Series: Undergraduate Texts in Mathematics
2009, XII, 248 p. 50 illus., Hardcover
ISBN: 978-0-387-87836-2
Due: August 2009
Applications covered: optimization, finance, statistical mechanics, birth and death processes, and gambling systems
Hands-on approach is used via realistic problems demonstrated with examples and numerical simulations
A wealth of completely solved example problems provide the reader with a sourcebook to follow toward the solution of their own computational problems
Each chapter ends with a large collection of homework problems illustrating and directing the material
Monte Carlo methods are among the most used and useful computational tools available today, providing efficient and practical algorithims to solve a wide range of scientific and engineering problems. Applications covered in this book include optimization, finance, statistical mechanics, birth and death processes, and gambling systems.
Explorations in Monte Carlo Methods provides a hands-on approach to learning this subject. Each new idea is carefully motivated by a realistic problem, thus leading from questions to theory via examples and numerical simulations. Programming exercises are integrated throughout the text as the primary vehicle for learning the material. Each chapter ends with a large collection of problems illustrating and directing the material.
This book is suitable as a textbook for students of engineering and the sciences, as well as mathematics. The problem-oriented approach makes it ideal for an applied course in basic probability and for a more specialized course in Monte Carlo methods. Topics include probability distributions, counting combinatorial objects, simulated annealing, genetic algorithms, option pricing, gamblers ruin, statistical mechanics, sampling, and random number generation.
Introduction to Monte Carlo Methods.- Some Probability Distributions and their Uses.- Markov Chain Monte Carlo.- Optimization by Monte Carlo Methods.- Random Walks.- Generating Uniform Random Numbers.- Perron Frobenius Theorem.-
Series: Progress in Probability , Vol. 61
2010, Approx. 250 p., Hardcover
ISBN: 978-3-0346-0029-3
Due: September 2009
Over the last fifteen years fractal geometry has established itself as a substantial mathematical theory in its own right. The interplay between fractal geometry, analysis and stochastics has highly influenced recent developments in mathematical modeling of complicated structures. This process has been forced by problems in these areas related to applications in statistical physics, biomathematics and finance.
This book is a collection of survey articles covering many of the most recent developments, like Schramm-Loewner evolution, fractal scaling limits, exceptional sets for percolation, and heat kernels on fractals. The authors were the keynote speakers at the conference "Fractal Geometry and Stochastics IV" at Greifswald in September 2008.
Key topics: Heat semigroups in metric measure spaces.- Schramm-Loewner evolution and multifractal analysis.- Random trees and graphs, dynamical percolation.- Random walks on self-similar groups.- Fractal processes and superprocesses.- Iterated function schemes and transformations of fractals.
Series: CMS Books in Mathematics
2010, Approx. 310 p. 60 illus., Hardcover
ISBN: 978-1-4419-0599-4
Due: March 2010
A valuable source of geometric problems
User-friendly exposition providing insight into the latest research and up-to-date bibliography
The goal of this book is to provide focused material for a semester long graduate level course, which can be regarded as a brief introduction to Discrete Geometry. There are a large number of exercises and the book can also act as a short problem book aimed at advanced undergraduate and graduate students as well as researchers. This text is centered around three major problems of Discrete Geometry. The first is the problem of Densest Sphere Packings, which has more than 100 years of mathematically rich history. The second major problem is typically quoted under the roughly 50 years old Illumination Conjecture of V. Boltyanski and H. Hadwiger, and the third topic is centered around another nearly 50 years old conjecture called the Kneser-Poulsen Conjecture. All three topics witnessed very recent breakthrough results, explaining their major role in this book.
Preface.- Introduction.- Sphere Packings.- Finite Packings by Translates of Convex Bodies.- The Illumination Conjecture and Its Relatives.- The Kneser-Poulsen Conjecture.- Ball-Polyhedra.- Selected Proofs.- Index.-
Series: Chapman & Hall/CRC Numerical Analy & Scient Comp. Series
ISBN: 9781439810965
Publication Date: 07/05/2009
Pages: 304
Decomposition Methods for Differential Equations: Theory and Applications describes the analysis of numerical methods for evolution equations based on temporal and spatial decomposition methods. It covers real-life problems, the underlying decomposition and discretization, the stability and consistency analysis of the decomposition methods, and numerical results.
The book focuses on the modeling of selected multi-physics problems, before introducing decomposition analysis. It presents time and space discretization, temporal decomposition, and the combination of time and spatial decomposition methods for parabolic and hyperbolic equations. The author then applies these methods to numerical problems, including test examples and real-world problems in physical and engineering applications. For the computational results, he uses various software tools, such as MATLABŪ, R3T, WIAS-HiTNIHS, and OPERA-SPLITT.
Exploring iterative operator-splitting methods, this book shows how to use higher-order discretization methods to solve differential equations. It discusses decomposition methods and their effectiveness, combination possibility with discretization methods, multi-scaling possibilities, and stability to initial and boundary values problems.
Preface. Introduction. Modeling: Multi-Physics Problems. Abstract Decomposition and Discretization Methods. Time-Decomposition Methods for Parabolic Equations. Decomposition Methods for Hyperbolic Equations. Spatial Decomposition Methods. Numerical Experiments. Summary and Perspectives. Notation. Appendices. Literature. References. Index
Series: Textbooks in Mathematics
ISBN: 9781420069556
Publication Date: 08/07/2009
Pages: 433
Written for sophomore/junior level students, this textbook begins with formal proof reasoning and writing, providing an understanding of the underlying language of mathematics. The author then describes deductive mathematical systems where theorems are proved using different mathematical techniques, allowing students to compare and contrast the methods. He covers the role of conjectures along with their proofs and disproof, followed by rational numbers and their properties. He then covers the method used for developing the system of real numbers from rational numbers. The book includes more than 75 examples and more than 600 problems. A solutions manual is available upon qualifying course adoptions.
Logic. Deductive Mathematical Systems and Proofs. Set Theory. Relations. Functions. Mathematical Induction. Cardinalities of Sets. Proofs from Group Theory. Proofs from Affine and Projective Planes.