In this paper, we define a new concept of FACTORIZATION for a bounded bilinear mapping $f:X\times Y\to Z$, depended on  a natural number $n$ and a cardinal number $\kappa$; which is called $n$-FACTORIZATION property of level $\kappa$. Then we study the relation between $n$-FACTORIZATION property of  level $\kappa$ for $X^*$ with respect to $f$ and automatically boundedness and $w^*$-$w^*$-continuity and also strong Arens irregularity. These results may help us to prove some previous  problems related to strong Arens irregularity more easier than old. These include some results proved by Neufang in ~\cite{neu1} and ~\cite{neu}.  Some applications to certain bilinear mappings on convolution algebras, on a locally compact group, are also included. Finally, some solutions related to  the Ghahramani-Lau conjecture is raised.

The subject of this paper is to study distribution of the prime factors p and their exponents, which we denote by vp (n!), in standard FACTORIZATION of n! into primes. We show that for each θ > 0 the primes p not exceeding nθ eventually assume almost all value of the sum ∑p⩽nθ vp(n!). Also, we introduce the notion of θ-truncated factorial, defined by n!θ =∏p⩽nθ  pvp (n!) and we show that the growth of log n!1/2 is almost half of growth of log n!1.

Classes of Abaffy-Broyden-Spedicato (ABS) methods have been introduced for solving linear systems of equations. The algorithms are powerful methods for developing matrix FACTORIZATIONs and many fundamental numerical linear algebra processes. Here, we show how to apply the ABS algorithms to devise algorithms to compute the WZ and ZW FACTORIZATIONs of a nonsingular matrix as well as the WTW and ZTZ FACTORIZATIONs of a symmetric positives definite matrix. We also describe the QZ and the QW FACTORIZATIONs, with Q orthogonal, and show how to appropriate the parameters of the ABS algorithms to compute these FACTORIZATIONs.