The development of algebraic geometry over groups, geometric group theory and group-based cryptography, has led to there being a tremendous recent interest in infinite group theory. This volume presents a good collection of papers detailing areas of current interest.

This book contains surveys and research articles on the state-of-the-art in finitely presented groups for researchers and graduate students. Overviews of current trends in exponential groups and of the classification of finite triangle groups and finite generalized tetrahedron groups are complemented by new results on a conjecture of Rosenberger and an approximation theorem. A special emphasis is on algorithmic techniques and their complexity, both for finitely generated groups and for finite Z-algebras, including explicit computer calculations highlighting important classical methods. A further chapter surveys connections to mathematical logic, in particular to universal theories of various classes of groups, and contains new results on countable elementary free groups. Applications to cryptography include overviews of techniques based on representations of p-groups and of non-commutative group actions. Further applications of finitely generated groups to topology and artificial intelligence complete the volume. All in all, leading experts provide up-to-date overviews and current trends in combinatorial group theory and its connections to cryptography and other areas.

This is the second volume of a series of books in various aspects of Mathematical Physics. Mathematical Physics has made great strides in recent years, and is rapidly becoming an important dis cipline in its own right. The fact that physical ideas can help create new mathematical theories, and rigorous mathematical theo rems can help to push the limits of physical theories and solve problems is generally acknowledged. We believe that continuous con tacts between mathematicians and physicists and the resulting dialogue and the cross fertilization of ideas is a good thing. This series of studies is published with this goal in mind. The present volume contains contributions which were original ly presented at the Second NATO Advanced Study Institute on Mathe matical Physics held in Istanbul in the Summer of 1972. The main theme was the application of group theoretical methods in general relativity and in particle physics. Modern group theory, in par ticular, the theory of unitary irreducibl~ infinite-dimensional representations of Lie groups is being increasingly important in the formulation and solution of dynamical problems in various bran ches of physics. There is moreover a general trend of approchement of the methods of general relativity and elementary particle physics. We hope it will be useful to present these investigations to a larger audience.

This is a commemoration volume to honor Professor M Veltman on the ocassion of his 60th birthday. It contains articles on Gauge field theories, a subject to which Prof. Veltman has made many important and seminal contributions. Some of the contributions are based on invited talks given at the Conference held in Ann Arbor, Michigan, May 16 - 18 1991. The articles in the book cover a wide range of topics from formal and phenomenological to the experimental aspects of Gauge theories.

Award-winning monograph of the Ferran Sunyer i Balaguer Prize 2001. Subgroup growth studies the distribution of subgroups of finite index in a group as a function of the index. In the last two decades this topic has developed into one of the most active areas of research in infinite group theory; this book is a systematic and comprehensive account of the substantial theory which has emerged. As well as determining the range of possible 'growth types', for finitely generated groups in general and for groups in particular classes such as linear groups, a main focus of the book is on the tight connection between the subgroup growth of a group and its algebraic structure. A wide range of mathematical disciplines play a significant role in this work: as well as various aspects of infinite group theory, these include finite simple groups and permutation groups, profinite groups, arithmetic groups and Strong Approximation, algebraic and analytic number theory, probability, and p-adic model theory. Relevant aspects of such topics are explained in self-contained 'windows'.

"The development of algebraic geometry over groups, geometric group theory and group-based cryptography, has led to there being a tremendous recent interest in infinite group theory. This volume presents a good collection of papers detailing areas of current interest."--

270 Expert contributions on aspects of landslide hazards, encompassing geological modeling and soil and rock mechanics, landslide processes, causes and effects, and damage avoidance and limitation strategies. Reference source for academics and professionals in geo-mechanical and geo-technical engineering, and others involved with research, des

In February 1981, the classification of the finite simple groups (Dl)* was completed,t. * representing one of the most remarkable achievements in the history or mathematics. Involving the combined efforts of several hundred mathematicians from around the world over a period of 30 years, the full proof covered something between 5,000 and 10,000 journal pages, spread over 300 to 500 individual papers. The single result that, more than any other, opened up the field and foreshadowed the vastness of the full classification proof was the celebrated theorem of Walter Feit and John Thompson in 1962, which stated that every finite group of odd order (D2) is solvable (D3)-a statement expressi ble in a single line, yet its proof required a full 255-page issue of the Pacific 10urnal of Mathematics [93]. Soon thereafter, in 1965, came the first new sporadic simple group in over 100 years, the Zvonimir Janko group 1 , to further stimulate the 1 'To make the book as self-contained as possible. we are including definitions of various terms as they occur in the text. However. in order not to disrupt the continuity of the discussion. we have placed them at the end of the Introduction. We denote these definitions by (DI). (D2), (D3). etc.

Why should mathematical logic be grounded on the basis of some formal requirements in the way that it has been developed since its classical emergence as a hybrid field of mathematics and logic in the 19th century or earlier? Contrary to conventional wisdom, the foundation of mathematic logic has been grounded on some false (or dogmatic) assumptions which have much impoverished the pursuit of knowledge. This is not to say that mathematical logic has been useless. Quite on the contrary, it has been quite influential in shaping the way that reality is to be understood in numerous fields of knowledge—by learning from the mathematical study of logic and its reverse, the logical study of mathematics. In the final analysis, the future of mathematical logic will depend on how its foundational crisis is to be resolved, and "the contrastive theory of rationality" (in this book) is to precisely show how and why it can be done by taking a contrastive turn, subject to the constraints imposed upon by "existential dialectic principles" at the ontological level (to avoid any reductionistic fallacy) and other ones (like the perspectives of culture, society, nature, and the mind). The contrastive theory of rationality thus shows a better way to ground mathematical logic (beyond both classical and non-classical logics) for the future advancement of knowledge and, if true, will alter the way of how mathematical logic is to be understood, with its enormous implications for the future of knowledge and its "post-human" fate.

Based on lectures given in honour of Stephen Hawking's sixtieth birthday, this book comprises contributions from some of the world's leading theoretical physicists. It begins with a section containing chapters by successful scientific popularisers, bringing to life both Hawking's work and other exciting developments in physics. The book then goes on to provide a critical evaluation of advanced subjects in modern cosmology and theoretical physics. Topics covered include the origin of the universe, warped spacetime, cosmological singularities, quantum gravity, black holes, string theory, quantum cosmology and inflation. As well as providing a fascinating overview of the wide variety of subject areas to which Stephen Hawking has contributed, this book represents an important assessment of prospects for the future of fundamental physics and cosmology.