Series: Advanced Courses in Mathematics - CRM Barcelona
2015, Approx. 125 p.
Softcover
ISBN 978-3-0348-0886-6
Due: February 14, 2015
Focusses on the analysis side of h-harmonics and Dunkl transforms
Written in a concise yet informative style
No previous knowledge on reflection groups required
As a unique case in this Advanced Courses book series, the authors have jointly written this introduction to h-harmonics and Dunkl transforms. These are extensions of the ordinary spherical harmonics and Fourier transforms, in which the usual Lebesgue measure is replaced by a reflection-invariant weighted measure.
The theory, originally introduced by C. Dunkl, has been expanded on by many authors over the last 20 years. These notes provide an overview of what has been developed so far. The first chapter gives a brief recount of the basics of ordinary spherical harmonics and the Fourier transform. The Dunkl operators, the intertwining operators between partial derivatives and the Dunkl operators are introduced and discussed in the second chapter. The next three chapters are devoted to analysis on the sphere, and the final two chapters to the Dunkl transform.
The authorsf focus is on the analysis side of both h-harmonics and Dunkl transforms. The need for background knowledge on reflection groups is kept to a bare minimum.
Preface.- Spherical harmonics and Fourier transform.- Dunkl operators associated with reflection groups.- h-Harmonics and analysis on the sphere.- Littlewood?Paley theory and the multiplier theorem.- Sharp Jackson and sharp Marchaud inequalities.- Dunkl transform.- Multiplier theorems for the Dunkl transform.- Bibliography.
Series: Universitext
2015, XII, 320 p.
Softcover
ISBN 978-3-319-14764-2
Due: May 10, 2015
Provides a relatively quick path to a special topic in differential geometry: principal bundles, whose properties are then described in some detail
Presents important, modern geometric ideas in a form accessible to students and researchers in other areas
No previous knowledge of differential geometry is assumed
This introductory graduate level text provides a relatively quick path to a special topic in classical differential geometry: principal bundles, whose properties, including connections and their curvature, are described in some detail. The topic of principal bundles in differential geometry has become classic, even standard, material in the modern graduate mathematics curriculum, which has been treated in various texts.
The unique approach taken in this text presents the material in a way that is intuitive for both students of mathematics and of physics. From a physics standpoint, the topic of principle bundles is essentially gauge theory expressed in geometric language. This book presents important, modern geometric ideas in a form readily accessible to students and researchers in both the physics and mathematics communities, providing each with an understanding and appreciation of the language and ideas of the other.
Introduction.- Basics of Manifolds.- Vector Bundles.- Vectors and Covectors.- Differential Forms.- Lie Derivatives.- Lie Groups.- Frobenius Theorem.- Principle Bundles.- Connections on Principle Bundles.- Curvature of a Connection.- Classical Electromagnetism.- Yang-Mills Theory.- Gauge Theory.- The Dirac Monopole.- Instantons.- What Next?.- Discussion of the Exercises.
Series: Lecture Notes in Mathematics, Vol. 2132
2015, Approx. 200 p.
Softcover
ISBN 978-3-319-14740-6
Due: May 11, 2015
This is the first book dedicated to the subject
Most of the material has never appeared in book form before (and all references to external results have been restricted to material which can be found in books)
The exposition is lively and detailed, in contrast to the typical style of mathematical papers
The monograph contains many annotated open problems, alternative routes and suggestions for new lines of research.
This monograph contains a detailed exposition of the up-to-date theory of separably injective spaces: new and old results are put into perspective with concrete examples (such as \ell_\infty/c_0 and C(K) spaces, where K is a finite height compact space or an F-space, ultrapowers of \mathcal L_\infty spaces and spaces of universal disposition).
It is no exaggeration to say that the theory of separably injective Banach spaces is strikingly different from that of injective spaces. For instance, separably injective Banach spaces are not necessarily isometric to, or complemented subspaces of, spaces of continuous functions on a compact space. Moreover, in contrast to the scarcity of examples and general results concerning injective spaces, we know of many different types of separably injective spaces and there is a rich theory around them. The monograph is completed with a preparatory chapter on injective spaces, a chapter on higher cardinal versions of separable injectivity and a lively discussion of open problems and further lines of research.
A primer on injective Banach spaces.- Separably injective Banach spaces.-
Spaces of universal disposition.- Ultraproducts of type L.- injectivity.-
Other weaker forms of injectivity.- Open Problems.
Series: Lecture Notes in Mathematics, Vol. 2135
2015, Approx. 130 p.
Softcover
ISBN 978-3-319-14758-1
Due: April 14, 2015
In this monograph we present a review of a number of recent results on the motion of a classical body immersed in an infinitely extended medium and subjected to the action of an external force. We investigate this topic in the framework of mathematical physics by focusing mainly on the class of purely Hamiltonian systems, for which very few results are available. We discuss two cases: when the medium is a gas and when it is a fluid. In the first case, the aim is to obtain microscopic models of viscous friction. In the second, we seek to underline some non-trivial features of the motion.
Far from giving a general survey on the subject, which is very rich and complex from both a phenomenological and theoretical point of view, we focus on some fairly simple models that can be studied rigorously, thus providing a first step towards a mathematical description of viscous friction. In some cases, we restrict ourselves to studying the problem at a heuristic level, or we present the main ideas, discussing only some aspects of the proof if it is prohibitively technical.
1. Introduction.- 2. Gas of point particles.- 3. Vlasov approximation.- 4. Motion of a body immersed in a Vlasov system.- 5. Motion of a body immersed in a Stokes ?uid.- A In?nite Dynamics.
2014 176 pages
Series: Textbooks in Mathematics
978-1-49-870637-7
Convexity is an ancient idea going back to Archimedes. Used sporadically in the mathematical literature over the centuries, today it is a flourishing area of research and a mathematical subject in its own right. Convexity is used in optimization theory, functional analysis, complex analysis, and other parts of mathematics.
Convex Analysis introduces analytic tools for studying convexity and provides analytical applications of the concept. The book includes a general background on classical geometric theory which allows readers to obtain a glimpse of how modern mathematics is developed and how geometric ideas may be studied analytically.
Featuring a user-friendly approach, the book contains copious examples and plenty of figures to illustrate the ideas presented. It also includes an appendix with the technical tools needed to understand certain arguments in the book, a tale of notation, and a thorough glossary to help readers with unfamiliar terms. This book is a definitive introductory text to the concept of convexity in the context of mathematical analysis and a suitable resource for students and faculty alike.