Florin-Petre Boca, Cardiff University, UK
Rotation C*-Algebras and Almost Mathieu Operators
A publication of the Theta Foundation.
Description
This book delivers a swift, yet concise, introduction to some aspects of rotation $C^*$-algebras and almost Mathieu operators. The two topics come from different areas of analysis: operator algebras and the spectral theory of Schrodinger operators, but can be approached in a unified way. The book does not try to be the definitive treatise on the subject, but rather presents a survey highlighting the important results and demonstrating this unified approach.
For each real number $\alpha$, the rotation $C^*$-algebra $A_\alpha$ can be abstractly defined as the universal $C^*$-algebra generated by two elements $U$ and $V$ subject to the relation $UV = e^{2\pi i \alpha} VU$. When $\alpha$ is an integer, $A_\alpha$ is isomorphic to the commutative $C^*$-algebra of continuous functions on a two-dimensional torus. When $\alpha$ is not an integer, the algebra is sometimes called a non-commutative 2-torus. In this respect, some of the methods you will find here can be regarded as a sort of non-commutative Fourier analysis. An almost Mathieu operator is a type of self-adjoint operator on the Hilbert space $\ell^2 = \ell^2(\mathbf{Z})$.
The exposition is geared toward a wide audience of mathematicians: researchers and advanced students interested in operator algebras, operator theory and mathematical physics. Readers are assumed to be acquainted with some functional analysis, such as definitions and basic properties of $C^*$-algebras and von Neumann algebras, some general results from ergodic theory, as well as the Fourier transform (harmonic analysis) on elementary abelian locally compact groups of the form $\mathbf{R}^d \times \mathbf{Z}^k \times \mathbf{T}^1 \times F$, where $F$ is a finite group.
Much progress has been made on these topics in the last twenty years. The present book will introduce you to the subjects and to the significant results.
Contents
Prerequisites on rotation $C^*$-algebras
Almost Mathieu operators and automorphisms of $A_\alpha$
Perturbations of the spectrum of $H_{\alpha, \lambda}$
The spectrum of almost Mathieu operators for rational $\alpha$
The absence of isolated points in the spectrum of $H_{\alpha, \lambda}$
Lyapunov exponents and pure point spectrum
The Lebesgue measure of $\mathrm{spec}_{(p/q,\lambda)}$
Some estimates for the Lebesgue measure of $\mathrm{spec}(H_{(p/q,\lambda)})$
Spectral computations for certain non-self-adjoint operators
Projections in rotation $C^*$-algebras
The approximation of irrational rotation $C^*$-algebras
The approximation of irrational non-commutative spheres
Subject index
Notation
Details:
Publisher: Theta Foundation
Distributor: American Mathematical Society
Publication Year: 2001
ISBN: 973-99097-7-9
Paging: 172 pp.
Binding: Hardcover
A R Rajwade
Convex Polyhedra with Regularity Conditions
and Hilbert's Third Problem
A publication of the Hindustan Book Agency.
Description
Since antiquity, people knew that there are only five regular solids, i.e. polyhedra whose all faces are regular polygons and all solid angles are also regular. These solids are, of course, the tetrahedron, the octahedron, the cube, the icosahedron, and the dodecahedron. Later, much attention was drawn to the question of how to describe polyhedra with other types of regularity conditions. The author puts together many facts known in this direction. He formulates four regularity conditions (two for faces and two for solid angles) and for any combination of their conditions lists all the corresponding polyhedra. In this way, he obtains such very interesting classes of solids as 13 semiregular solids, or 8 deltahedra, or 92 regularly faces polyhedra, etc. In later chapters the author presents some related topics of geometry of solids, like star polyhedra and plane tessellations. In the concluding chapter, a complete solution of the Hilbert 3rd problem is given.
Supplied with many figures, the book can be easily read by anyone interested in this beautiful classical geometry.
Contents
Introduction
Definitions and notations
Theorems of Euler and Descartes
The regularity restrictions and the five bodies of Plato
Metrical properties of the five Platonic polyhedra
The fourteen bodies of Archimedes
Another method of enumerating the semi-regular polyhedra
The eight Deltahedra
Finiteness of the number of convex regular faced polyhedra (RFP) and the remaining cases of regularity restrictions
Star polyhedra and plane tessellations
A theorem of Johnson and Grunbaum
Description of the ninety-two RFP and their derivation from the simple ones
Hilbert's third problem
Bibliography
Index
Details:
Publisher: Hindustan Book Agency
Distributor: American Mathematical Society
Publication Year: 2001
ISBN: 81-85931-28-3
Paging: 120 pp.
