Series: Applied Mathematical Sciences, Vol. 181
2012, 2012, XI, 394 p. 77 illus.
Hardcover, ISBN 978-1-4471-2917-2
Due: May 31, 2012
Intertwines classic topics with new results to provide extensive coverage of the area
An ample amount of worked out examples and user-friendly algorithms are supplied for both theory and application
Includes complete Maple programs, allowing the reader to reconstruct the majority of formulas and study using concrete models
Dynamical system theory has developed rapidly over the past fifty years. It is a subject upon which the theory of limit cycles has a significant impact for both theoretical advances and practical solutions to problems. Hopf bifurcation from a center or a focus is integral to the theory of bifurcation of limit cycles, for which normal form theory is a central tool. Although Hopf bifurcation has been studied for more than half a century, and normal form theory for over 100 years, efficient computation in this area is still a challenge with implications for Hilbertfs 16th problem.
This book introduces the most recent developments in this field and provides major advances in fundamental theory of limit cycles. Split into two parts, the first focuses on the study of limit cycles bifurcating from Hopf singularity using normal form theory with later application to Hilbertfs 16th problem, while the second considers near Hamiltonian systems using Melnikov function as the main mathematical tool.
Classic topics with new results are presented in a clear and concise manner and are accompanied by the liberal use of illustrations throughout. Containing a wealth of examples and structured algorithms that are treated in detail, a good balance between theoretical and applied topics is demonstrated. By including complete Maple programs within the text, this book also enables the reader to reconstruct the majority of formulas provided, facilitating the use of concrete models for study.
Through the adoption of an elementary and practical approach, this book will be of use to graduate mathematics students wishing to study the theory of limit cycles as well as scientists, across a number of disciplines, with an interest in the applications of periodic behavior.
Hopf Bifurcation and Normal Form Computation.- Comparison of Methods for Computing Focus Values.- Application (I)?Hilbertfs 16th Problem.- Application (II)?Practical Problems.- Fundamental Theory of the Melnikov Function Method.- Limit Cycle Bifurcations Near a Center.- Limit Cycles Near a Homoclinic or Heteroclinic Loop.- Finding More Limit Cycles Using Melnikov Functions.- Limit Cycle Bifurcations in Equivariant Systems.
2012, 2012, XVI, 156 p. 43 illus.
ISBN 978-0-8176-8303-0
The second edition includes two additional chapters, focusing on the more advanced topics of the book
Presents differential forms from a geometric perspective accessible at the sophomore undergraduate level
Each new concept is presented with a natural picture that students can easily grasp; algebraic properties then follow
Designed to support three distinct, classroom tested, course tracks
Contains excellent motivation, numerous illustrations and solutions to selected problems
The modern subject of differential forms subsumes classical vector calculus. This text presents differential forms from a geometric perspective accessible at the advanced undergraduate level. The author approaches the subject with the idea that complex concepts can be built up by analogy from simpler cases, which, being inherently geometric, often can be best understood visually.
Each new concept is presented with a natural picture that students can easily grasp; algebraic properties then follow. This facilitates the development of differential forms without assuming a background in linear algebra. Throughout the text, emphasis is placed on applications in 3 dimensions, but all definitions are given so as to be easily generalized to higher dimensions.
The second edition includes a completely new chapter on differential geometry, as well as other new sections, new exercises and new examples. Additional solutions to selected exercises have also been included. The work is suitable for use as the primary textbook for a sophomore-level class in vector calculus, as well as for more upper-level courses in differential topology and differential geometry.
Series: International Series in Operations Research & Management Science, Vol. 171
2012, 2012, XIV, 199 p. 82 illus., 31 in color.
Hardcover, ISBN 978-1-4614-3231-9
Due: March 31, 2012
Concise, readable synthesis of important yet scattered research to date
Discusses problems yet to be fully solved
Provides many examples not supplied in original studies
This research monograph summarizes a line of research that maps certain classical problems of discrete mathematics and operations research - such as the Hamiltonian cycle and the Travelling Salesman problems ? into convex domains where continuum analysis can be carried out. Arguably, the inherent difficulty of these, now classical, problems stems precisely from the discrete nature of domains in which these problems are posed. The convexification of domains underpinning the reported results is achieved by assigning probabilistic interpretation to key elements of the original deterministic problems. In particular, approaches summarized here build on a technique that embeds Hamiltonian Cycle and Traveling Salesman problems in a structured singularly perturbed Markov decision process. The unifying idea is to interpret subgraphs traced out by deterministic policies (including Hamiltonian cycles, if any) as extreme points of a convex polyhedron in a space filled with randomized policies.
The above, innovative, approach has now evolved to the point where there are many, both theoretical and algorithmic, results that exploit the nexus between graph theoretic structures and both probabilistic and algebraic entities of related Markov chains. The latter include moments of first return times, limiting frequencies of visits to nodes, or the spectra of certain matrices traditionally associated with the analysis of Markov chains. However, these results and algorithms are dispersed over more than fifteen research papers appearing in journals catering to disparate audiences such as: MOR, Random Structures and Algorithms, SIAM J. on Discrete Mathematics, Optimization, J. of Mathematical Analysis and Applications and some others. Furthermore, because of the evolution of this topic and specific orientation of these journals, the published manuscripts are often written in a very terse manner and use disparate notation. As such it is difficult for new researchers to make use of the many advances reported in these papers.
