Universal minimal flows of automorphism groups
It is shown that the Lagrange's equations for a Lagrangian system on a Lie algebroid are obtained as the equations for the critical points of the action functional defined on a Banach manifold of curves. The theory of Lagrangian reduction and the relation with the method of Lagrange multipliers are also studied.
In this paper we answer three open problems on varieties of topological groups by invoking Lie group theory. We also reprove in the present context that locally compact groups with arbitrarily small invariant identity neighborhoods can be approximated by Lie groups
We explore (weak) continuity properties of group operations. For this purpose, the Novak number and developability number are applied. It is shown that if is a regular right (left) semitopological group with such that all left (right) translations are feebly continuous, then is a topological group. This extends several results in literature.
Non-abelian Polish groups arising as countable products of countable groups can be tame in arbitrarily complicated ways. This contrasts with some results of Solecki who revealed a very different picture in the abelian case.