Singularities of Caustics and Wave Fronts

Author: Vladimir Arnold

Publisher: Springer Science & Business Media

ISBN: 9401133301

Category: Mathematics

Page: 259

View: 8268


One service mathematics has rendered the 'Et moi ...) si j'avait su comment en revenir, human race. It has put common sense back je n'y serais point aile.' Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non The series is divergent; therefore we may be sense'. ErieT. Bell able to do something with it. O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

Differential Geometry from a Singularity Theory Viewpoint

Author: Shyuichi Izumiya,Maria del Carmen Romero Fuster,Maria Aparecida Soares Ruas,Farid Tari

Publisher: World Scientific

ISBN: 9814590460

Category: Mathematics

Page: 400

View: 4923


Differential Geometry from a Singularity Theory Viewpoint provides a new look at the fascinating and classical subject of the differential geometry of surfaces in Euclidean spaces. The book uses singularity theory to capture some key geometric features of surfaces. It describes the theory of contact and its link with the theory of caustics and wavefronts. It then uses the powerful techniques of these theories to deduce geometric information about surfaces embedded in 3, 4 and 5-dimensional Euclidean spaces. The book also includes recent work of the authors and their collaborators on the geometry of sub-manifolds in Minkowski spaces. Contents:The Case for the Singularity Theory ApproachSubmanifolds of the Euclidean SpaceSingularities of Germs of Smooth MappingsContact Between Submanifolds of ℝnLagrangian and Legendrian SingularitiesSurfaces in the Euclidean 3-SpaceSurfaces in the Euclidean 4-SpaceSurfaces in the Euclidean 5-SpaceSpacelike Surfaces in the Minkowski Space-TimeGlobal Viewpoint Readership: Advanced undergraduates and post-graduate students, and researchers in the fields of differential geometry and singularity theory. Key Features:The book is unique in its nature. It provides a coherent approach for studying the geometry of sub-manifolds of various ambient spaces from the singularity theory point of viewThe book informs the reader about the progress in the field of extrinsic differential geometry and singularity theory. The information is new and has not been treated in previous textbooksThe book gathers scattered work from various research articles, most of which are recent, and describes techniques that could be used to tackle problems in other areas of mathematicsKeywords:Contact;Extrinsic Geometry;Genericity;Caustics;Singularities;Surfaces;Transversality;Wave Fronts

Singularities in Geometry and Topology

Proceedings of the Trieste Singularity Summer School and Workshop, ICTP, Trieste, Italy, 15 August - 3 September 2005

Author: Jean-Paul Brasselet

Publisher: World Scientific

ISBN: 981270681X

Category: Mathematical physics

Page: 916

View: 1937


Singularity theory appears in numerous branches of mathematics, as well as in many emerging areas such as robotics, control theory, imaging, and various evolving areas in physics. The purpose of this proceedings volume is to cover recent developments in singularity theory and to introduce young researchers from developing countries to singularities in geometry and topology. The contributions discuss singularities in both complex and real geometry. As such, they provide a natural continuation of the previous school on singularities held at ICTP (1991), which is recognized as having had a major influence in the field.

Real and Complex Singularities

Author: Ana Claudia Nabarro,Juan J. Nuño-Ballesteros,Raúl Oset Sinha,Maria Aparecida Soares Ruas

Publisher: American Mathematical Soc.

ISBN: 1470422050

Category: Differential geometry -- Classical differential geometry -- Classical differential geometry

Page: 355

View: 6150


This volume is a collection of papers presented at the XIII International Workshop on Real and Complex Singularities, held from July 27–August 8, 2014, in São Carlos, Brazil, in honor of María del Carmen Romero Fuster's 60th birthday. The volume contains the notes from two mini-courses taught during the workshop: on intersection homology by J.-P. Brasselet, and on non-isolated hypersurface singularities and Lê cycles by D. Massey. The remaining contributions are research articles which cover topics from the foundations of singularity theory (including classification theory and invariants) to topology of singular spaces (links of singularities and semi-algebraic sets), as well as applications to topology (cobordism and Lefschetz fibrations), dynamical systems (Morse-Bott functions) and differential geometry (affine geometry, Gauss-maps, caustics, frontals and non-Euclidean geometries). This book is published in cooperation with Real Sociedad Matemática Española (RSME)

Singularities of Differentiable Maps, Volume 1

Classification of Critical Points, Caustics and Wave Fronts

Author: V.I. Arnold,S.M. Gusein-Zade,Alexander N. Varchenko

Publisher: Springer Science & Business Media

ISBN: 0817683402

Category: Mathematics

Page: 282

View: 2575


​Singularity theory is a far-reaching extension of maxima and minima investigations of differentiable functions, with implications for many different areas of mathematics, engineering (catastrophe theory and the theory of bifurcations), and science. The three parts of this first volume of a two-volume set deal with the stability problem for smooth mappings, critical points of smooth functions, and caustics and wave front singularities. The second volume describes the topological and algebro-geometrical aspects of the theory: monodromy, intersection forms, oscillatory integrals, asymptotics, and mixed Hodge structures of singularities. The first volume has been adapted for the needs of non-mathematicians, presupposing a limited mathematical background and beginning at an elementary level. With this foundation, the book's sophisticated development permits readers to explore more applications than previous books on singularities.

