In this paper, we compute the number of distinct centralizers some classes finite rings. We then characterize all rings with $n$ for any positive integer $n \le 5$. Further give connections between a ring and its commutativity degree.

Traditional analysis and usage of operation commutativity relies on pairwise commutativity relationships. In contrast, this paper presentsmethod group commutativity, which specifies the conditions under which operations in amethod group will commute. Method group commutativity can be practically applied to efficiently support distributed object caching and concurrency control. A formal definiti...

‎Suppose $n$ is a fixed positive integer‎. ‎We introduce the relative n-th non-commuting graph $Gamma^{n} _{H,G}$‎, ‎associated to the non-abelian subgroup $H$ of group $G$‎. ‎The vertex set is $Gsetminus C^n_{H,G}$ in which $C^n_{H,G} = {xin G‎ : ‎[x,y^{n}]=1 mbox{~and~} [x^{n},y]=1mbox{~for~all~} yin H}$‎. ‎Moreover‎, ‎${x,y}$ is an edge if $x$ or $y$ belong to $H$ and $xy^{n}eq y^{n}x$ or $x...

Atomic objects have suitable properties to realize reliable distributed computing. Using semantic information of atomic objects can make reliable distributed computing highly concurrent. This paper describes a commutativity based concurrency control algorithm in a multiversion object for highly concurrent distributed computing. Weihl proposed two concurrency control algorithms based on a commut...

In this paper, it is pointed out that the Berry’s phase is a good index of degree of no-commutativity of the quantum statistical model. Intrinsic relations between the ‘complex structure’ of the Hilbert space and Berry’s phase is also discussed.

Although the explicit commutativitiy conditions for second-order linear time-varying systems have been appeared in some literature, these are all for initially relaxed systems. This paper presents explicit necessary and sufficient commutativity conditions for commutativity of second-order linear time-varying systems with non-zero initial conditions. It has appeared interesting that the second r...