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...

The real numbers are governed by a collection of rules that have to do with addition, multiplication, and inequalities. In the rules below, x, y, z 2 R. (In other words, x, y, and z are real numbers.) Rules of addition. • (x + y) + z = x + (y + z) (Law of associativity) • x + y = y + x (Law of commutativity) • x + 0 = x (Law of identity) • x + x = 0 (Law of inverses) Rules of multiplication. • ...

Algebraic structures in which the property of commutativity is substituted by mediality are introduced. We consider (associative) graded algebras and instead almost (generalized or $\varepsilon$-commutativity) we introduce ("commutativity-to-mediality" ansatz). Higher twisted products "deforming" brackets (being medial analog Lie brackets) defined. Toyoda's theorem connects (universal) with abe...

It is shown in first order perturbation theory that anharmonic oscillators in non-commutative space behave smoothly in the commutative limit just as harmonic oscillators do. The non-commutativity provides a method for converting a problem in degenerate perturbation theory to a non-degenerate problem. In the last few years theories in non-commutative space [1]-[5] have been studied extensively. ...

In this paper, we use the structure theory to prove an analog to a well-known theorem of Herstein as follows: Let R be a ring with center C such that for all x,y ? R either [x,y]= 0 or x-x [x,y]? C for some non negative integer n= n(x,y) dependingon x and y. Then R is commutative.