Circle-valued Morse theory [[electronic resource] /] / Andrei V. Pajitnov

• Record Nr.: UNINA9910454619003321
• Autore: Pajitnov Andrei V
• Pubbl/distr/stampa: Berlin ; ; New York, : De Gruyter, c2006
• ISBN: 1-282-19426-7; 9786612194269; 3-11-019797-9
• Descrizione fisica: 1 online resource (464 pages)
• Collana: De Gruyter studies in mathematics, , 0179-0986 ; ; 32
• Classificazione: SK 350
• Disciplina: 514/.74
• Soggetti: Morse theory; Manifolds (Mathematics); Electronic books.
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Note generali: Description based upon print version of record.
• Nota di bibliografia: Includes bibliographical references (p. [437]-444) and index.
• Nota di contenuto: Front matter -- Contents -- Preface -- Introduction -- Part 1. Morse functions and vector fields on manifolds -- CHAPTER 1. Vector fields and C0 topology -- CHAPTER 2. Morse functions and their gradients -- CHAPTER 3. Gradient flows of real-valued Morse functions -- Part 2. Transversality, handles, Morse complexes -- CHAPTER 4. The Kupka-Smale transversality theory for gradient flows -- CHAPTER 5. Handles -- CHAPTER 6. The Morse complex of a Morse function -- Part 3. Cellular gradients -- CHAPTER 7. Condition (C) -- CHAPTER 8. Cellular gradients are C0-generic -- CHAPTER 9. Properties of cellular gradients -- Part 4. Circle-valued Morse maps and Novikov complexes -- CHAPTER 10. Completions of rings, modules and complexes -- CHAPTER 11. The Novikov complex of a circle-valued Morse map -- CHAPTER 12. Cellular gradients of circle-valued Morse functions and the Rationality Theorem -- CHAPTER 13. Counting closed orbits of the gradient flow -- CHAPTER 14. Selected topics in the Morse-Novikov theory -- Backmatter
• Sommario/riassunto: In the early 1920's M. Morse discovered that the number of critical points of a smooth function on a manifold is closely related to the topology of the manifold. This became a starting point of the Morse theory which is now one of the basic parts of differential topology. ; Circle-valued Morse theory originated from a problem in hydrodynamics studied by S. P. Novikov in the early 1980's. Nowadays, it is a constantly growing field of contemporary mathematics with applications and connections to many geometrical problems such as Arnold's conjecture in the theory of Lagrangian intersections, fibrations of manifolds over the circle, dynamical zeta functions, and the theory of knots and links in the three-dimensional sphere. The aim of the book is to give a systematic treatment of geometric foundations of the subject and recent research results. The book is accessible to first year graduate students specializing in geometry and topology.

Mathematics of the weather : polygonal spline local-Galerkin methods on spheres / / Jürgen Steppeler and Jinxi Li

