Volume 73 , issue 4 ( 2021 )back
AN EXTENSION OF THE CONCEPT OF $\boldsymbol\gamma$-CONTINUITY FOR MULTIFUNCTIONS223$-$242
M. Przemski

Abstract

A function $f:(X,\tau) \rightarrow (Y,\tau^{\ast})$ between topological spaces is called $\gamma$-continuous if $f^{-1}(W) \subset Cl(Int(f^{-1}(W))) \cup Int(Cl(f^{-1}(W)))$ for each open $W \subset Y$, where $Cl$ (resp. $Int$ ) denotes the closure (resp. interior) operator on X. When we use the other possible operators obtained by multiple composing $Cl$ and $Int$, then this condition boils down to the definitions of known types of generalized continuity. The case of multifunctions is quite different. The appropriate condition have two forms: $F^{+}(W) \subset Cl(Int(F^{+}(W))) \cup Int(Cl(F^{+}(W)))$ called $u.\gamma .c.$ or, $F^{-}(W) \subset Cl(Int(F^{-}(W))) \cup Int(Cl(F^{-}(W)))$ called $l.\gamma .c.$, where F$^{+}$(W) = $\left\{x ın X: F(x) \subset W \right\}$ and F$^{-}$(W) = $\left\{x ın X: F(x) \cap W \neq\emptyset\right\}$. So, one can consider the simultaneous use of the two different inverse images namely, $F^{+}(W)$ and $F^{-}(W)$. We will show that in this case the usage of all possible multiple compositions of $Cl$ and $Int$ leads to the new different types of continuity for multifunctions, which together with the previous defined types of continuity forms a collection which is complete in a certain topological sense.

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Keywords: Multifunction; upper semi continuity; quasi-continuity; $\gamma$-continuity.

MSC: 54C05, 54C08, 54C60, 54A05, 58C07

ON JET LIKE BUNDLES OF VECTOR BUNDLES243$-$255
M. Doupovec, J. Kurek, W. M. Mikulski

Abstract

We describe completely the so called jet like functors of a vector bundle $E$ over an $m$-dimensional manifold $M$, i.e.\ bundles $FE$ over $M$ canonically depending on $E$ such that $F(E_1\times_M E_2)=FE_1\times_MFE_2$ for any vector bundles $E_1$ and $E_2$ over $M$. Then we study how a linear vector field on $E$ can induce canonically a vector field on $FE$.

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Keywords: Bundle functor; gauge bundle functor; natural transformation; (gauge) natural operator; vector bundle; module bundle; jet.

MSC: 58A05, 58A20, 58A32

CONICS FROM THE ADJOINT REPRESENTATION OF $SU(2)$256$-$267
M. Crasmareanu

Abstract

The aim of this paper is to introduce and study the class of conics provided by the symmetric matrices of the adjoint representation of the Lie group $SU(2)=S^3$. This class depends on three real parameters as components of a point of sphere $S^2$ and various relationships between these parameters give special subclasses of conics. A symmetric matrix inspired by one giving by Barning as Pythagorean triple preserving matrix and associated hyperbola are carefully analyzed. We extend this latter hyperbola to a class of hyperbolas with integral coefficients. A complex approach is also included.

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Keywords: Conic; adjoint representation of $SU(2)$; complex variable.

MSC: 11D09, 51N20, 30C10, 22E47

AN INTRODUCTION TO $\mathfrak{U}$-METRIC SPACE AND NON-LINEAR CONTRACTION WITH APPLICATION TO THE STABILITY OF FIXED POINT EQUATION268$-$281
K. Roy, M. Saha, D. Dey

Abstract

In this paper, we introduce the notion of $U$-metric space of $n$-tuples which generalizes several known metric-type spaces. We study topological properties of such newly constructed spaces and prove Cantor's intersection-like theorem. Banach contraction principle theorem is proved in this space and we apply the theorem to obtain the stability of a fixed point equation.

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Keywords: $\mathfrak{U}$-metric space; Cantor's intersection like theorem; fixed point; stability of fixed point equation.

MSC: 47H10, 54H25

CONFORMAL YAMABE SOLITON AND $*$-YAMABE SOLITON WITH TORSE FORMING POTENTIAL VECTOR FIELD282$-$292
S. Roy, S. Dey, A. Bhattacharyya

Abstract

The goal of this paper is to study conformal Yamabe soliton and $*$-Yamabe soliton, whose potential vector field is torse forming. Here, we have characterized conformal Yamabe soliton admitting potential vector field as torse forming with respect to Riemannian connection, semi-symmetric metric connection and projective semi-symmetric connection on Riemannian manifold. We have also shown the nature of $*$-Yamabe soliton with torse forming vector field on Riemannian manifold admitting Riemannian connection. Lastly we have developed an example to corroborate some theorems regarding Riemannian connection on Riemannian manifold.

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Keywords: Yamabe soliton; conformal Yamabe soliton; $*$-Yamabe soliton; torse-forming vector field; torqued vector field; semisymmetric metric connection; projective semisymmetric connection.

MSC: 53C21, 53C25, 53B50, 35Q51