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Hilbert's tenth problem is one of 23 problems proposed by David Hilbert in 1900 at the International Congress of Mathematicians in Paris. These problems gave focus for the exponential development of mathematical thought over the following century. The tenth problem asked for a general algorithm to determine if a given Diophantine equation has a solution in integers. It was finally resolved in a series of papers written by Julia Robinson, Martin Davis, Hilary Putnam, and finally Yuri Matiyasevich in 1970. They showed that no such algorithm exists. This book is an exposition of this remarkable achievement. Often, the solution to a famous problem involves formidable background. Surprisingly, the solution of Hilbert's tenth problem does not. What is needed is only some elementary number theory and rudimentary logic. In this book, the authors present the complete proof along with the romantic history that goes with it. Along the way, the reader is introduced to Cantor's transfinite numbers, axiomatic set theory, Turing machines, and Gödel's incompleteness theorems. Copious exercises are included at the end of each chapter to guide the student gently on this ascent. For the advanced student, the final chapter highlights recent developments and suggests future directions. The book is suitable for undergraduates and graduate students. It is essentially self-contained.
This volume is the proceedings of the NATO Advanced Study Institute, "Diffusion in Materials", held at "Centre Paul Langevin", Aussois, during March 12-25, 1989. There were 105 participants of whom 24 were lecturers and members of the international advisory committee. In addition to the participants from NATO countries, a small number of participants came from Australia, Hungary, Poland and Tunisia. The principal aim of the organizing committee was to bring together scientists of wide interest and expertise in the field of diffusion and to familiarize the young workers in material science with the wide range of theoretical models and methods and of experimental techniques . The Institute was concerned with the study of diffusion and related phenomena in solids which are at the cutting edge of novel technologies. The discussion of basic theories of defects in solids and their transport, with their applications in the understanding of diffusion processes in "simple solids" was followed by the wide range of current theoretical models and methods, experimental techniques and their potential. The lectures on the diffusion in specific materials included : metals, dilute and concentrated alloys, simple and compound semiconductors, stoichiometric and non-stoichiometric oxides, high-Tc compounds, carbides, nitrides, silicates, conducting polymers and thin films, ionic, superionic, amorphous and irradiated materials.
Explores the Brazilian political economy with the use of computable general equilibrium income distribution models.