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The biennial meetings at São Carlos have helped create a worldwide community of experts and young researchers working on singularity theory, with a special focus on applications to a wide variety of topics in both pure and applied mathematics. The tenth meeting, celebrating the 60th birthdays of Terence Gaffney and Maria Aparecida Soares Ruas, was a special occasion attracting the best known names in the area. This volume contains contributions by the attendees, including three articles written or co-authored by Gaffney himself, and survey articles on the existence of Milnor fibrations, global classifications and graphs, pairs of foliations on surfaces, and Gaffney's work on equisingularity.
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A phenomenon which appears in nature, or human behavior, can sometimes be explained by saying that a certain potential function is maximized, or minimized. For example, the Hamiltonian mechanics, soapy films, size of an atom, business management, etc. In mathematics, a point where a given function attains an extreme value is called a critical point, or a singular point. The purpose of singularity theory is to explore the properties of singular points of functions and mappings. This is a volume on the proceedings of the fourth Japanese-Australian Workshop on Real and Complex Singularities held
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Singularity theory encompasses many different aspects of geometry and topology, and an overview of these is represented here by papers given at the International Singularity Conference held in 1991 at Lille. The conference attracted researchers from a wide variety of subject areas, including differential and algebraic geometry, topology, and mathematical physics. Some of the best known figures in their fields participated, and their papers have been collected here. Contributors to this volume include G. Barthel, J. W. Bruce, F. Delgado, M. Ferrarotti, G. M. Greuel, J. P. Henry, L. Kaup, B. Lichtin, B. Malgrange, M. Merle, D. Mond, L. Narvaez, V. Neto, A. A. Du Plessis, R. Thom and M. Vaquié. Research workers in singularity theory or related subjects will find that this book contains a wealth of valuable information on all aspects of the subject.
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Many key phenomena in physics and engineering are described as singularities in the solutions to the differential equations describing them. Examples covered thoroughly in this book include the formation of drops and bubbles, the propagation of a crack and the formation of a shock in a gas. Aimed at a broad audience, this book provides the mathematical tools for understanding singularities and explains the many common features in their mathematical structure. Part I introduces the main concepts and techniques, using the most elementary mathematics possible so that it can be followed by readers with only a general background in differential equations. Parts II and III require more specialised methods of partial differential equations, complex analysis and asymptotic techniques. The book may be used for advanced fluid mechanics courses and as a complement to a general course on applied partial differential equations.
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In this short note, we first show (1) if (n, p) lies inside Mather's nice region then any A-stable multigerm f : (R^n, S)·(R^p, 0) and any C! unfolding of f are A-simple, and (2) for any (n, p) there exists a non-negative integer i such that for any integer j ((i·j)) there exists an A-stable multigerm f : (R^n·R^j, S · {0}) · (R^p · R^j , (0, 0)) which is not A-simple. Next, we obtain a characterization of curves among multigerms of corank at most one from the view point of A-stabie multigerms and A-simple multigerms. It turns out that for any (n, p) such that n < p an asymmetric Cantor set is naturally constructed by using upper bounds for multiplicities of A-stable multigerms and upper bounds for multiplicities of A-simple multigerms, and the desired characterization of curves can be obtained by cardinalities of constructed asymmetric Cantor sets.
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In this short note, we first show (1) if (n, p) lies inside Mather's nice region then any A-stable multigerm f : (R^n, S)·(R^p, 0) and any C! unfolding of f are A-simple, and (2) for any (n, p) there exists a non-negative integer i such that for any integer j ((i·j)) there exists an A-stable multigerm f : (R^n·R^j, S · {0}) · (R^p · R^j , (0, 0)) which is not A-simple. Next, we obtain a characterization of curves among multigerms of corank at most one from the view point of A-stabie multigerms and A-simple multigerms. It turns out that for any (n, p) such that n < p an asymmetric Cantor set is naturally constructed by using upper bounds for multiplicities of A-stable multigerms and upper bounds for multiplicities of A-simple multigerms, and the desired characterization of curves can be obtained by cardinalities of constructed asymmetric Cantor sets.
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