Hence the main purpose of this book is to present a concise and yet, well written, synthesis of the majority of the theoretical and algorithmic results obtained so far. In addition the book will discuss numerous open questions and problems that arise from this body of work and which are yet to be fully solved. The authors believe that their approach casts the Hamiltonian Cycle and Traveling Salesman problems in a mathematical framework that permits analytical concepts and techniques, not used hitherto in their context, to be brought to bear to further clarify both the underlying difficulty of NP-completeness of these problems and the relative exceptionality of truly difficult instances.
Finally, the material is arranged in such a manner that the introductory chapters require very little mathematical background and discuss instances of graphs with interesting structures that motivated a lot of the research in this topic. More difficult results are introduced later but, unlike the research manuscripts where they were originally proved, are illustrated with numerous examples.
Illustrative Graphs.- Intriguing Properties.- Markov Chains.- Markov Decision Processes.- Determinants.- Traces.- Linear Programming Based Algorithms.- Interior Point and Cross-Entropy Algorithms.- Self-similar Structure and Hamiltonicity.- Graph Enumeration.
Series: Undergraduate Topics in Computer Science
2nd ed. 2012, 2012, XXI, 283 p. 17 illus.
Softcover, ISBN 978-1-4471-2499-3
Only minimal background in mathematics necessary
Careful selection of material that is really needed by students in the first two years of their university life in Computer Science and Information Sciences
Brings out the interplay between qualitative thinking and calculation
Teaches the material as a language for thinking in, as much as knowledge to be gained
This easy-to-follow textbook introduces the mathematical language, knowledge and problem-solving skills that undergraduate students need to enter the world of computer and information sciences. The language is in part qualitative, with concepts such as set, relation, function and recursion/induction; but it is also partly quantitative, with principles of counting and finite probability. Entwined with both are the fundamental notions of logic and their use for representation and proof. In ten chapters on these topics, the book guides the student through essential concepts and techniques.
The extensively revised second edition provides further clarification of matters that typically give rise to difficulty in the classroom and restructures the chapters on logic to emphasize the role of consequence relations and higher-level rules, as well as including more exercises and solutions.
*Teaches finite mathematics as a language for thinking, as much as knowledge
and skills to be acquired
*Uses an intuitive approach with a focus on examples for all general concepts
*Brings out the interplay between the qualitative and the quantitative
in all areas covered, particularly in the treatment of recursion and induction
*Balances carefully the abstract and concrete, principles and proofs, specific
facts and general perspectives
*Includes highlight boxes that raise common queries and clear away confusions
*Provides numerous exercises, with selected solutions, to test and deepen
the readerfs understanding
This clearly-written text/reference is a must-read for first-year undergraduate students of computing. Assuming only minimal mathematical background, it is ideal for both the classroom and independent study.
Dr. David Makinson is a Visiting Professor in the Department of Philosophy, Logic and Scientific Method at the London School of Economics, UK.
Collecting Things Together: Sets
Comparing Things: Relations
Associating One Item with Another: Functions
Recycling Outputs as Inputs: Induction and Recursion
Counting Things: Combinatorics
Weighing the Odds: Probability
Squirrel Math: Trees
Yea and Nay: Propositional Logic
Something about Everything: Quantificational Logic
Just Supposing: Proof and Consequence
Series: Advances in Applied Neurological Sciences
2012, 2012, Approx. 600 p. 10 illus., 5 in color.
Softcover, ISBN 978-3-642-27554-8
Due: April 14, 2012
New comprehensive analysis of general nonlinear DAEs
First book describing the projector approach for nonlinear, higher-index DAEs
Rigorous description, multitude of examples highlighting analytical and numerical challenges in this field Suitable for students, researchers and users from different application fields (e.g. circuit and electromagnetic simulation, mechanical engineering, system biology)
Differential algebraic equations (DAEs), including so-called descriptor systems, began to attract significant research interest in applied and numerical mathematics in the early 1980's, no more than about three decades ago. In this relatively short time, DAEs have become a widely acknowledged tool to model processes subjected to constraints, in order to simulate and to control processes in various application fields such as network simulation, chemical kinematics, mechanical engineering, system biology.
DAEs and their more abstract versions in infinite-dimensional spaces comprise a great potential for future mathematical modeling of complex coupled processes.
The purpose of the book is to expose the impressive complexity of general DAEs from an analytical point of view, to describe the state of the art as well as open problems and so to motivate further research to this versatile, extra-ordinary topic from a broader mathematical perspective.
The book elaborates a new general structural analysis capturing linear and nonlinear DAEs in a hierarchical way. The DAE structure is exposed by means of special projector functions. Numerical integration issues and computational aspects are treated also in this context.
Content Level ā Upper undergraduate
Keywords ā Abstract Differential-Algebraic Equations - Applied Analysis - Descriptor Systems - Differential-Algebraic Equations - Ordinary Differential Equiations
Related subjects ā Applications - Computational Science & Engineering - Dynamical Systems & Differential Equations