Subring subgroups in Chevalley groups with doubly laced root systems

AbstractWe study the lattice L of subgroups of a Chevalley group G(Φ,A) over a ring A, containing its elementary subgroup E(Φ,S) over a subring S⊆A. The standard description asserts that for any H∈L there exists a unique subring R⊇S of A such that H contains E(Φ,R) as a normal subgroup.The standard description was obtained by Ya. Nuzhin for algebraic field extensions. On the other hand, it was expected that the standard description does not hold for transcendental extensions. This was recently proved by the author for simply laced root systems.Now, suppose that Φ is doubly laced, i.e. Φ=Bl, Cl or F4, and that 2 is invertible in S. In the article we prove that in these settings the standard description of L holds for an arbitrary pair of rings S⊆A

On the Mathematical Theory of Records

summary:In the present work, we briefly analyze the development of the mathematical theory of records. We first consider applications associated with records. We then view distributional and limit results for record values and times. We further present methods of generation of continuous records. In the end of this work, we discuss some tests based on records