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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract
Suppose one has an amount of money M to be invested for a year at an annual interest rate I compounded continuously, i.e, total worth of the investment at the end of the year is the limit of what results from compounding 2, 4, 8, 16,…, in each instance at equal time intervals. During the year, the net worth is continually growing, the rate of earning at each time is proportional to the current value. As the year progresses, the value of the account changes, but the law governing earning does not change. This is an example of an autonomous system. One-parameter semigroups in the problems to follow deal with such autonomous systems.
J. W. Neuberger

Chapter 2. The Idea of a Semigroup

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Chapter 3. Translation Semigroups

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Chapter 4. Linear Continuous Semigroups

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Chapter 5. Strongly Continuous Linear Semigroups

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Chapter 6. An Application to the Heat Equation

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Chapter 7. Some Problems in Analysis

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Chapter 8. Combining Semigroups, Linear Continuous Case

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Chapter 9. Combining Semigroups, Nonlinear Continuous Case

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Chapter 10. Some Connections Between Resolvents and Linear Semigroups

Abstract
This group of problems may be thought of as a continuation of those in Chapter 5. A review of Chapter 5 might be in order.
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Chapter 11. Combining Semigroups, Strongly Continuous Linear Case

Without Abstract
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Chapter 12. Splitting Method, Numerics

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Chapter 13. Semigroups of Steepest Descent, Abstract Linear Case

Without Abstract
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Chapter 14. Semigroups of Steepest Descent for Differential Equations

Abstract
The first problem in this chapter seeks to make the point that for a given linear transformation A on a finite-dimensional space to itself, an adjoint for A depends on a choice of inner products, one for the domain space and one for the range space.
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Chapter 15. Numerics for Semigroups of Steepest Descent

Abstract
At first we work with some numerical problems. We use the same example as in Chapter 14 but in a discrete form.
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Chapter 16. Semigroups and Families of Sobolev Spaces

Without Abstract
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Chapter 17. Nonlinear Semigroups Studied by Linear Methods

Abstract
In this chapter, denote by X a separable complete metric space, that is to say, a Polish space.
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Chapter 18. Measures and Linear Extension of Nonlinear Semigroups

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Chapter 19. Local Semigroups and Lie Generators

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Chapter 20. Boundary (Supplementary) Conditions for Partial Differential Equations

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Chapter 21. Quasianalyticity and Semigroups

Abstract
This chapter contains a development for linear semigroups which is of interest in the theory of probability and other areas.
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Chapter 22. Continuous Newton’s Method and Semigroups

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Chapter 23. Generalized Semigroups Without Forward Uniqueness

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Chapter 24. Nonlinear Semigroups and Monotone Operators

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Chapter 25. Notes

Abstract
Semigroups, semiflows, semidynamical systems—all are the same. Once when I gave a talk at the University of Alaska in Fairbanks, I mentioned that I once heard that the Inuit had something like eighty different words for snow. One of my hosts countered that the Inuit had over a hundred words for snow.
J. W. Neuberger

Backmatter

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