Mathematik für Physiker Band 3

Variationsrechnung - Differentialgeometrie - Mathematische Grundlagen der allgemeinen Relativitätstheorie

Author: Helmut Fischer,Helmut Kaul

Publisher: Springer-Verlag

ISBN: 3658004754

Category: Science

Page: 408

View: 8389


Wie in den ersten beiden Bänden ihres Werkes stellen die Autoren auch im abschließenden dritten Band mathematische Grundlagen der Physik in gut zugänglicher und ansprechender Form dar. Das Buch eignet sich sowohl für das Selbststudium als auch zur Begleitung von Vorlesungen.

Mathematical Methods of Classical Mechanics

Author: V.I. Arnol'd

Publisher: Springer Science & Business Media

ISBN: 1475720637

Category: Mathematics

Page: 520

View: 5501


This book constructs the mathematical apparatus of classical mechanics from the beginning, examining basic problems in dynamics like the theory of oscillations and the Hamiltonian formalism. The author emphasizes geometrical considerations and includes phase spaces and flows, vector fields, and Lie groups. Discussion includes qualitative methods of the theory of dynamical systems and of asymptotic methods like averaging and adiabatic invariance.

Revue de mathématique élémentaires

Author: N.A

Publisher: N.A


Category: Mathematics

Page: N.A

View: 2599


Elemente der Mathematik (EL) publishes survey articles about important developments in the field of mathematics; stimulating shorter communications that tackle more specialized questions; and papers that report on the latest advances in mathematics and applications in other disciplines. The journal does not focus on basic research. Rather, its articles seek to convey to a wide circle of readers - teachers, students, engineers, professionals in industry and administration - the relevance, intellectual challenge and vitality of mathematics today. The Problems Section, covering a diverse range of exercises of varying degrees of difficulty, encourages an active grappling with mathematical problems.

Fifth International Conference on Mathematical and Numerical Aspects of Wave Propagation

Author: Alfredo Bermœdez

Publisher: SIAM

ISBN: 9780898714708

Category: Science

Page: 1038

View: 4324


This conference was held in Santiago de Compostela, Spain, July 10-14, 2000. This volume contains papers presented at the conference covering a broad range of topics in theoretical and applied wave propagation in the general areas of acoustics, electromagnetism, and elasticity. Both direct and inverse problems are well represented. This volume, along with the three previous ones, presents a state-of-the-art primer for research in wave propagation. The conference is conducted by the Institut National de Recherche en Informatique et en Automatique with the cooperation of SIAM.

Advances in Microlocal Analysis

Author: H.G. Garnir

Publisher: Springer Science & Business Media

ISBN: 9789027721952

Category: Mathematics

Page: 390

View: 2225


The 1985 Castel vecchio-Pas coli NATO Advanced Study Institute is aimed to complete the trilogy with the two former institutes I organized : "Boundary Value Problem for Evolution Partial Differential Operators", Liege, 1976 and "Singularities in Boundary Value Problems", Maratea, 1980. It was indeed necessary to record the considerable progress realized in the field of the propagation of singularities of Schwartz Distri butions which led recently to the birth of a new branch of Mathema tical Analysis called Microlocal Analysis. Most of this theory was mainly built to be applied to distribution solutions of linear partial differential problems. A large part of this institute still went in this direction. But, on the other hand, it was also time to explore the new trend to use microlocal analysis In non linear differential problems. I hope that the Castelvecchio NATO ASI reached its purposes with the help of the more famous authorities in the field. The meeting was held in Tuscany (Italy) at Castelvecchio-Pascoli, little village in the mountains north of Lucca on September 2-12, 1985. It was hosted by "11 Ciocco" an international vacation Center, In a comfortable hotel located in magnificent mountain surroundings and provided with all conference and sport facilities.

Catastrophe Theory

Author: Vladimir I. Arnol'd

Publisher: Springer Science & Business Media

ISBN: 3642581242

Category: Mathematics

Page: 150

View: 9579


The new edition of this non-mathematical review of catastrophe theory contains updated results and many new or expanded topics including delayed loss of stability, shock waves, and interior scattering. Three new sections offer the history of singularity and its applications from da Vinci to today, a discussion of perestroika in terms of the theory of metamorphosis, and a list of 93 problems touching on most of the subject matter in the book.