We establish a correspondence between inverse sumset theorems (which can be viewed as classifications of approximate (abelian) groups) and inverse theorems for the Gowers norms (which can be viewed as classifications of approximate poly-nomials). In particular, we show that the inverse sumset theorems of Fre˘ ıman type are equivalent to the known inverse results for the Gowers U 3 norms, and mo...

Bit-parallel finite field multiplication in F2m using polynomial basis can be realized in two steps: polynomial multiplication and reduction modulo the irreducible polynomial. In this article, we prove that the modular polynomial reduction can be done with (r − 1)(m − 1) bit additions, where r is the Hamming weight of the irreducible polynomial. We also show that a bit-parallel squaring operati...

Definition 2.1. A commutative ring R is Noetherian if every chain of ideals in R I0 ⊂ I1 ⊂ I2 ⊂ · · · terminates after a finite number of steps (i.e., there is an interger k such that Is = Ik if s ≥ k). Remark 2.2. Polynomial rings R [X1, . . . , Xn] are Noetherian if R is. In particular, S (p) = C [p∗] is Noetherian. Theorem 2.3. If R is a Noetherian ring, and M is a finitely generated R-modul...

A Class of new methods based on a septic non-polynomial spline function for the numerical solution of problems in calculus of variations is presented. The local truncation errors and the methods of order 2th, 4th, 6th, 8th, 10th, and 12th, are obtained. The inverse of some band matrixes are obtained which are required in proving the convergence analysis of the presented method. Convergence anal...