Differentiable Periodic Maps

MEMS Vibratory Gyroscopes provides a solid foundation in the theory and fundamental operational principles of micromachined vibratory rate gyroscopes, and introduces structural designs that provide inherent robustness against structural and environmental variations. In part one, the dynamics of the vibratory gyroscope sensing element is developed, common micro-fabrication processes and methods commonly used in inertial sensor production are summarized, design of mechanical structures for both linear and torsional gyroscopes are presented, and electrical actuation and detection methods are discussed along with details on experimental characterization of MEMS gyroscopes. In part two, design concepts that improve robustness of the micromachined sensing element are introduced, supported by constructive computational examples and experimental results illustrating the material. MEMS Vibratory Gyroscopes is a must have book for engineers in both industry and academia who specialize in the design and manufacture of gyroscopes. Readers will find: A unique balance between theory and practical design issues. Comprehensive and detailed information outlining the mathematical models of the mechanical structure and system-level sensor design. Solid background Information on mechanical and electrical design, fabrication, packaging, testing and characterization. About The MEMs Reference Shelf: "The MEMs Reference Shelf is a series devoted to Micro-Electro-Mechanical Systems (MEMs) which combine mechanical, electrical, optical, or fluidic elements on a common microfabricated substrate to create sensors, actuators, and microsystems. The series, authored by leading MEMs practitioners, strives to provide a framework where basic principles, known methodologies and new applications are integrated in a coherent and consistent manner." STEPHEN D. SENTURIA Massachusetts Institute of Technology, Professor of Electrical Engineering, Emeritus

This research tract contains an exposition of our research on bordism and differentiable periodic maps done in the period 1960-62. The research grew out of the conviction, not ours alone, that the subject of transformation groups is in need of a large infusion of the modern methods of algebraic topology. This conviction we owe at least in part to Armand Borel; in particular Borel has maintained the desirability of methods in transformation groups that use differentiability in a key fashion [9, Introduction], and that is what we try to supply here. We do not try to relate our work to Smith theory, the homological study of periodic maps due to such a large extent to P. A. Smith; for a modern development of that subject which expands it greatly see the Borel Seminar notes [9]. It appears to us that our work is independent of Smith theory, but in part inspired by it. We owe a particular debt to G. D. Mostow, who pointed out to us some time ago that it followed from Smith theory that an involution on a compact manifold, or a map of prime period [italic lowercase]p on a compact orientable manifold, could not have precisely one fixed point. It was this fact that led us to believe it worthwhile to apply cobordism to periodic maps.

The aim of this thesis is to give a complete exposition of the main topological and ergodic properties of the expanding maps and provide some new results about their periodic points sets. It is common practice in topological dynamics to examine the results of differentiable dynamics and try to establish them in a not necessarily differentiable hyperbolic map.

The volume develops a thorough theory of singular fibers of generic differentiable maps. This is the first work that establishes the foundational framework of the global study of singular differentiable maps of negative codimension from the viewpoint of differential topology. The book contains not only a general theory, but also some explicit examples together with a number of very concrete applications. This is a very interesting subject in differential topology, since it shows a beautiful interplay between the usual theory of singularities of differentiable maps and the geometric topology of manifolds.