Wanchai Tapanyo | Searcharxiv

arXiv2022

gcd-Pairs in $\mathbb{Z}_{n}$ and their graph representations

This research introduces a gcd-pair in $\mathbb{Z}_n$ which is an unordered pair $\{[a]_n, [b]_n\}$ of elements in $ \mathbb{Z}_n $ such that $0\leq a,b < n$ and the greatest common divisor $\gcd(a,b)$ divides $ n $. The properties of gcd-pairs in $ \mathbb{Z}_n $ and their graph representations are investigated. We also provide the counting formula of gcd-pairs in $ \mathbb{Z}_n $ and its subsets. The algorithms to find, count and check gcd-pairs in $ \mathbb{Z}_{n}$ are included.

math.CO↗

arXiv2019

On Completion of \textit{C}*-algebra-valued metric spaces

The concept of a $ C $*-algebra-valued metric space was introduced in 2014. It is a generalization of a metric space by replacing the set of real numbers by a $ C $*-algebra. In this paper, we show that $ C $*-algebra-valued metric spaces are cone metric spaces in some point of view which is useful to extend results of the cone case to $ C $*-algebra-valued metric spaces. Then the completion theorem of $ C $*-algebra-valued metric spaces is obtained. Moreover, the completion theorem of $ C $*-algebra-valued normed spaces is verified and the connection with Hilbert $ C $*-modules, generalized inner product spaces, is also provided.

math.FA↗