we survey the recent investigations on (bounded, sequential) approximate connesamenability and pseudo-connes amenability for dual banach algebras. we will discuss thecore problems concerning these notions and address the signi cance of any solutions to themto the development of the eld.

in this paper, we study the arens regularity properties of module actions and we extend some proposition from baker, dales, lau and others into general situations. we establish some relationships between the topological centers of module actions and factorization properties of them with some results in group algebras. in 1951 arens shows that the second dual of banach algebra endowed with the e...

Let and be randomly chosen subsets of the first positive integers cardinalities , such that their sumset has size . We show asymptotically almost surely are fully contained in arithmetic progressions with same common difference approximately also prove a counting theorem for pairs sets arbitrary abelian groups. The results hold Our main tool is an asymmetric version method hypergraph containers...

Given integers i, j, k, L,M , we establish a new double bounded q−series identity from which the three parameter (i, j, k) key identity of Alladi-Andrews-Gordon for Göllnitz’s (big) theorem follows if L,M → ∞. When L = M , the identity yields a strong refinement of Göllnitz’s theorem with a bound on the parts given by L. This is the first time a bounded version of Göllnitz’s (big) theorem has b...