A general mechanism for "breaking" commutativity in algebras is proposed: if the underlying set taken to be not a crisp set, but rather an obscure/fuzzy membership function, reflecting degree of truth that element belongs can incorporated into commutation relations. The special "deformations" and $\varepsilon $-commutativity are introduced such way equal degrees result "nondeformed" case. We al...

4. Other cases. The corresponding theory for a non-algebraic curve is complicated by the fact that the number of polynomials of the nth degree in the orthogonal system increases with n, but is compensatingly simplified by the observation that the representation corresponding to (2) holds for all values of x and y, so that (except for the assumption that the domain of orthogonality is all on one...

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.

We give a simple proof of George Andrews’s balanced 5F4 evaluation using two fundamental principles: the nth difference of a polynomial of degree less than n is zero, and a polynomial of degree n that vanishes at n+1 points is identically zero.

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...

As we all know, every derivation of a polynomial that is nth degree another has common factor with this polynomial. Then, Eduardo Casas-Alvero came up conjecture think opposite the axiom just talked before. That to proof if We are going use different method attack and prove it when n small.