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Vector Fields on Singular Varieties

Author: Jean-Paul Brasselet

Publisher: Springer Science & Business Media

Published: 2009-12-17

Total Pages: 242

ISBN-13: 3642052045

Vector fields on manifolds play a major role in mathematics and other sciences. In particular, the Poincaré-Hopf index theorem gives rise to the theory of Chern classes, key manifold-invariants in geometry and topology. It is natural to ask what is the ‘good’ notion of the index of a vector field, and of Chern classes, if the underlying space becomes singular. The question has been explored by several authors resulting in various answers, starting with the pioneering work of M.-H. Schwartz and R. MacPherson. We present these notions in the framework of the obstruction theory and the Chern-Weil theory. The interplay between these two methods is one of the main features of the monograph.

Bifurcations of Planar Vector Fields

Author: Freddy Dumortier

Publisher: Springer

Published: 2006-12-08

Total Pages: 234

ISBN-13: 3540384332

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The book reports on recent work by the authors on the bifurcation structure of singular points of planar vector fields whose linear parts are nilpotent. The bifurcation diagrams of the most important codimension-three cases are studied in detail. The results presented reach the limits of what is currently known on the bifurcation theory of planar vector fields. While the treatment is geometric, special analytical tools using abelian integrals are needed, and are explicitly developed. The rescaling and normalization methods are improved for application here. The reader is assumed to be familiar with the elements of Bifurcation and Dynamical Systems Theory. The book is addressed to researchers and graduate students working in Ordinary Differential Equations and Dynamical Systems, as well as anyone modelling complex multiparametric phenomena.

Vector Fields with Applications to Thermodynamics and Irreversibility

Author: Luis Manuel Braga da Costa Campos

Publisher: CRC Press

Published: 2022-11-30

Total Pages: 956

ISBN-13: 1000416038

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Vector Fields with Applications to Thermodynamics and Irreversibility is part of the series "Mathematics and Physics for Science and Technology", which combines rigorous mathematics with general physical principles to model practical engineering systems with a detailed derivation and interpretation of results. Volume V presents the mathematical theory of partial differential equations and methods of solution satisfying initial and boundary conditions, and includes applications to: acoustic, elastic, water, electromagnetic and other waves; the diffusion of heat, mass and electricity; and their interactions. This is the first book of the volume. The second book of volume V continues this book on thermodynamics, focusing on the equation of state and energy transfer processes including adiabatic, isothermal, isobaric and isochoric. These are applied to thermodynamic cycles, like the Carnot, Atkinson, Stirling and Barber-Brayton cycles, that are used in thermal devices, including refrigerators, heat pumps, and piston, jet and rocket engines. In connection with jet propulsion, adiabatic flows and normal and oblique shock waves in free space and nozzles with variable cross-section are considered. The equations of fluid mechanics are derived for compressible two-phase flow in the presence of shear and bulk viscosity, thermal conduction and mass diffusion. The thermodynamic cycles are illustrated by detailed calculations modelling the operation of piston, turbojet and rocket engines in various ambient conditions, ranging from sea level, the atmosphere of the earth at altitude and vacuum of space, for the propulsion of land, sea, air and space vehicles. The book is intended for graduate students and engineers working with mathematical models and can be applied to problems in mechanical, aerospace, electrical and other branches of engineering dealing with advanced technology, and also in the physical sciences and applied mathematics. This book: Simultaneously covers rigorous mathematics, general physical principles and engineering applications with practical interest Provides interpretation of results with the help of illustrations Includes detailed proofs of all results L.M.B.C. Campos was chair professor and the Coordinator of the Scientific Area of Applied and Aerospace Mechanics in the Department of Mechanical Engineering and also the director (and founder) of the Center for Aeronautical and Space Science and Technology until retirement in 2020. L.A.R.Vilela is currently completing an Integrated Master's degree in Aerospace Engineering at Institute Superior Tecnico (1ST) of Lisbon University.

Harmonic Vector Fields

Author: Sorin Dragomir

Publisher: Elsevier

Published: 2011-10-26

Total Pages: 529

ISBN-13: 0124158269

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An excellent reference for anyone needing to examine properties of harmonic vector fields to help them solve research problems. The book provides the main results of harmonic vector ?elds with an emphasis on Riemannian manifolds using past and existing problems to assist you in analyzing and furnishing your own conclusion for further research. It emphasizes a combination of theoretical development with practical applications for a solid treatment of the subject useful to those new to research using differential geometric methods in extensive detail. A useful tool for any scientist conducting research in the field of harmonic analysis Provides applications and modern techniques to problem solving A clear and concise exposition of differential geometry of harmonic vector fields on Reimannian manifolds Physical Applications of Geometric Methods

Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields

Author: John Guckenheimer

Publisher: Springer Science & Business Media

Published: 2013-11-21

Total Pages: 475

ISBN-13: 1461211409

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An application of the techniques of dynamical systems and bifurcation theories to the study of nonlinear oscillations. Taking their cue from Poincare, the authors stress the geometrical and topological properties of solutions of differential equations and iterated maps. Numerous exercises, some of which require nontrivial algebraic manipulations and computer work, convey the important analytical underpinnings of problems in dynamical systems and help readers develop an intuitive feel for the properties involved.

