Bøger i C.I.M.E. Foundation Subseries serien

Lectures Given at the 1st 1985 Session of the Centro Internazionale Matematico Estivo, (CIME)

The contributions making up this volume are expanded versions of the courses given at the C.I.M.E. Summer School on the Theory of Moduli.

The fundamental problem in control engineering is to providerobust performance to uncertain plants. H -control theorybegan in the early eighties as an attempt to lay downrigorous foundations on the classical robust controlrequirements. It now turns out that H -control theory is atthe crossroads of several important directions of researchspace or polynomial approach to control and classicalinterpolation theory; harmonic analysis and operator theory;minimax LQ stochastic control and integral equations. Thebook presents the underlying fundamental ideas, problems andadvances through the pen of leading contributors to thefield, for graduate students and researchers in bothengineering and mathematics. From the Contents: C. Foias: Commutant Lifting Techniquesfor Computing Optimal H Controllers.- B.A. Francis:Lectures on H Control and Sampled-Data Systems.- J.W. Helton: Two Topics in Systems Engineering Frequency DomainDesign and Nonlinear System.- H. Kwakernaak: The PolynomialApproach to H -Optimal Regulation.- J.B. Pearson: A ShortCourse in l - Optimal Control

This volume contains the text of four sets of lecturesdelivered at the third session of the Summer Schoolorganized by C.I.M.E. (Centro Internazionale MatematicoEstivo). These texts are preceded by an introduction writtenby C. Cercignani and M. Pulvirenti which summarizes thepresent status in the area of Nonequilibrium Problems inMany-Particle Systems and tries to put the contents of thedifferent sets of lectures in the right perspective, inorder to orient the reader. The lectures deal with theglobal existence of weak solutions for kinetic models andrelated topics, the basic concepts of non-standard analysisand their application to gas kinetics, the kinetic equationsfor semiconductors and the entropy methods in the study ofhydrodynamic limits. CONTENTS: C. Cercignani, M. Pulvirenti: NonequilibriumProblems in Many-Particle Systems. An Introduction.- L. Arkeryd: Some Examples of NSA in Kinetic Theory.- P.L. Lions: Global Solutions of Kinetic Models and RelatedProblems.- P.A. Markowich: Kinetic Models forSemiconductors.- S.R.S. Varadhan: Entropy Methods inHydrodynamic Scaling.

The Session was intended to give a broad survey of the mathematical problems arising in the chaotic transition of deterministic dynamical systems, both in classical and quantum mechanics.The lectures of Mather and Forni thoroughly cover the area- preserving twist maps, and include an up-to-date version of the Aubry-Mather theory.The lectures of Bellissard describe the quantum aspects: classical limit, localization, spectral properties of the relevant Schrodinger operators and thus represent an exhaus- tive introduction to the mathematics of "e;quantum chaos"e;. The lectures of Degli Esposti et al. reviews equidistribu- tion of unstable periodic orbits and classical limit of quantized toral symplectomorphisms.

Nowadays we are facing numerous and important imaging problems: nondestructive testing of materials, monitoring of industrial processes, enhancement of oil production by efficient reservoir characterization, emerging developments in noninvasive imaging techniques for medical purposes - computerized tomography (CT), magnetic resonance imaging (MRI), positron emission tomography (PET), X-ray and ultrasound tomography, etc. In the CIME Summer School on Imaging (Martina Franca, Italy 2002), leading experts in mathematical techniques and applications presented broad and useful introductions for non-experts and practitioners alike to many aspects of this exciting field. The volume contains part of the above lectures completed and updated by additional contributions on other related topics.

This book is a collection of lecture notes for the CIME course on "Multiscale and Adaptivity: Modeling, Numerics and Applications," held in Cetraro (Italy), in July 2009.Complex systems arise in several physical, chemical, and biological processes, in which length and time scales may span several orders of magnitude. Traditionally, scientists have focused on methods that are particularly applicable in only one regime, and knowledge of the system on one scale has been transferred to another scale only indirectly. Even with modern computer power, the complexity of such systems precludes their being treated directly with traditional tools, and new mathematical and computational instruments have had to be developed to tackle such problems. The outstanding and internationally renowned lecturers, coming from different areas of Applied Mathematics, have themselves contributed in an essential way to the development of the theory and techniques that constituted the subjects of the courses.

The issue of regularity has played a central role in the theory of Partial Differential Equations almost since its inception, and despite the tremendous advances made it still remains a very fruitful research field.In particular considerable strides have been made in regularity estimates for degenerate and singular elliptic and parabolic equations over the last several years, and in many unexpected and challenging directions.Because of all these recent results, it seemed high time to create an overview that would highlight emerging trends and issues in this fascinating research topic in a proper and effective way.The course aimed to show the deep connections between these topics and to open new research directions through the contributions of leading experts in all of these fields.

The four papers collected in this book discuss advanced results in analytic number theory, including recent achievements of sieve theory leading to asymptotic formulae for the number of primes represented by suitable polynomials; counting integer solutions to Diophantine equations, using results from algebraic geometry and the geometry of numbers; the theory of Siegel¿s zeros and of exceptional characters of L-functions; and an up-to-date survey of the axiomatic theory of L-functions introduced by Selberg.