This volume contains papers presented at the Twelfth Taniguchi Symposium on the Theory of Condensed Matter, which was held at Kashikojima (in Ise Shima National Park), Japan, November 14-19, 1989. The general purpose of the Taniguchi Symposia is to encourage important developing, rather than established, fields in condensed matter theory. The topic of the present sym posium, Quasicrystais, is quite typical. In 1984, Shechtman, Blech, Gratias and Cahn discovered the icosahedral symmetry of a diffraction pattern and Levine and Steinhardt independently presented the notion of quasicrystals. Before these discoveries, Roger Penrose of Oxford University had invented a space-filling non-periodic tiling, now called Penrose tiling. These factors form a new field that had become mathematically viable by the end of 1984, and many important new ideas are still being created. In standard textbooks of solid-state science, the first chapter used to be devoted to symmetry and periodicity in crystals. Now, the textbooks should be revised; quasi-periodicity and its physical properties should be added in several chapters and almost all standard conceptions should be reconsidered. However, the facts that are known about quasiperiodicity are not enough to complete even an introductory chapter of a textbook. Revision should be extended to generalized crystallography, defects, crystal growth, electronic structure, spectral theory and localization, electron transport, spin statistics, etc. These are all topics treated in this volume.

The basics of complex functions will be explained for students of Engineering Sciences, with the aim of being able to use 'complex function theory' as a tool. The goal is not rigor as mathematics, but ease of use that may suit the application. Explanations are based on concrete examples rather than abstract general theory. The book starts from very beginning of complex numbers, and extends theory of Introduction to Elliptic Function and Hypergeometric Differential Equations.

The basics of complex functions will be explained for students of Engineering Sciences, with the aim of being able to use 'complex function theory' as a tool. The goal is not rigor as mathematics, but ease of use that may suit the application. Explanations are based on concrete examples rather than abstract general theory. The book starts from very beginning of complex numbers, and extends theory of Introduction to Elliptic Function and Hypergeometric Differential Equations.

This book is aimed at researchers who are working in a field of quasicrystals to provide a reference to recent developments and ideas in the field and also at graduate students, who intend to study quasicrystals, to provide introduction of ideas. Topics in this book cover an entire field of quasicrystals, both experimental and theoretical, including new developments: the state of the art in quasicrystallography, new families of quasicrystals, phasons in aperiodic solids, ab initio studies on stability mechanism, quantum transport phenomena, elastic/plastic properties and surface of quasicrystals.
. Comprehensive reviews by experts in the field
. Complete reference of original papers and new topics
. Intelligible introduction of quasicrystals by experts