ISBN13: 9780195385861
Hardback, 576 pages
Sep 2009,
Linearity plays a critical role in the study of elementary differential equations; linear differential equations, especially systems thereof, demonstrate a fundamental application of linear algebra. In Differential Equations with Linear Algebra , we explore this interplay between linear algebra and differential equations and examine introductory and important ideas in each, usually through the lens of important problems that involve differential equations. Written at a sophomore level, the text is accessible to students who have completed multivariable calculus. With a systems-first approach, the book is appropriate for courses for majors in mathematics, science, and engineering that study systems of differential equations.
Because of its emphasis on linearity, the text opens with a full chapter devoted to essential ideas in linear algebra. Motivated by future problems in systems of differential equations, the chapter on linear algebra introduces such key ideas as systems of algebraic equations, linear combinations, the eigenvalue problem, and bases and dimension of vector spaces. This chapter enables students to quickly learn enough linear algebra to appreciate the structure of solutions to linear differential equations and systems thereof in subsequent study and to apply these ideas regularly.
The book offers an example-driven approach, beginning each chapter with one or two motivating problems that are applied in nature. The following chapter develops the mathematics necessary to solve these problems and explores related topics further. Even in more theoretical developments, we use an example-first style to build intuition and understanding before stating or proving general results. Over 100 figures provide visual demonstration of key ideas; the use of the computer algebra system Maple and Microsoft Excel are presented in detail throughout to provide further perspective and support students' use of technology in solving problems. Each chapter closes with several substantial projects for further study, many of which are based in applications.
Preface
1. Essentials of Linear Algebra 2. First-Order Differential Equations 3. Linear Systems of Differential Equations 4. Higher-Order Differential Equations 5. Laplace Tranforms 6. Nonlinear Systems of Differential Equations 7. Numerical Methods for Differential Equations 8. Series Solutions for Differential Equations A. Review of Integration Techniques B. Complex Numbers C. Roots of Polynomials D. Linear Transformations E. Solutions to Selected Exercises
256 pages | 15 black and white line drawings | 216x138mm
978-0-19-923778-4 | Hardback | September 2009 (estimated
No one can forget the horrific images of the destructive power of the tsunami that engulfed Southeast Asia on Boxing Day in 2004, or the chaos wrought by Hurricane Katrina. Could these 'megadisasters' have been predicted?
This book is about the science and mathematics that underlies efforts to understand and predict megadisasters. There are similarities in the variety of cataclysms that we are prone to, whether hurricanes, tsunamis, sudden changes of climate, or stock market crashes. These are all events that are associated with complex systems, with many variables, and their science and mathematics is that of 'chaotic systems'. Their behaviour is very difficult to predict. Other kinds of megadisasters are the risk of a massive asteroid impact, and the development of pandemics.
Understanding and predicting these phenomena involve developing complex mathematical models, and we have a long way to go. In this book, Diacu describes the struggles of mathematicians and scientists over the centuries to get to grips with the nature of volcanoes, hurricanes, and other complex phenomena and prevent future tragedies. But he also includes human stories that remind us of their terrifying power and the experience of being caught up in them.
Readership: General readers of popular science, particularly those interested in mathematics, mathematical modelling, chaos, statistics, and in natural disasters and geophysics.
1: Introduction: Glimpsing the future
2: Walls of Water: Tsunamis
3: Land in Upheaval: Earthquakes
4: Rivers of Fire: Volcanic eruptions
5: Giant Whirlwinds: Hurricanes, Cyclones and typhoons
6: Mutant Seasons: Sudden climate changes
7: Earth in Collision: Cosmic impacts
8: Economic Breakdown: Financial crashes
9: Tiny Killers: Pandemics
10: Models and Prediction: How far can we go?
