23 research outputs found
On the Whitehead determinant for semi-local rings
AbstractWe answer the question: when the Whitehead determinant of a semi-local ring is the abelization of the multiplicative group
Prestabilization for K1 of banach algebras
AbstractThe injective stability for the general linear group modulo elementary matrices begins at one plus the stable range of the ring of entries. At one step earlier, the kernel of stabilization is perhaps larger than the group of elementary matrices. Using a Dieudonné-style determinant, it is shown that this kernel is generated by matrices of the form (1 + XY)(1 + XY), under certain conditions on the ring of entries, or in the relative case, on the ideal. For any ideal of stable rank one, the kernel is given in terms of generators (X + Z + XYZ)(X + Z + ZYX). Under somewhat stronger conditions, the kernel is shown to be a commutator subgroup
Normal Subgroups of Classical Groups over von Neumann Regular Rings
AbstractLet R be a von Neumann regular ring. We obtain a complete description of all subgroups of the pseudo-orthogonal groups O2nR for n ≥ 3 which are normalized by elementary orthogonal matrices. We also prove the normality of EO2n(R, J) in O2nR when R is an abelian von Neumann regular ring and n ≥ 1
The characteristic exponents of the falling ball model
We study the characteristic exponents of the Hamiltonian system of () point masses freely falling in the vertical half line
under constant gravitation and colliding with each other and
the solid floor elastically. This model was introduced and first studied
by M. Wojtkowski. Hereby we prove his conjecture: All relevant characteristic
(Lyapunov) exponents of the above dynamical system are nonzero, provided that
(i. e. the masses do not increase as we go up) and
Upgrading the Local Ergodic Theorem for planar semi-dispersing billiards
The Local Ergodic Theorem (also known as the `Fundamental Theorem') gives
sufficient conditions under which a phase point has an open neighborhood that
belongs (mod 0) to one ergodic component. This theorem is a key ingredient of
many proofs of ergodicity for billiards and, more generally, for smooth
hyperbolic maps with singularities. However the proof of that theorem relies
upon a delicate assumption (Chernov-Sinai Ansatz), which is difficult to check
for some physically relevant models, including gases of hard balls. Here we
give a proof of the Local Ergodic Theorem for two dimensional billiards without
using the Ansatz.Comment: 17 pages, 2 figure