This work aims to prove a desingularization theorem for analytic families of two-dimensional vector fields, under the hypothesis that all its singularities have a non-vanishing first jet. Application to problems of singular perturbations and finite cyclicity are discussed in the last chapter.

In a coherent, exhaustive and progressive way, this book presents the tools for studying local bifurcations of limit cycles in families of planar vector fields. A systematic introduction is given to such methods as division of an analytic family of functions in its ideal of coefficients, and asymptotic expansion of non-differentiable return maps and desingularisation. The exposition moves from classical analytic geometric methods applied to regular limit periodic sets to more recent tools for singular limit sets. The methods can be applied to theoretical problems such as Hilbert's 16th problem, but also for the purpose of establishing bifurcation diagrams of specific families as well as explicit computations. - - - The book as a whole is a well-balanced exposition that can be recommended to all those who want to gain a thorough understanding and proficiency in the recently developed methods. The book, reflecting the current state of the art, can also be used for teaching special courses. (Mathematical Reviews)

​This volume was produced in conjunction with the Thematic Program in o-Minimal Structures and Real Analytic Geometry, held from January to June of 2009 at the Fields Institute. Five of the six contributions consist of notes from graduate courses associated with the program: Felipe Cano on a new proof of resolution of singularities for planar analytic vector fields; Chris Miller on o-minimality and Hardy fields; Jean-Philippe Rolin on the construction of o-minimal structures from quasianalytic classes; Fernando Sanz on non-oscillatory trajectories of vector fields; and Patrick Speissegger on pfaffian sets. The sixth contribution, by Antongiulio Fornasiero and Tamara Servi, is an adaptation to the nonstandard setting of A.J. Wilkie's construction of o-minimal structures from infinitely differentiable functions. Most of this material is either unavailable elsewhere or spread across many different sources such as research papers, conference proceedings and PhD theses. This book will be a useful tool for graduate students or researchers from related fields who want to learn about expansions of o-minimal structures by solutions, or images thereof, of definable systems of differential equations. ​

In a coherent, exhaustive and progressive way, this book presents the tools for studying local bifurcations of limit cycles in families of planar vector fields. A systematic introduction is given to such methods as division of an analytic family of functions in its ideal of coefficients, and asymptotic expansion of non-differentiable return maps and desingularisation. The exposition moves from classical analytic geometric methods applied to regular limit periodic sets to more recent tools for singular limit sets. The methods can be applied to theoretical problems such as Hilbert's 16th problem, but also for the purpose of establishing bifurcation diagrams of specific families as well as explicit computations.

Generalizes the Le cycles and numbers to the case of hyper surfaces inside arbitrary analytic spaces. This book defines the Le-Vogel cycles and numbers, and prove that the Le-Vogel numbers control Thom's $a_f$ condition. It describes the relationship between the Euler characteristic of the Milnor fibre and the Le-Vogel numbers.

The authors analyse two topological invariants of an embedding of an arrangement of rational plane curves in the projective complex plane, namely, the cohomology ring of the complement and the characteristic varieties. Their main result states that the cohomology ring of the complement to a rational arrangement is generated by logarithmic 1 and 2-forms and its structure depends on a finite number of invariants of the curve (its combinatorial type).

Investigates the homotopy theory of the suspensions of the real projective plane. This book computes the homotopy groups up to certain range. It also studies the decompositions of the self smashes and the loop spaces with some applications to the Stiefel manifolds.

This book provides an introduction to dynamical systems with multiple time scales. The approach it takes is to provide an overview of key areas, particularly topics that are less available in the introductory form. The broad range of topics included makes it accessible for students and researchers new to the field to gain a quick and thorough overview. The first of its kind, this book merges a wide variety of different mathematical techniques into a more unified framework. The book is highly illustrated with many examples and exercises and an extensive bibliography. The target audience of this book are senior undergraduates, graduate students as well as researchers interested in using the multiple time scale dynamics theory in nonlinear science, either from a theoretical or a mathematical modeling perspective.

A formula for the odd-primary v1-periodic homotopy groups of a finite H-space in terms of its K-theory and Adams operations has been obtained by Bousfield. This work applys this theorem to give explicit determinations of the v1-periodic homotopy groups of (E8,5) and (E8,3), thus completing the determination of all odd-primary v1-periodic homotopy groups of all compact simple Lie groups, a project suggested by Mimura in 1989.

Includes a paper that deals the connective K homology and cohomology of finite groups $G$. This title uses the methods of algebraic geometry to study the ring $ku DEGREES*(BG)$ where $ku$ denotes connective complex K-theory. It describes the variety in terms of the category of abelian $p$-subgroups of $G$ for primes $p$ dividing the group