Stable Approximate Evaluation of Unbounded Operators

Stable Approximate Evaluation of Unbounded Operators
Author: C. W. Groetsch
Publisher: Springer Science & Business Media
Total Pages: 134
Release: 2007
Genre: Mathematics
ISBN: 3540399429

Spectral theory of bounded linear operators teams up with von Neumann’s theory of unbounded operators in this monograph to provide a general framework for the study of stable methods for the evaluation of unbounded operators. An introductory chapter provides numerous illustrations of unbounded linear operators that arise in various inverse problems of mathematical physics. Before the general theory of stabilization methods is developed, an extensive exposition of the necessary background material from the theory of operators on Hilbert space is provided. Several specific stabilization methods are studied in detail, with particular attention to the Tikhonov-Morozov method and its iterated version.

Eigenvalues, Embeddings and Generalised Trigonometric Functions

Eigenvalues, Embeddings and Generalised Trigonometric Functions
Author: Jan Lang
Publisher: Springer
Total Pages: 232
Release: 2011-03-17
Genre: Mathematics
ISBN: 3642184294

The main theme of the book is the study, from the standpoint of s-numbers, of integral operators of Hardy type and related Sobolev embeddings. In the theory of s-numbers the idea is to attach to every bounded linear map between Banach spaces a monotone decreasing sequence of non-negative numbers with a view to the classification of operators according to the way in which these numbers approach a limit: approximation numbers provide an especially important example of such numbers. The asymptotic behavior of the s-numbers of Hardy operators acting between Lebesgue spaces is determined here in a wide variety of cases. The proof methods involve the geometry of Banach spaces and generalized trigonometric functions; there are connections with the theory of the p-Laplacian.

Random Perturbation of PDEs and Fluid Dynamic Models

Random Perturbation of PDEs and Fluid Dynamic Models
Author: Franco Flandoli
Publisher: Springer
Total Pages: 187
Release: 2011-03-02
Genre: Mathematics
ISBN: 3642182313

The book deals with the random perturbation of PDEs which lack well-posedness, mainly because of their non-uniqueness, in some cases because of blow-up. The aim is to show that noise may restore uniqueness or prevent blow-up. This is not a general or easy-to-apply rule, and the theory presented in the book is in fact a series of examples with a few unifying ideas. The role of additive and bilinear multiplicative noise is described and a variety of examples are included, from abstract parabolic evolution equations with non-Lipschitz nonlinearities to particular fluid dynamic models, like the dyadic model, linear transport equations and motion of point vortices.

Computational Approach to Riemann Surfaces

Computational Approach to Riemann Surfaces
Author: Alexander I. Bobenko
Publisher: Springer Science & Business Media
Total Pages: 268
Release: 2011-02-12
Genre: Mathematics
ISBN: 3642174124

This volume offers a well-structured overview of existent computational approaches to Riemann surfaces and those currently in development. The authors of the contributions represent the groups providing publically available numerical codes in this field. Thus this volume illustrates which software tools are available and how they can be used in practice. In addition examples for solutions to partial differential equations and in surface theory are presented. The intended audience of this book is twofold. It can be used as a textbook for a graduate course in numerics of Riemann surfaces, in which case the standard undergraduate background, i.e., calculus and linear algebra, is required. In particular, no knowledge of the theory of Riemann surfaces is expected; the necessary background in this theory is contained in the Introduction chapter. At the same time, this book is also intended for specialists in geometry and mathematical physics applying the theory of Riemann surfaces in their research. It is the first book on numerics of Riemann surfaces that reflects the progress made in this field during the last decade, and it contains original results. There are a growing number of applications that involve the evaluation of concrete characteristics of models analytically described in terms of Riemann surfaces. Many problem settings and computations in this volume are motivated by such concrete applications in geometry and mathematical physics.

Fatou, Julia, Montel

Fatou, Julia, Montel
Author: Michèle Audin
Publisher: Springer
Total Pages: 342
Release: 2011-01-29
Genre: Mathematics
ISBN: 3642178545

How did Pierre Fatou and Gaston Julia create what we now call Complex Dynamics, in the context of the early twentieth century and especially of the First World War? The book is based partly on new, unpublished sources. Who were Pierre Fatou, Gaston Julia, Paul Montel? New biographical information is given on the little known mathematician that was Pierre Fatou. How did the WW1 injury of Julia influence mathematical life in France? From the reviews of the French version: "Audin’s book is ... filled with marvelous biographical information and analysis, dealing not just with the men mentioned in the book’s title but a large number of other players, too ... [It] addresses itself to scholars for whom the history of mathematics has a particular resonance and especially to mathematicians active, or even with merely an interest, in complex dynamics. ... presents it all to the reader in a very appealing form." (Michael Berg, The Mathematical Association of America, October 2009)

Some Mathematical Models from Population Genetics

Some Mathematical Models from Population Genetics
Author: Alison Etheridge
Publisher: Springer
Total Pages: 129
Release: 2011-01-05
Genre: Mathematics
ISBN: 3642166326

This work reflects sixteen hours of lectures delivered by the author at the 2009 St Flour summer school in probability. It provides a rapid introduction to a range of mathematical models that have their origins in theoretical population genetics. The models fall into two classes: forwards in time models for the evolution of frequencies of different genetic types in a population; and backwards in time (coalescent) models that trace out the genealogical relationships between individuals in a sample from the population. Some, like the classical Wright-Fisher model, date right back to the origins of the subject. Others, like the multiple merger coalescents or the spatial Lambda-Fleming-Viot process are much more recent. All share a rich mathematical structure. Biological terms are explained, the models are carefully motivated and tools for their study are presented systematically.

Affine Density in Wavelet Analysis

Affine Density in Wavelet Analysis
Author: Gitta Kutyniok
Publisher: Springer Science & Business Media
Total Pages: 149
Release: 2007-06-07
Genre: Mathematics
ISBN: 3540729496

This volume provides a thorough and comprehensive treatment of irregular wavelet frames. It introduces and employs a new notion of affine density as a highly effective tool for examining the geometry of sequences of time-scale indices. Coverage includes non-existence of irregular co-affine frames, the Nyquist phenomenon for wavelet systems, and approximation properties of irregular wavelet frames.

Spin Glasses

Spin Glasses
Author: Erwin Bolthausen
Publisher: Springer
Total Pages: 190
Release: 2007-01-11
Genre: Science
ISBN: 3540409084

This book serves as a concise introduction to the state-of-the-art of spin glass theory. The collection of review papers are written by leading experts in the field and cover the topic from a wide variety of angles. The book will be useful to both graduate students and young researchers, as well as to anyone curious to know what is going on in this exciting area of mathematical physics.

Iterative Regularization Methods for Nonlinear Ill-Posed Problems

Iterative Regularization Methods for Nonlinear Ill-Posed Problems
Author: Barbara Kaltenbacher
Publisher: Walter de Gruyter
Total Pages: 205
Release: 2008-09-25
Genre: Mathematics
ISBN: 311020827X

Nonlinear inverse problems appear in many applications, and typically they lead to mathematical models that are ill-posed, i.e., they are unstable under data perturbations. Those problems require a regularization, i.e., a special numerical treatment. This book presents regularization schemes which are based on iteration methods, e.g., nonlinear Landweber iteration, level set methods, multilevel methods and Newton type methods.