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This is the first comprehensive monograph on the mathematical theory of the solitaire game “The Tower of Hanoi” which was invented in the 19th century by the French number theorist Édouard Lucas. The book comprises a survey of the historical development from the game’s predecessors up to recent research in mathematics and applications in computer science and psychology. Apart from long-standing myths it contains a thorough, largely self-contained presentation of the essential mathematical facts with complete proofs, including also unpublished material. The main objects of research today are the so-called Hanoi graphs and the related Sierpiński graphs. Acknowledging the great popularity of the topic in computer science, algorithms and their correctness proofs form an essential part of the book. In view of the most important practical applications of the Tower of Hanoi and its variants, namely in physics, network theory, and cognitive (neuro)psychology, other related structures and puzzles like, e.g., the “Tower of London”, are addressed. Numerous captivating integer sequences arise along the way, but also many open questions impose themselves. Central among these is the famed Frame-Stewart conjecture. Despite many attempts to decide it and large-scale numerical experiments supporting its truth, it remains unsettled after more than 70 years and thus demonstrates the timeliness of the topic. 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.

Mathematical recreations. --- Mathematics -- Humor. --- Mathematics. --- Puzzles. --- Mathematical recreations --- Mathematics --- Physical Sciences & Mathematics --- Elementary Mathematics & Arithmetic --- Mathematical Theory --- History --- History. --- Mathematical puzzles --- Number games --- Recreational mathematics --- Recreations, Mathematical --- Algorithms. --- Sequences (Mathematics). --- Game theory. --- Combinatorics. --- Mathematics, general. --- History of Mathematical Sciences. --- Sequences, Series, Summability. --- Game Theory, Economics, Social and Behav. Sciences. --- Algorithm Analysis and Problem Complexity. --- Puzzles --- Scientific recreations --- Games in mathematics education --- Magic squares --- Magic tricks in mathematics education --- Computer software. --- Mathematical sequences --- Numerical sequences --- Algebra --- Math --- Science --- Software, Computer --- Computer systems --- Combinatorics --- Mathematical analysis --- Algorism --- Arithmetic --- Games, Theory of --- Theory of games --- Mathematical models --- Annals --- Auxiliary sciences of history --- Foundations

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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 is 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 text is particularly strong on the spectral theory of Sturm-Liouville equations, which has given rise to a major branch of modern analysis. Among other current aspects of the theory discussed are oscillation theory for differential equations and Jacobi matrices, approximation of singular boundary value problems by regular ones, applications to systems of differential equations, extension of the theory to partial differential equations and to non-linear problems, and various generalizations of Borg's inverse theory. A unique feature of the book is a comprehensive catalogue of Sturm-Liouville differential equations covering more than fifty examples, together with their spectral properties. Many of these examples are connected with special functions and with problems in mathematical physics and applied mathematics. 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.

Sturm-Liouville equation --- Differential equations --- Differential operators. --- Qualitative theory --- Operators, Differential --- Operator theory --- 517.91 Differential equations --- Liouville-Sturm equation --- Boundary value problems --- Global analysis (Mathematics). --- Differential equations, partial. --- Mathematics. --- Analysis. --- Partial Differential Equations. --- Mathematics, general. --- Math --- Science --- Partial differential equations --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Mathematical analysis. --- Analysis (Mathematics). --- Partial differential equations. --- 517.1 Mathematical analysis --- Mathematical analysis