Weak systems of Gandy, Jensen and Devlin

In Part I, we formulate and examine some systems that have arisen in the study of the constructible hierarchy; we find numerous transitive models for them, among which are supertransitive models containing all ordinals that show that Devlin's system BS lies strictly between Gandy's systems PZ and BST'; and we use our models to show that BS fails to handle even the simplest rudimentary functions, and is thus inadequate for the use intended for it in Devlin's treatise. In Part II we propose and study an enhancement of the underlying logic of these systems, build further models to show where the previous hierarchy of systems is preserved by our enhancement; and consider three systems that might serve for Devlin's purposes: one the enhancement of a version of BS, one a formulation of Gandy-Jensen set theory, and the third a subsystem common to those two. In Part III we give new proofs of results of Boffa by constructing three models in which, respectively, TCo, AxPair and AxSing fail; we give some sufficient conditions for a set not to belong to the rudimentary closure of another set, and thus answer a question of McAloon; and we comment on Gandy's numerals and correct and sharpen other of his observations.

Reduction, Linearization and stability of relative equilibria for mechanical systems on riemannian manifolds

Consider a Riemannian manifold equipped with an infinitesimal isometry. For this setup, a unified treatment is provided, solely in the language of Riemannian geometry, of techniques in reduction, linearization, and stability of relative equilibria. In particular, for mechanical control systems, an explicit characterization is given for the manner in which reduction by an infinitesimal isometry, and linearization along a controlled trajectory "commute." As part of the development, relationships are derived between the Jacobi equation of geodesic variation and concepts from reduction theory, such as the curvature of the mechanical connection and the effective potential. As an application of our techniques, fiber and base stability of relative equilibria are studied. The paper also serves as a tutorial of Riemannian geometric methods applicable in the intersection of mechanics and control theory.