This text consists of a sequence of problems which develop a variety of aspects in the field of semigroupsof operators. Many of the problems are not found easily in other books. Written in the Socratic/Moore method, this is a problem book without the answers presented. To get the most out of the content requires high motivation from the reader to work out the exercises. The reader is given the opportunity to discover important developments of the subject and to quickly arrive at the point of independent research. The compactness of the volume and the reputation of the author lends this consider set of problems to be a 'classic' in the making.
This text is highly recommended for us as supplementary material for 3 graduate level courses.
-Preface.-1. Introduction.-2. The idea of a semigroup.-3. Translation semigroups.-4. Linear continuous semigroups.-5.Strongly continuous linear semigroups.-6. An Application to the Heat Equation.-7. Some Problems in Analysis.-8.Semigroups of steepest descent.-9. Numerics of semigroups of steepest descent.-10. Nonlinear semigroups studied by linear methods.-11. Measures and linear extension of nonlinear semigroups.-12. Local semigroups and Lie generators.-13. Quasi-analyticity of semigroups.-14. Continuous Newton's method and semigroups-15. Generalized semigroups without forward uniqueness.-16. Semigroups of nonlinear contractions and monotone operators.-17. Notes.-18. References.
From the reviews:
"The book under review is a problem book, consisting of 448 problems ... . The final chapter contains extensive notes with comments on the problems, which throw some light to the background and give hints and references to the literature. ... For an advanced graduate student, the book should serve as valuable supplement ... as well as a starting point for independent research. Intricate research problems and up-to-date treatment make the text highly recommended reading even for experts in the field." (Marjeta Kramar Fijavz, Zentralblatt MATH, Vol. 1235, 2012)
"Solving the problems in the book is a very interesting enterprise, but the most beneficial thing is that these problems lead to very enlightening discussions and research problems. ... this is a very interesting book which can prove useful to graduate students and mathematicians ... . the Notes and the extended list of References are great invitations for more study and research." (Mihaela Poplicher, The Mathematical Association of America, April, 2012)
Über den Autor
John W. Neuberger is a Regents Professor at the University of North Texas, Denton, TX. He received his PhD at 22 from the University of Texas, completing both undergraduate and graduate work in 6 years. Neuberger has been a strong advocate of the Moore (Socratic) method of teaching during his long career in mathematics and is well respected in the fields of PDEs, numerical analysis, functional analysis, real variables, superconductivity, and algebraic geometry. His motto is: "when a man learns to teach himself, there is nothing more we can do for him."
Written in the Socratic/Moore method style and provision of references, enables the motivated student to arrive at the point of independent research
Student who works through the problems will have a range of introduction to aspects of one-parameter semigroups of transformations
Problems include wide applicability to probability and the Heat equation