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Introduction
3

with kinetic transport theory and Chapter 10 ‘closes the loop’ dealing with
modern PDEs and variational tools of image analysis and processing. We remark
that some of the techniques discussed in Chapter 10 were used to elaborate the
images shown in this book. Finally, Chapter 11 – being somewhat isolated from
a thematic point of view but not from the mathematical background – deals with
socio-economic modeling, also based on kinetic equations.
Finally, a comment on mathematical depth (or lack of it …) in the presen-
tation of the topics is in order. There is only a thin line between mathematical
superficiality and excess of mathematical detail for a project like this. The au-
thor has done his best to stay on this line, keeping in mind that the basic idea
in writing this book has been to TOUCH on certain topics in applied PDEs
and – in the best of all cases – to arouse the reader’s interest to go deeper in
certain directions of modern PDE research. For this, historic facts are presented,
references to the scientific literature are included in all Chapters (also these are
strongly biased by the author’s scientific background and taste, no completeness
can be expected …), important open problems are pointed out and also refer-
ences to related internet sites are given. The author is fully aware of the fact that
webpages are not set up for eternity, so it may very well be that some links will
not be in operation anymore when the reader tries them out at some later point
of time, although they were checked out thoroughly at the time of writing this
book. However, it seemed too much of an omission to forsake internet-based
information in the context of this book.