The solitaire game "The Tower of Hanoi" was invented in the 19th century by the French number theorist Édouard Lucas. The book presents its mathematical theory and offers a survey of the historical development from predecessors up to recent research. In addition to long-standing myths, it provides a detailed overview of the essential mathematical facts with complete proofs, and also includes unpublished material, e.g., on some captivating integer sequences. The main objects of research today are the so-called Hanoi graphs and the related Sierpinski graphs. Acknowledging the great popularity of the topic in computer science, algorithms, together with their correctness proofs, form an essential part of the book. In view of the most important practical applications, namely in physics, network theory and cognitive (neuro)psychology, the book also addresses other structures related to the Tower of Hanoi and its variants.The updated second edition includes, for the first time in English, the breakthrough reached with the solution of the "The Reve's Puzzle" in 2014. This is a special case of the famed Frame-Stewart conjecture which is still open after more than 75 years. Enriched with elaborate illustrations, connections to other puzzles and challenges for the reader in the form of (solved) exercises as well as problems for further exploration, this book is enjoyable reading for students, educators, game enthusiasts and researchers alike.Excerpts from reviews of the first edition: "The book is an unusual, but very welcome, form of mathematical writing: recreational mathematics taken seriously and serious mathematics treated historically. I don't hesitate to recommend this book to students, professional research mathematicians, teachers, and to readers of popular mathematics who enjoy more technical expository detail."Chris Sangwin, The Mathematical Intelligencer 37(4) (2015) 87f. "The book demonstrates that the Tower of Hanoi has a very rich mathematical structure, and as soon as we tweak the parameters we surprisingly quickly find ourselves in the realm of open problems."László Kozma, ACM SIGACT News 45(3) (2014) 34ff. "Each time I open the book I discover a renewed interest in the Tower of Hanoi. I am sure that this will be the case for all readers."Jean-Paul Allouche, Newsletter of the European Mathematical Society 93 (2014) 56.
This contributed volume contains a collection of articles on state-of-the-art developments on the construction of theoretical integral techniques and their application to specific problems in science and engineering. Written by internationally recognized researchers, the chapters in this book are based on talks given at the Thirteenth International Conference on Integral Methods in Science and Engineering, held July 21-25, 2014, in Karlsruhe, Germany. A broad range of topics is addressed, from problems of existence and uniqueness for singular integral equations on domain boundaries to numerical integration via finite and boundary elements, conservation laws, hybrid methods, and other quadrature-related approaches. This collection will be of interest to researchers in applied mathematics, physics, and mechanical and electrical engineering, as well as graduate students in these disciplines and other professionals for whom integration is an essential tool.
Possible world models were introduced by Saul Kripke in the early 1960s. Basically, a possible worlds model is nothing but a graph with labelled nodes and labelled edges. Such graphs provide semantics for various modal logics: logics of necessity and possibility (alethic logics), logics of time (temporal logics), logics of knowledge and belief (epistemic and doxastic logics), logics of programs and of action (dynamic logics), logics of obligation (deontic logics), as well as for logics for describing ontologies (description logics). They have also turned out useful for other nonclassical logics such as intuitionistic logics, conditional logics, and several paraconsistent and relevant logics. All these logics have been studied intensively in philosophical and mathematical logic and in computer science, and have been applied increasingly in various domains such as program semantics, artificial intelligence, and more recently in the semantic web.All these logics were not only studied semantically but also proof theoretically. The proof systems for modal logics come in various styles: Hilbert style, natural deduction, sequents, and resolution. However, it is fair to say that the most uniform and most successful such systems are tableaux systems. Given a logic and a formula, they allow one to check whether there is a model in that logic. This basically amounts to trying to build a model for the formula by building a tree. In this book we follow a more general approach and try to build a graph, the advantage being that a graph is closer to a Kripke model than a tree. This book provides a step-by-step introduction to possible worlds semantics (and by that to modal and other nonclassical logics) via the tableaux method. It is accompanied by a piece of software called LoTREC (www.irit.fr/Lotrec). LoTREC allow one to check whether a given formula is true at a given world of a given model and to check whether a given formula is satisfiable in a given logic. The latter can be done immediately if the tableau system for that logic has already been implemented in LoTREC. For logics for which this has not been done yet LoTREC offers the possibility to implement a tableau system in a relatively easy way: users need not be computer scientists in order to implement new modal logics, thanks to a simple, graph-based, interactive language.
In the context of network theory, Complex networks can be de?ned as a collection of nodes connected by edges representing various complex int- actions among the nodes. Almost any large-scale system, be it natural or man-made, can be viewed as a complex network of interacting entities, which is dynamically evolving over time. Naturally occurring networks include - ological, ecological and social networks (e. g. , metabolic networks, gene r- ulatory networks, protein interaction networks, signaling networks, epidemic networks, food webs, scienti?c collaboration networks and acquaintance n- works), whereas man-made networks include communication networks and transportation infrastructures (e. g. , the Internet, the World Wide Web, pe- to-peer networks, power grids and airline networks). This edited volume is a sequel to the workshop Dynamics on and of C- plex Networks (http://www. cel. iitkgp. ernet. in/?eccs07/) held as a satellite event of the fourth European Conference on Complex Systems in Dresden, Germany from October 1-5, 2007. The primary aim of this workshop was to systematically explore the statistical dynamics "on" and "of" complex n- works that prevail across a large number of scienti?c disciplines. Dynamics on networks refers to the di?erent types of processes, for instance, prolife- tion and di?usion, that take place on networks. The functionality/e?ciency of these processes is strongly tied to the underlying topology as well as the dynamic behavior of the network.
This is a collection of survey articles based on lectures presented at a colloquium and workshop in Geneva in 2003 to commemorate the 200th anniversary of the birth of Charles François Sturm. It aims at giving an overview of the development of Sturm-Liouville theory from its historical roots to present day research. It is the first time that such a comprehensive survey has been made available in compact form. The contributions come from internationally renowned experts and cover a wide range of developments of the theory. The book can therefore serve both as an introduction to Sturm-Liouville theory and as background for ongoing research. The volume is addressed to researchers in related areas, to advanced students and to those interested in the historical development of mathematics. The book will also be of interest to those involved in applications of the theory to diverse areas such as engineering, fluid dynamics and computational spectral analysis.