© 2011

A Sequence of Problems on Semigroups


  • 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


Part of the Problem Books in Mathematics book series (PBM)

Table of contents

  1. Front Matter
    Pages i-vi
  2. J. W. Neuberger
    Pages 1-2
  3. J. W. Neuberger
    Pages 3-7
  4. J. W. Neuberger
    Pages 9-11
  5. J. W. Neuberger
    Pages 13-16
  6. J. W. Neuberger
    Pages 17-21
  7. J. W. Neuberger
    Pages 23-26
  8. J. W. Neuberger
    Pages 27-30
  9. J. W. Neuberger
    Pages 31-33
  10. J. W. Neuberger
    Pages 47-49
  11. J. W. Neuberger
    Pages 61-63
  12. J. W. Neuberger
    Pages 65-69
  13. J. W. Neuberger
    Pages 81-85

About this book


A Sequence of Problems on Semigroups consists of an arrangement of problems which are designed to develop a variety of aspects to understanding the area of one-parameter semigroups of operators. Written in the Socratic/Moore method, this is a problem book with neither the proofs nor the answers presented. To get the most out of the content requires high motivation to work out the exercises. However, the reader is given the opportunity to discover important developments of the subject and to quickly arrive at the point of independent research.

Many of the problems are not found easily in other books and they vary in level of difficulty. A few open research questions are also presented. The compactness of the volume and the reputation of the author lends this concise set of problems to be a 'classic' in the making. This text is highly recommended for use as supplementary material for three graduate level courses.


linear continuous semigroups nonlinear semigroups semigroups semigroups and application to the heat equation semigroups and probability

Authors and affiliations

  1. 1.Dept. Mathematics, MathematicsUniversity of North TexasDentonUSA

About the authors

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."

Bibliographic information

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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 Fijavž, 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)