Telcs, Andras

The Art of Random Walks

Series: Lecture Notes in Mathematics, Vol. 1885
2006, Approx. 200 p., Softcover
ISBN: 3-540-33027-5
Due: May 18, 2006

About this book

Einstein proved that the mean square displacement of Brownian motion is proportional to time. He also proved that the diffusion constant depends on the mass and on the conductivity (sometimes referred to Einstein’s relation). The main aim of this book is to reveal similar connections between the physical and geometric properties of space and diffusion. This is done in the context of random walks in the absence of algebraic structure, local or global spatial symmetry or self-similarity. The authors study the heat diffusion at this general level and discuss the following topics:

The multiplicative Einstein relation,

Isoperimetric inequalities,

Heat kernel estimates

Elliptic and parabolic Harnack inequality.

Written for:

Researchers and graduate students in probability theory and random walks or Markov chains

Table of contents

1 Introduction.- Basic Definitions and Preliminaries. Part I Potential Theory and Isoperimetric Inequalities: Some Elements of Potential Theory.- Isoperimetric Inequalities.- Polynomial Volume Growth. Part II Local Theory: Motivation of the Local Approach.- Einstein Relation.- Upper Estimates.- Lower Estimates.- Two-sided Estimates.- Closing Remarks.- Parabolic Harnack Inequality.- Semi-local Theory.- Subject Index.- References.

Steinbach, Olaf

Numerical Approximation Methods for Elliptic Boundary Value Problems
Finite and Boundary Elements

Series: Texts in Applied Mathematics
2006, Approx. 400 p., Hardcover
ISBN: 0-387-31312-5
Due: July 2006

About this textbook

This book presents a unified theory of the Finite Element Method and the Boundary Element Method for a numerical solution of second order elliptic boundary value problems. This includes the solvability, stability, and error analysis as well as efficient methods to solve the resulting linear systems. Applications are the potential equation, the system of linear elastostatics and the Stokes system. While there are textbooks on the finite element method, this is one of the first books on Theory of Boundary Element Methods.

Written for:

Graduate students and researchers

Table of contents

Boundary Value Problems - Function Spaces - Variational Methods - Variational Formulations for Boundary Value Problems - Fundamental Solutions of Partial Differential Equations - Boundary Integral Operators - Boundary Integral Equations - Numerical Methods for Variational Problems - Finite Elements - Boundary Elements - Boundary Element Methods - Preconditioned Iterative Solvers - Fast Boundary Element Methods - Domain Decomposition Methods

Syropoulos, Apostolos

Hypercomputation
Computing Beyond the Church-Turing Barrier

Series: Monographs in Computer Science
2006, XVI, 216 p. 14 illus., Hardcover
ISBN: 0-387-30886-5
Due: July 2006

About this book

Hypercomputation is a relatively new theory of computation that is about computing methods and devices that transcend the so-called Church-Turing thesis. This book will provide a thorough description of the field of hypercomputation covering all attempts at devising conceptual hypermachines and all new promising computational paradigms that may eventually lead to the construction of a hypermachine.

Readers of this book will get a deeper understanding of what computability is and why the Church-Turing thesis poses an arbitrary limit to what can be actually computed. Hypercomputing is in and of itself quite a novel idea and as such the book will be interesting in its own right. The most important features of the book, however, will be the thorough description of the various attempts of hypercomputation: from trial-and-error machines to the exploration of the human mind, if we treat it as a computing device.

Written for:

Researchers, engineers, and professionals in Computer Science, Mathematics and Physics

Table of contents

Preface.- Introduction.- On the Church-Turing Thesis.- Early Hypercomputers.- Infinite Time Turing Machines.- Interactive Computing.- Hyperminds.- Computing Real Numbers.- Relativistic and Quantum Hypercomputation.- Natural Computation and Hypercomputation.- Appendix A. Interactability and Hypercomputation.- Appendix B.- Socio-Economical Implications.- Appendix C. A Precis of Topology and Differential Geometry.- Bibliography.- People Index.- Subject Index.

Freudenburg, Gene

Locally Nilpotent Derivations and G_a-Actions

Series: Encyclopaedia of Mathematical Sciences, Vol. 136
Volume package: Invariant Theory
2006, Approx. 280 p., Hardcover
ISBN: 3-540-29521-6
Due: July 2006

About this book

This book explores the theory and application of locally nilpotent derivations, which is a subject of growing interest and importance not only among those in commutative algebra and algebraic geometry, but also in fields such as Lie algebras and differential equations.

The author provides a unified treatment of the subject, beginning with 16 First Principles on which the entire theory is based. These are used to establish classical results, such as Rentschler’s Theorem for the plane, right up to the most recent results, such as Makar-Limanov’s Theorem for locally nilpotent derivations of polynomial rings. Topics of special interest include: progress in the dimension three case, finiteness questions (Hilbert’s 14th Problem), algorithms, the Makar-Limanov invariant, and connections to the Cancellation Problem and the Embedding Problem. The reader will also find a wealth of pertinent examples and open problems and an up-to-date resource for research.