Binding: Hardcover
Luis Barreira, Instituto Superior Tecnico, Lisbon, Portugal,
and Yakov B. Pesin, Pennsylvania State University, University Park, PA
Lyapunov Exponents and Smooth Ergodic Theory
Description
This book is a systematic introduction to smooth ergodic theory. The topics discussed include the general (abstract) theory of Lyapunov exponents and its applications to the stability theory of differential equations, stable manifold theory, absolute continuity, and the ergodic theory of dynamical systems with nonzero Lyapunov exponents (including geodesic flows).
The authors consider several nontrivial examples of dynamical systems with nonzero Lyapunov exponents to illustrate some basic methods and ideas of the theory.
This book is self-contained. The reader needs a basic knowledge of real analysis, measure theory, differential equations, and topology. The authors present basic concepts of smooth ergodic theory and provide complete proofs of the main results. They also state some more advanced results to give readers a broader view of smooth ergodic theory. This volume may be used by those nonexperts who wish to become familiar with the field.
Contents
Introduction
Lyapunov stability theory of differential equations
Elements of nonuniform hyperbolic theory
Examples of nonuniformly hyperbolic systems
Local manifold theory
Ergodic properties of smooth hyperbolc measures
Bibliography
Index
Details:
Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: University Lecture Series,Volume: 23
Publication Year: 2002
ISBN: 0-8218-2921-1
Paging: 151 pp.
Binding: Softcover
Edited by: Bradd Hart and Matthew Valeriote
McMaster University, Hamilton, ON, Canada
Lectures on Algebraic Model Theory
Description
In recent years, model theory has had remarkable success in solving important problems as well as in shedding new light on our understanding of them. The three lectures collected here present recent developments in three such areas: Anand Pillay on differential fields, Patrick Speissegger on o-minimality and Matthias Clasen and Matthew Valeriote on tame congruence theory.
Contents
Differential fields
Differential fields
Lecture son o-minimality
Lectures on o-minimality
Tame congruence theory
The structure of finite algebras
Varieties
Bibliography
Details:
Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: Fields Institute Monographs, Volume: 15
Publication Year: 2001
ISBN: 0-8218-2705-7
Paging: 111 pp.
Binding: Hardcover
Georgi V. Smirnov, University of Porto, Portugal
Introduction to the Theory of Differential Inclusions
Description
A differential inclusion is a relation of the form $\dot x \in F(x)$, where $F$ is a set-valued map associating any point $x \in R^n$ with a set $F(x) \subset R^n$. As such, the notion of a differential inclusion generalizes the notion of an ordinary differential equation of the form $\dot x = f(x)$. Therefore, all problems usually studied in the theory of ordinary differential equations (existence and continuation of solutions, dependence on initial conditions and parameters, etc.) can be studied for differential inclusions as well. Since a differential inclusion usually has many solutions starting at a given point, new types of problems arise, such as investigation of topological properties of the set of solutions, selection of solutions with given properties, and many others.
Differential inclusions play an important role as a tool in the study of various dynamical processes described by equations with a discontinuous or multivalued right-hand side, occurring, in particular, in the study of dynamics of economical, social, and biological macrosystems. They also are very useful in proving existence theorems in control theory.
This text provides an introductory treatment to the theory of differential inclusions. The reader is only required to know ordinary differential equations, theory of functions, and functional analysis on the elementary level.
Chapter 1 contains a brief introduction to convex analysis. Chapter 2 considers set-valued maps. Chapter 3 is devoted to the Mordukhovich version of nonsmooth analysis. Chapter 4 contains the main existence theorems and gives an idea of the approximation techniques used throughout the text. Chapter 5 is devoted to the viability problem, i.e., the problem of selection of a solution to a differential inclusion that is contained in a given set. Chapter 6 considers the controllability problem. Chapter 7 discusses extremal problems for differential inclusions. Chapter 8 presents stability theory, and Chapter 9 deals with the stabilization problem.
Contents
Foundations
Convex analysis
Set-valued analysis
Nonsmooth analysis
Differential inclusions
Existence theorems
Viability and invariance
Controllability
Optimality
Stability
Stabilization
Comments
Bibliography
Index
Details:
Publisher: American Mathematical Society
Distributor: American Mathematical Society
Series: Graduate Studies in Mathematics,Volume: 41
Publication Year: 2002
ISBN: 0-8218-2977-7
Paging: 226 pp.
Binding: Hardcover