Time-Varying Vector Fields and Their Flows

Author: Saber Jafarpour

Publisher: Springer

Published: 2014-10-27

Total Pages: 0

ISBN-13: 9783319101385

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This short book provides a comprehensive and unified treatment of time-varying vector fields under a variety of regularity hypotheses, namely finitely differentiable, Lipschitz, smooth, holomorphic, and real analytic. The presentation of this material in the real analytic setting is new, as is the manner in which the various hypotheses are unified using functional analysis. Indeed, a major contribution of the book is the coherent development of locally convex topologies for the space of real analytic sections of a vector bundle, and the development of this in a manner that relates easily to classically known topologies in, for example, the finitely differentiable and smooth cases. The tools used in this development will be of use to researchers in the area of geometric functional analysis.

Geometry of Vector Fields

Author: Yu. Aminov

Publisher: CRC Press

Published: 2000-02-23

Total Pages: 190

ISBN-13: 9789056992019

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Presenting a classical approach to the foundations and development of the geometry of vector fields, this volume space, three orthogonal systems, and applications in mechanics. Other topics, including vector fields, Pfaff forms and systems in n-dimensional space, foliations and Godbillon-Vey invariant, are also considered. There is much interest in the study of geometrical objects in n-dimensional Euclidean space, and this volume provides a useful and comprehensive presentation.

Structural, Syntactic, and Statistical Pattern Recognition

Author: Niels da Vitoria Lobo

Publisher: Springer Science & Business Media

Published: 2008-11-24

Total Pages: 1029

ISBN-13: 3540896880

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This book constitutes the refereed proceedings of the 12th International Workshop on Structural and Syntactic Pattern Recognition, SSPR 2008 and the 7th International Workshop on Statistical Techniques in Pattern Recognition, SPR 2008, held jointly in Orlando, FL, USA, in December 2008 as a satellite event of the 19th International Conference of Pattern Recognition, ICPR 2008. The 56 revised full papers and 42 revised poster papers presented together with the abstracts of 4 invited papers were carefully reviewed and selected from 175 submissions. The papers are organized in topical sections on graph-based methods, probabilistic and stochastic structural models for PR, image and video analysis, shape analysis, kernel methods, recognition and classification, applications, ensemble methods, feature selection, density estimation and clustering, computer vision and biometrics, pattern recognition and applications, pattern recognition, as well as feature selection and clustering.

Manifolds, Vector Fields, and Differential Forms

Author: Gal Gross

Publisher: Springer Nature

Published: 2023-04-25

Total Pages: 348

ISBN-13: 3031254090

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This textbook serves as an introduction to modern differential geometry at a level accessible to advanced undergraduate and master's students. It places special emphasis on motivation and understanding, while developing a solid intuition for the more abstract concepts. In contrast to graduate level references, the text relies on a minimal set of prerequisites: a solid grounding in linear algebra and multivariable calculus, and ideally a course on ordinary differential equations. Manifolds are introduced intrinsically in terms of coordinate patches glued by transition functions. The theory is presented as a natural continuation of multivariable calculus; the role of point-set topology is kept to a minimum. Questions sprinkled throughout the text engage students in active learning, and encourage classroom participation. Answers to these questions are provided at the end of the book, thus making it ideal for independent study. Material is further reinforced with homework problems ranging from straightforward to challenging. The book contains more material than can be covered in a single semester, and detailed suggestions for instructors are provided in the Preface.

The Volume of Vector Fields on Riemannian Manifolds

Author: Olga Gil-Medrano

Publisher: Springer Nature

Published: 2023-07-31

Total Pages: 131

ISBN-13: 3031368576

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This book focuses on the study of the volume of vector fields on Riemannian manifolds. Providing a thorough overview of research on vector fields defining minimal submanifolds, and on the existence and characterization of volume minimizers, it includes proofs of the most significant results obtained since the subject’s introduction in 1986. Aiming to inspire further research, it also highlights a selection of intriguing open problems, and exhibits some previously unpublished results. The presentation is direct and deviates substantially from the usual approaches found in the literature, requiring a significant revision of definitions, statements, and proofs. A wide range of topics is covered, including: a discussion on the conditions for a vector field on a Riemannian manifold to determine a minimal submanifold within its tangent bundle with the Sasaki metric; numerous examples of minimal vector fields (including those of constant length on punctured spheres); a thorough analysis of Hopf vector fields on odd-dimensional spheres and their quotients; and a description of volume-minimizing vector fields of constant length on spherical space forms of dimension three. Each chapter concludes with an up-to-date survey which offers supplementary information and provides valuable insights into the material, enhancing the reader's understanding of the subject. Requiring a solid understanding of the fundamental concepts of Riemannian geometry, the book will be useful for researchers and PhD students with an interest in geometric analysis.