• Record Nr.: UNISA996495168203316
• Autore: Steppeler Jürgen
• Pubbl/distr/stampa: Cham, Switzerland : , : Springer, , [2022]; ©2022
• ISBN: 3-031-07238-3
• Descrizione fisica: 1 online resource (328 pages)
• Collana: Springer atmospheric sciences
• Disciplina: 551.51011
• Soggetti: Atmospheric models; Atmospheric models - Mathematics; Physical geography; Models matemàtics; Geografia física; Llibres electrònics
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di bibliografia: Includes bibliographical references (pages 311-316) and index.
• Nota di contenuto: Intro -- Foreword -- Preface -- Acknowledgments -- Contents -- Acronyms -- Introduction -- Numerics -- Discretization on Spherical Grids -- Efficiency of the Computational Grid -- Numerical Methods -- Validations of Numerical Methods Using NWP Models -- Verifications of Numerical Methods for Climate Modeling -- Simple Finite Difference ; Procedures -- The Runge-Kutta and Other Time Discretization Schemes -- Homogeneous and Inhomogeneous Difference Schemes -- Some Further Properties of Finite Difference Schemes -- The Von Neumann Method of Stability Analysis -- Dynamic Equations of Toy Models -- Diffusion -- The Boussinesq Model of Convection Between Heated Plates -- The Lorenz Paradigmatic Model -- Local-Galerkin Schemes in 1D -- Functional Representations, Amplitudes, and Basis Functions -- The Classic Galerkin Procedure -- Spectral Elements -- The L-Galerkin Scheme: o3o3 -- The L-Galerkin Scheme: o2o3 -- Splines of High Smoothness -- A Conserving Second-Order Scheme Using a Homogeneous FD Scheme -- Boundaries and Diffusion -- Transfer Function Analysis -- A Numerical Test for Irregular Resolution -- Internal Boundaries for Vertical Discretization -- Open Boundary Condition -- The L-Galerkin scheme: o3o5C1C2 -- The L-Galerkin Scheme: o4o5C1C2 -- The Interface to Physics in High-Order L-Galerkin Schemes -- Polygonal Spline Solutions Using Distributions and Discontinuities -- Von Neumann Analysis of Some onom Schemes -- 2D Basis Functions for Triangular and Rectangular Meshes -- Rhomboidal Basis Functions and Sparse Grids for the Regular Grid Case -- Euclid's Lemma -- Triangular Basis Functions and Full Grids -- Triangular Basis Functions for the Rectangular Case -- The Corner Derivative Representation -- An Irregular Structured Quadrilateral Grid with Triangular Cells -- An Example of a Regularization Operator.; Finite Difference Schemes on Sparse and Full Grids -- Non-conserving Schemes for Full Grids -- Alternative Methods to Compute Derivatives -- Baumgardner's Cloud Derivative Method -- Third-Order Differencing for Corner Points with a Second-Degree Polynomial Representation -- Enhanced Stencil Order -- The Full Triangular o3o3 Method -- Sparse Grids -- L-Galerkin Schemes for Sparse Triangular Meshes -- Totally Irregular Triangular and Quadrilateral Mesh: Hexagons and Other Polygons -- Staggered Grid Systems and Their Basis Function Representation -- A Simple Cut-Cell System Based on the Staggered Low-Order Basis Functions -- A Conserving Version of the Cut-Cell Scheme -- Full and Sparse Hexagonal Grids in the Plane -- Indices and Basis Functions of Hexagonal Grids in a Plane -- Numerical Methods of Hexagonal Grids on the Plane -- Hexagonal Options -- Isotropy of the Hexagonal Grid in Comparison to Rhomboidal Grid -- Platonic and Semi-Platonic Solids -- Cubed Sphere, Icosahedron, and Examples of Semi-Platonic Solids -- Geometric Properties of Spherical Grids -- Equations of Motion on the Spherical Grid and Non-conserving Finite Difference Schemes -- Further Spherical Test Problems -- Conserving L-Galerkin Schemes on the Sphere -- A Simple Non-conserving Homogeneous Order Discretization on the Sphere -- Hexagonal Grids on the Sphere -- Numerical Tests -- 1D Homogeneous Advection Test for onom Methods, SEM2 and SEM3 -- A Numerical Example of Open Boundary Condition for a Fast Wave -- The T64 Solid for Discretization by Quadrilateral Cells -- Shallow Water Tests on the Sphere: Solid Body Rotation, Solid Body Flow, Advection, and Williamson Test No. 6 -- 2D Mountain Wave Test -- The Kalman Filter Data Analysis -- Test of the o3o3 Scheme on the Cubed Sphere Grid Using the Shallow Water Version of the HOMME Model.; Projections of Semi-platonic Solids to Triangular Surfaces -- CPU Time Used with a 3D Version of o3o3 Scheme -- 1D Example for the QUASAR System -- Operations of Linear Spaces -- Summary and Outlook -- Computer Aspects of Parallel Processing -- Numerical Weather Prediction for Small Research Groups and Owners of Private PCs -- New Applications for NWP -- Large Eddy Simulation -- The Elliptical and the Potato Shaped Earth -- Data Assimilation -- Global Models for ; Forecasting and Climate Research -- Linear Algebra -- Examples of Program -- Dispersion Analysis of o2o3 and o3o3 Methods -- 1D Homogeneous Advection Test -- Appendix A -- Neighborhood Relations for the Full Triangular Grid and a Compact Storage System -- The Serendipity Interpolation on the Sphere -- The Quasi-arithmetic Rendition QUASAR to Obtain a Sparse Field Representation -- Glossary -- References -- Index.