2009, Approx. 485 p., Hardcover
ISBN: 978-90-481-2392-6
Due: April 2009
The chosen problems often originate from real physical situations
They are worked out in detail so that the reader may perfectly integrate all the notions presented in the summary of the course
The aim of this work is to bridge the gap between the well-known Newtonian mechanics and the studies on chaos, ordinarily reserved to experts. Several topics are treated: Lagrangian, Hamiltonian and Jacobi formalisms, studies of integrable and quasi-integrable systems. The chapter devoted to chaos also enables a simple presentation of the KAM theorem. All the important notions are recalled in summaries of the lectures. They are illustrated by many original problems, stemming from real-life situations, the solutions of which are worked out in great detail for the benefit of the reader.
This book will be of interest to undergraduate students as well as others whose work involves mechanics, physics and engineering in general.
Foreword
Synoptic Tables.
Chapter 1 : The Lagrangian formulation (1 1 problems)
Chapter 2 : Lagrangian systems (14 problems)
Chapter 3 : The Hamilton's principle (15 problems)
Chapter 4 : The Hamiltonian formalism (17 problems)
Chapter 5 : The Hamilton-Jacobi formalism (1 1 problems)
Chapter 6 : Integrable systems (18 problems)
Chapter 7 : Quasi-integrable systems (9 problems)
Chapter 8 : From order to chaos (12 problems).
Bibliography.
Series: Oberwolfach Seminars , Vol. 35
2009, VIII, 104 p. 30 illus., Softcover
ISBN: 978-3-0346-0047-7
About this textbook
Contains polished notes of three introductory courses to tropical geometry
Based on a seminar at the Mathematical Research Center in Oberwolfach in October 2004
Tropical geometry is algebraic geometry over the semifield of tropical numbers, i.e., the real numbers and negative infinity enhanced with the (max,+)-arithmetics. Geometrically, tropical varieties are much simpler than their classical counterparts. Yet they carry information about complex and real varieties.
These notes present an introduction to tropical geometry and contain some applications of this rapidly developing and attractive subject. It consists of three chapters which complete each other and give a possibility for non-specialists to make the first steps in the subject which is not yet well represented in the literature. The intended audience is graduate, post-graduate, and Ph.D. students as well as established researchers in mathematics.
Series: Understanding Complex Systems
2009, Approx. 360 p., Hardcover
ISBN: 978-3-540-92266-7
Due: May 12, 2009
The science of graphs and networks has become by now a well-established tool for modelling and analyzing a variety of systems with a large number of interacting components. Starting from the physical sciences, applications have spread rapidly to the natural and social sciences, as well as to economics, and are now further extended, in this volume, to the concept of innovations, viewed broadly.
In an abstract, systems-theoretical approach, innovation can be understood as a critical event which destabilizes the current state of the system, and results in a new process of self-organization leading to a new stable state.
The contributions to this anthology address different aspects of the relationship between innovation and networks. The various chapters incorporate approaches in evolutionary economics, agent-based modeling, social network analysis and econophysics and explore the epistemic tension between insights into economics and society-related processes, and the insights into new forms of complex dynamics.
Introduction: Network Perspective on Innovation: Innovative Networks- Network Innovation.- Knowledge Networks: Structure and Dynamics.- Death of Distance in Science? A Gravity Approach to Research Collaboration.- Evolution and Dynamics of Networks in "Regional Innovation Systems".- Agent-based modelling of Innovation Networks-the Fairytale of Spillovers.- Structural Holes, Innovation and the Distribution of Ideas.- Social Network Analysis- A Physics Approach.- Modelling, Evolving Innovation Networks.- Propagation of Innovations in Complex Patterns of Interaction.- Modelling Self-organization and Innovation Processes in Networks.
Series: Publications of the Scuola Normale Superiore
Subseries: Lecture Notes (Scuola Normale Superiore) , Vol. 8
2009, XVI, 237 p., Softcover
ISBN: 978-88-7642-341-3
These lecture notes originate from a course delivered at the Scuola Normale in Pisa in 2006. Generally speaking, the prerequisites do not go beyond basic mathematical material and are accessible to many undergraduates. The contents mainly concern diophantine problems on affine curves, in practice describing the integer solutions of equations in two variables. This case historically suggested some major ideas for more general problems. Starting with linear and quadratic equations, the important connections with Diophantine Approximation are presented and Thue's celebrated results are proved in full detail. In later chapters more modern issues on heights of algebraic points are dealt with, and applied to a sharp quantitative treatment of the unit equation. The book also contains several Supplements, hinted exercises and an Appendix on recent work on heights.
1. Classical Diophantine Equations: linear and quadratic equations, Pell Equation, Diophantine Approximation, congruences. Supplements on Pell equations and irrationality of exp(n) and pi. Notes.- 2. Thue's theorems on Diophantine Equations and rational approximations: Description of strategy and detailed proofs. Later refinements. Supplements on integral points on curves and Rungefs theorem. Notes.- 3. Heights and Diophantine equations over number fields: Product formulas, Weil and Mahler heights, Diophantine approximation in number fields, the S-unit equation and its applications. Supplements on the abc-theorem in function fields and on multiplicative dependence of algebraic functions and their values. Notes.- 4. Heights on subvarieties of G_m^n: Torsion points on plane curves and algebraic points of small height on subvarieties of G_m^n. Structure of algebraic subgroups. Theorems of Zhang and Bilu and applications to the S-unit equation. Supplements on discrete and closed subgroups of R^n and on the Skolem-Mahler-Lech theorem. Notes.- 5. The S-unit equation. A sharp quantitative S-unit theorem; explicit Padef approximations and the counting of large solutions; counting of small solutions. Applications of the quantitative S-unit theorem. Notes.- Appendix by F. Amoroso: Bounds for the height: Generalized Lehmer problem, Dobrowolski lower bounds. Heights of varieties and extensions of lower bounds to higher dimensions; sharp quantitative Zhang's theorem.
Series: Publications of the Scuola Normale Superiore
Subseries: Lecture Notes (Scuola Normale Superiore) , Vol. 9
Originally published as a monograph
2009, Approx. 190 p., Softcover
ISBN: 978-88-7642-342-0
Due: April 2009
This book contains the lecture notes of a basic course in interpolation theory. In the mathematical literature there are many good books on the subject, but none of them is very elementary, and in many cases the basic principles are hidden below great generality. In this second edition, the principles of interpolation theory are illustrated aiming at simplification rather than at generality. The abstract theory is reduced as far as possible, and many examples and applications are given, especially to operator theory and to regularity in partial differential equations. Moreover the treatment is self-contained, the only prerequisite being the knowledge of basic functional analysis.
1 Real interpolation.- 2 Complex interpolation.- 3 Interpolation and domains of operators.- 4 Powers of positive operators.- 5 Interpolation and semigroups.- 6 Analytic semigroups and interpolation.- Appendix: The Bochner integral.