2009, X, 287 p. 173 illus. With 20 tables., Hardcover
ISBN: 978-3-211-75539-6
Algorithmic composition ? composing by means of formalizable methods ? has a century old tradition not only in occidental music history. This is the first book to provide a detailed overview of prominent procedures of algorithmic composition in a pragmatic way rather than by treating formalizable aspects in single works. In addition to an historic overview, each chapter presents a specific class of algorithm in a compositional context by providing a general introduction to its development and theoretical basis and describes different musical applications. Each chapter outlines the strengths, weaknesses and possible aesthetical implications resulting from the application of the treated approaches. Topics covered are: markov models, generative grammars, transition networks, chaos and self-similarity, genetic algorithms, cellular automata, neural networks and artificial intelligence are covered. The comprehensive bibliography makes this work ideal for the musician and the researcher alike.
Series: Monographs in Theoretical Computer Science. An EATCS Series
2009, X, 615 p., Hardcover
ISBN: 978-3-642-01491-8
Due: July 2, 2009
Weighted finite automata are classical nondeterministic finite automata in which the transitions carry weights. These weights may model, for example, the cost involved when executing a transition, the resources or time needed for this, or the probability or reliability of its successful execution. Weights can also be added to classical automata with infinite state sets like pushdown automata, and this extension constitutes the general concept of weighted automata. Since their introduction in the 1960s they have stimulated research in related areas of theoretical computer science, including formal language theory, algebra, logic, and discrete structures. Moreover, weighted automata and weighted context-free grammars have found application in natural-language processing, speech recognition, and digital image compression.
This book covers all the main aspects of weighted automata and formal power series methods, ranging from theory to applications. The contributors are the leading experts in their respective areas, and each chapter presents a detailed survey of the state of the art and pointers to future research. The chapters in Part I cover the foundations of the theory of weighted automata, specifically addressing semirings, power series, and fixed point theory. Part II investigates different concepts of weighted recognizability. Part III examines alternative types of weighted automata and various discrete structures other than words. Finally, Part IV deals with applications of weighted automata, including digital image compression, fuzzy languages, model checking, and natural-language processing.
Computer scientists and mathematicians will find this book an excellent survey and reference volume, and it will also be a valuable resource for students exploring this exciting research area.
Part I, Foundations: Semirings and Formal Power Series.- Fixed Point Theory. Part II, Concepts of Weighted Recognizability: Finite Automata.- Rational and Recognisable Power Series.- Weighted Automata and Weighted Logics.- Weighted Automata Algorithms. Part III, Weighted Discrete Structures: Algebraic Systems and Pushdown Automata.- Lindenmayer Systems.- Weighted Tree Automata and Tree Transducers.- Traces, Series-Parallel Posets, and Pictures: A Weighted Study. Part IV, Applications: Digital Image Compression.- Fuzzy Languages.- Model Checking Linear-Time Properties of Probabilistic Systems.- Applications of Weighted Automata in Natural Language Processing. Index.
Series: Understanding Complex Systems
2009, Approx. 220 p., Hardcover
ISBN: 978-3-642-00936-5
Due: September 2009
The study of chaotic behaviour in nonlinear, dynamical systems is now a well established research domain with ramifications into all fields of sciences, spanning a vast range of applications, from celestial mechanics, via climate change, to the functioning of brownian motors in cells.
A more recent discovery is that chaos can be controlled and, under appropriate conditions, can actually be constructive in the sense of becoming a control parameter itself for the system under investigation, stochastic resonance being a prime example.
The present work is putting emphasis on the latter aspects, and after recalling the paradigm changes introduced by the concept of chaos, leads the reader skillfully through the basics of chaos control by detailing relevant algorithms for both Hamiltonian and dissipative systems amongst others.
The main part of the book is then devoted to the issue of synchronization in chaotic systems, an introduction to stochastic resonance and a survey of ratchet models.
This short and concise primer is particularly suitable for postgraduate students and non-specialist scientists from related areas, wishing to enter the field quickly and efficiently.
Introduction.- Paradigm for Chaos.- Main Features of Chaotic Systems.- Reconstruction of Dynamical Systems.- Controlling Chaos.- Synchronization of Chaotic Systems.- Stochastic Resonance.- The Appearance of Regular Fluxes Without Gradients.- References.- Index.