we give a new proof of the well known wedderburn's little theorem (1905) that a finite‎ ‎division ring is commutative‎. ‎we apply the concept of frobenius kernel in frobenius representation theorem in finite group‎ ‎theory to build a proof‎.

The notion of a Frobenius coring is introduced, and it is shown that any such coring produces a tower of Frobenius corings and Frobenius extensions.

A group G is called a CI-group if, for any subsets S, T ⊂ G, whenever two Cayley graphs Cay(G, S) and Cay(G, T ) are isomorphic, there exists an element σ ∈ Aut(G) such that S = T . The problem of seeking finite CI-groups is a longstanding open problem in the area of Cayley graphs. This paper contributes towards a complete classification of finite CI-groups. First it is shown that the Frobenius...

This paper explores the interplay between the Frobenius functor and Serre’s vanishing conjecture over a Noetherian local ring R of prime characteristic p. We show that the Frobenius functor induces a diagonalizable map on certain Q-vector spaces, which are tensor products of Q with quotients of Grothendieck groups. This allows us to decompose an element (representing a bounded complex of finite...

Quantum homogeneous vector bundles are introduced by a direct description of their sections in the context of Woronowicz type compact quantum groups. The bundles carry natural topologies inherited from the quantum groups, and their sections furnish projective modules over algebras of functions on quantum homogeneous spaces. Further properties of the quantum homogeneous vector bundles are invest...

Based on the Frobenius series method, stresses analysis of the functionally graded rotating thick cylindrical pressure vessels (FGRTCPV) are examined. The vessel is considered in both plane stress and plane strain conditions. All of the cylindrical shell properties except the Poisson ratio are considered exponential function along the radial direction. The governing Navier equation for this pro...

We construct blocks of finite groups with arbitrarily large Morita Frobenius numbers, an invariant which determines the size minimal field definition associated basic algebra. This answers a question Benson and Kessar. also improves upon result second author where $\mathcal{O}$-Morita numbers are constructed.

We study co-Frobenius and more generally Quasi-co-Frobenius corings over arbitrary baserings and over PF baserings in particular. We generalize some results about (Quasi-) co-Frobenius coalgebras to the case of non-commutative base rings and give several new characterisations for co-Frobenius and Quasi-co-Frobenius corings, some of them are new even in the coalgebra situation. We construct Mori...

We begin with personal notes describing the atmosphere of ”Bogoliubov renormalization group” birth. Then we expose the history of RG discovery in the QFT and of the RG method devising in the mid-fifties. The third part is devoted to proliferation of RG ideas into diverse parts of theoretical physics. We conclude with discussing the perspective of RG method further development and its applicatio...