let  and  be banach algebras, ,  and . we define an -product on  which is a strongly splitting extension of  by . we show that these products form a large class of banach algebras which contains all module extensions and triangular banach algebras. then we consider spectrum, arens regularity, amenability and weak amenability of these products.

Let $mathcal{A}$ be a Banach algebra with BAI and $E$ be an introverted subspace of $mathcal{A}^prime$. In this paper we study the quotient Arens regularity of $mathcal{A}$ with respect to $E$ and prove that the group algebra $L^1(G)$ for a locally compact group $G$, is quotient Arens regular with respect to certain introverted subspace $E$ of $L^infty(G)$. Some related result are given as well.

The first half of this paper is mainly expository, and aims at introducing the regularity lemma of Szemerédi. Among others, we discuss an early application of the regularity lemma that relates the notions of universality and uniform distribution of edges, a form of ‘pseudorandomness’ or ‘quasi-randomness’. We then state two closely related variants of the regularity lemma for sparse graphs and ...

in this paper we will generalize  some of known results on the tight closure of an ideal to the tight closure of an ideal relative to a module .

Understanding how humans code for and respond to environmental uncertainty/regularity is a question shared by current computational and neurobiological approaches to human cognition. To date, studies investigating neurobiological systems that track input uncertainty have examined responses to uni-sensory streams. It is not known, however, whether there exist brain systems that combine informati...

Duplication is used to increase data availability in the presence of site or communication failures and to decrease retrieval costs by local access if possible. Our system is completely decentralized and nodes can be removed/added dynamically. P2P is a trendy technology used for file sharing. File duplication and Regularity protection are the methods used in P2P for elevated system performance....

In this paper d2 degree of a vertex and total d2 -degree of a vertex in an intuitionistic fuzzy graphs are defined. Also (2, (c1, c2))-regularity and totally (2, (c1, c2))-regularity of an intuitionistic fuzzy graphs are defined. A relation between (2, (c1, c2))-regularity and totally (2, (c1, c2))-regularity on intuitionistic fuzzy graph is studied. (2, (c1, c2))regularity on a path on four ve...

In this report, we review the connections between regularity theory, low rank approximation of matrices and tensors, and approximate constraint satisfaction problems. Our presentation of the algorithmic aspects of regularity theory follows parts of [1, 2, 7]; wherever possible we cast the material in the modern language advocated by Tao [15]. One main motivation is to understand the power and l...