In this paper by using some conditions, we show that the weak amenability of (2n)-th dual of a Banach algebra A for some n ⩾ 1 implies the weak amenability of A.

In this paper by using some conditions, we show that the weak amenability of (2n)-th dual of a Banach algebra A for some n ⩾ 1 implies the weak amenability of A.

A ring R is uniquely (nil) clean in case for any a 2 R there exists a uniquely idempotent e 2 R such that a e is invertible (nilpotent). Let C = ( A V W B ) be the Morita Context ring. We determine conditions under which the rings A;B are uniquely (nil) clean. Moreover we show that the center of a uniquely (nil) clean ring is uniquely (nil) clean.

In this paper, a fundamentally new method, based on the denition, is introduced for numerical computation of eigenvalues, generalized eigenvalues and quadratic eigenvalues of matrices. Some examples are provided to show the accuracy and reliability of the proposed method. It is shown that the proposed method gives other sequences than that of existing methods but they still are convergent to the desired eigenvalues, generalized eigenvalues and quadratic eigenvalues of matrices. These examples show an interesting phenomenon in the procedure: The diagonal matrix that converges to eigenvalues gives them in decreasing order in the sense of absolute value. Appendices A to C provide Matlab codes that implement the proposed algorithms. They show that the proposed algorithms are very easy to program.

In this paper, we intend to solve special kind of ordinary differential equations which is called Heun equations, by converting to a corresponding stochastic differential equation(S.D.E.). So, we construct a stochastic linear equation system from this equation which its solution is based on computing fundamental matrix of this system and then, this S.D.E. is solved by numerically methods. Moreover, its asymptotic stability and statistical concepts like expectation and variance of solutions are discussed. Finally, the attained solutions of these S.D.E.s compared with exact solution of corresponding differential equations.

In this paper, we represent an inexact inverse subspace iteration method for com- puting a few eigenpairs of the generalized eigenvalue problem Ax = Bx[Q. Ye and P. Zhang, Inexact inverse subspace iteration for generalized eigenvalue problems, Linear Algebra and its Application, 434 (2011) 1697-1715 ]. In particular, the linear convergence property of the inverse subspace iteration is preserved.

Let A be a Banach algebra and E be a Banach A-bimodule then S = A E, the l1-direct sum of A and E becomes a module extension Banach algebra when equipped with the algebras product (a; x):(a′; x′) = (aa′; a:x′ + x:a′). In this paper, we investigate △-amenability for these Banach algebras and we show that for discrete inverse semigroup S with the set of idempotents ES, the module extension Banach algebra S = l1(ES) l1(S) is △-amenable as a l1(ES)-module if and only if l1(ES) is amenable as Banach algebra.

Let a; b; k 2 K and u ; v 2 U(K). We show for any idempotent e 2 K, ( a 0 b 0 ) is e-clean i ( a 0 u(vb + ka) 0 ) is e-clean and if ( a 0 b 0 ) is 0-clean, ( ua 0 u(vb + ka) 0 ) is too.

Let G be a nite group. The prime graph of G is a graph (G) with vertex set (G), the set of all prime divisors of jGj, and two distinct vertices p and q are adjacent by an edge if G has an element of order pq. In this paper we prove that if (G) = (G2(5)), then G has a normal subgroup N such that (N) f2; 3; 5g and G=N = G2(5).