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 $n$ ($\ge 2$) point masses $m_1,\dots,m_n$ freely falling in the vertical half line $\{q|\, q\ge 0\}$ under constant gravitation and colliding with each other and the solid floor $q=0$ 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 $m_1\ge\dots\ge m_n$ (i. e. the masses do not increase as we go up) and $m_1\ne m_2$

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