Commutativity of operations is the better way to provide a high degree of concurrency on shared data types. In this short paper, we present a technique to increase concurrency using operational transformations. This technique enforces commutativity even though the operations do not naturally commute. We report our experience on (i) automatically verifying the correctness of this transformation-...

Commutative Yang-Mills theories in 1+1 dimensions exhibit an interesting interplay between geometrical properties and U(N) gauge structures: in the exact expression of a Wilson loop with n windings a non trivial scaling intertwines n and N . In the non-commutative case the interplay becomes tighter owing to the merging of space-time and “internal” symmetries in a larger group U(∞). We perform a...

the degree kirchhoff index of a connected graph $g$ is defined as‎ ‎the sum of the terms $d_i,d_j,r_{ij}$ over all pairs of vertices‎, ‎where $d_i$ is the‎ ‎degree of the $i$-th vertex‎, ‎and $r_{ij}$ the resistance distance between the $i$-th and‎ ‎$j$-th vertex of $g$‎. ‎bounds for the degree kirchhoff index of the line and para-line‎ ‎graphs are determined‎. ‎the special case of regular grap...

in this paper we study the structure and the commutativity of a ring r, in which for each x,y ? r, there exist two integers depending on x,y such that [x,y]k equals x n or y n.

then we term the elements x and y n-commutative. It is not difficult to verify that »-commutativity and (1—»)-commutativity are equivalent properties of the elements x and y, that ( — ̂ -commutativity implies ordinary commutativity, and that commuting elements are »-commutative. From any concept and property involving the fact that certain elements [or functions of elements] commute, one may der...

We introduce a family of hypergroups, called weakly complete, generalizing the construction complete hypergroups. Starting from given group G, our prescribes β-classes hypergroups and allows some hyperproducts not to be parts, based on suitably defined relation over G. The commutativity degree can related that underlying group. Furthermore, in analogy commutativity, we completeness finite analy...