIGHTED ORLICZ SPACES | 137$-$146 |

**Abstract**

In this paper, by some group action, we introduce a new type of weighted Orlicz spaces $L^\Phi_{w,v}(\Omega)$, where $w$ and $v$ are weights on $\Omega$ and $\Phi$ is a Young function. We study conditions under which $L^\Phi_{w,v}(G)$ is a convolution Banach algebra, where $G$ is a locally compact group.

**Keywords:** Locally compact group; weighted Orlicz algebra; Young function; convolution; spaceablity; inclusion.

**MSC:** 46E30, 47B37, 43A15