This book introduces graduate students in mathematics with concepts from topology and functional analysis, both linear and nonlinear. It is the fifth book in a series designed to train interested readers how to think properly using mathematical abstractions, and how to use the tools of mathematical analysis in applications.
This book introduces graduate students in mathematics with concepts from topology and functional analysis, both linear and nonlinear. It is the fifth book in a series designed to train interested readers how to think properly using mathematical abstractions, and how to use the tools of mathematical analysis in applications.Hinweis: Dieser Artikel kann nur an eine deutsche Lieferadresse ausgeliefert werden.
Author James K. Peterson is an Emeritus Professor at the School of Mathematical and Statistical Sciences, Clemson University. He tries hard to build interesting models of complex phenomena using a blend of mathematics, computation, and science. To this end, he has written four books on how to teach such things to biologists and cognitive scientists. These books grew out of his Calculus for Biologists courses offered to the biology majors from 2007 to 2015. He has taught the analysis courses since he started teaching both at Clemson and at his previous post at Michigan Technological University. In between, he spent time as a senior engineer in various aerospace firms and even did a short stint in a software development company. The problems he was exposed to were very hard, and not amenable to solution using just one approach. Using tools from many branches of mathematics, from many types of computational languages, and from first-principles analysis of natural phenomena was absolutely essential to make progress. In both mathematical and applied areas, students often need to use advanced mathematics tools they have not learned properly. So, he has recently written a series of five books on mathematical analysis to help researchers with the problem of learning new things after they have earned their degrees and are practicing scientists. Along the way, he has also written papers in immunology, cognitive science, and neural network technology, in addition to having grants from the NSF, NASA, and the US Army. He also likes to paint, build furniture, and write stories.
Inhaltsangabe
I. Introduction. 1. Introduction. II. Some Algebraic Topology. 2. Basic Metric Space Topology. 3. Forms and Curves. 4. The Jordan Curve Theorem. III. Deeper Topological Ideas. 5. Vector Spaces and Topology. 6. Locally Convex Spaces and Seminorms. 7. A New Look at Linear Functionals. 8. Deeper Results on Linear Functionals. 9. Stone-Weierstrass Results. IV. Topological Degree Theory. 10. Brouwer Degree Theory. 11 Leray-Schauder Degree. 12. Coincidence Degree. V. Manifolds. 13. Manifolds. 14. Smooth Functions on Manifolds. 15. The Global Structure of Manifolds. VI. Emerging Topologies. 16. Asynchronous Computation. 17. Signal Models and Autoimmune Disease. 18. Bar Code Computations in Consciousness Models. VII. Summing It All Up. 19. Summing It All Up. VIII. References. IX. Detailed Index.
I. Introduction. 1. Introduction. II. Some Algebraic Topology. 2. Basic Metric Space Topology. 3. Forms and Curves. 4. The Jordan Curve Theorem. III. Deeper Topological Ideas. 5. Vector Spaces and Topology. 6. Locally Convex Spaces and Seminorms. 7. A New Look at Linear Functionals. 8. Deeper Results on Linear Functionals. 9. Stone-Weierstrass Results. IV. Topological Degree Theory. 10. Brouwer Degree Theory. 11 Leray-Schauder Degree. 12. Coincidence Degree. V. Manifolds. 13. Manifolds. 14. Smooth Functions on Manifolds. 15. The Global Structure of Manifolds. VI. Emerging Topologies. 16. Asynchronous Computation. 17. Signal Models and Autoimmune Disease. 18. Bar Code Computations in Consciousness Models. VII. Summing It All Up. 19. Summing It All Up. VIII. References. IX. Detailed Index.
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