The Modular Isomorphism Problem asks if an isomorphism of group algebras two finite p-groups G and H over a field characteristic p, implies isomorhism the groups H. We survey history problem, explain strategies which were developed to study it present recent negative solution problem. problem is also compared other problems for rings various question remaining open are included.

We define a new function-valued inner product on L2(G), called ?-bracket product, where G is a locally compact abelian group and ? is a topological isomorphism on G. We investigate the notion of ?-orthogonality, Bessel's Inequality and ?-orthonormal bases with respect to this inner product on L2(G).

We present a reduction between subgraph isomorphism and minimal partial subgraph isomorphism. We also provide a new reduction between graph isomorphism and minimal partial graph isomorphism. The new reduction is more efficient and simpler than the previous reduction and can be generalized to subgraph isomorphism. In addition, we show that a reduction from graph isomorphism that makes only one c...

in this paper we study the concept of ph-biatness ofa banach algebra a, where ph is a continuous homomorphism on a.we prove that if ph is a continuous epimorphism on a and a hasa bounded approximate identity and a is ph- biat, then a is ph-amenable. in the case where ph is an isomorphism on a we showthat the ph- amenability of a implies its ph-biatness.