Written for:

Graduate students and researchers

Table of contents

0 Introduction.- 1 First Principles.- 2 Further Properties of Locally Nilpotent Derivations.- 3 Polynomial Rings.- 4 Dimension Two.- 5 Dimension Three.- 6 Linear Actions of Vector Groups.- 7 Non-Finitely Generated Kernels.- 8 Algorithms.- 9 The Makar-Limanov and Derksen Invariants.- 10 Slices, Embeddings and Cancellation.- 11 Epilogue.- References.- Index.

Aubert, Gilles, Kornprobst, Pierre

Mathematical Problems in Image Processing
Partial Differential Equations and the Calculus of Variations

Series: Applied Mathematical Sciences, Vol. 147
2006, Approx. 410 p. 100 illus., Hardcover
ISBN: 0-387-32200-0
Due: June 2006

About this book

Partial differential equations (PDEs) and variational methods were introduced into image processing about fifteen years ago. Since then, intensive research has been carried out. The goals of this book are to present a variety of image analysis applications, the precise mathematics involved and how to discretize them.

Thus, this book is intended for two audiences. The first is the mathematical community by showing the contribution of mathematics to this domain. It is also the occasion to highlight some unsolved theoretical questions. The second is the computer vision community by presenting a clear, self-contained and global overview of the mathematics involved in image procesing problems. This work will serve as a useful source of reference and inspiration for fellow researchers in Applied Mathematics and Computer Vision, as well as being a basis for advanced courses within these fields.

During the four years since the publication of the first edition, there has been substantial progress in the range of image processing applications covered by the PDE framework. The main goals of the second edition are to update the first edition by giving a coherent account of some of the recent challenging applications, and to update the existing material. In addition, this book provides the reader with the opportunity to make his own simulations with a minimal effort. To this end, programming tools are made available, which will allow the reader to implement and test easily some classical approaches.

Written for:

Researchers and graduate students

Table of contents

Foreword.- Preface to the Second Edition.- Preface.- Guide to the Main Mathematical Concepts and Their Application.- Notation and Symbols.- Introduction.- Mathematical Preliminaries.- Image Restoration.- The Segmentation Problem.- Other Challenging Applications.- A Introduction to Finite Difference Methods.- B Experiment Yourself!.- References.- Index

Athreya, Krishna B., Lahiri, Soumen N.

Measure Theory and Probability Theory with Applications

Series: Springer Texts in Statistics
2006, XIV, 610 p., Hardcover
ISBN: 0-387-32903-X
Due: August 2006

About this textbook

This is a graduate level textbook on measure theory and probability theory. It also includes a modest coverage of important topics of current research interest such as Markov chain Monte Carlo (MCMC) methods and bootstrap methods. The book can be used as a text for a two semester sequence of courses in measure theory and probability theory, with an option to include supplemental material on stochastic processes and special topics. It is intended primarily for first year Ph.D. students in mathematics and statistics although mathematically advanced students from engineering and economics would also find the book useful. Prerequisites are kept to the minimal level of an understanding of basic real analysis concepts such as limits, continuity, differentiability, Riemann integration, and convergence of sequences and series. A review of this material is included in the appendix.

The book starts with an informal introduction that provides some heuristics into the abstract concepts of measure and integration theory, which are then rigorously developed. The first part of the book can be used for a standard real analysis course for both mathematics and statistics Ph.D. students as it provides full coverage of topics such as the construction of Lebesgue-Stieltjes measures on real line and Euclidean spaces, the basic convergence theorems, L^p spaces, signed measures, Radon-Nikodym theorem, Lebesgue's decomposition theorem and the fundamental theorem of Lebesgue integration on R, product spaces and product measures, and Fubini-Tonelli theorems. It also provides an elementary introduction to Banach and Hilbert spaces, convolutions, Fourier series and Fourier and Plancherel transforms. Thus part I would be particularly useful for students in a typical Statistics Ph.D. program if a separate course on real analysis is not a standard requirement.

Part II (chapters 6-13) provides full coverage of standard graduate level probability theory. It starts with Kolmogorov's probability model and Kolmogorov's existence theorem. It then treats thoroughly the laws of large numbers including renewal theory and ergodic theorems with applications and then weak convergence of probability distributions, characteristic functions, the Levy-Cramer continuity theorem and the central limit theorem as well as stable laws. It ends with conditional expectations and conditional probability, and an introduction to the theory of discrete time martingales.

Part III (chapters 14-18) covers Markov chains with countable and general state spaces, Brownian motion and jump Markov processes, resampling methods and branching processes.

Table of contents

Measures and integration: an informal introduction.- Measures.- Integration.- LP spaces.- Differentiation.- Product measures, convolutions, and transforms.- Probability spaces.- Independence.- Laws of large numbers.- Convergence in distribution.- Characteristic functions.- Central limit theorems.- Conditional expectation and conditional probability.- Discrete parameter martingales.- Markov chains and MCMC.- Stochastic processes.- Limit theorems for dependent processes.- The bootstrap.- Branching process