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Miloslav Znojil
Reviewer number:
Item's zbl-Number:
DE 016 689 575
Aldrovandi, R.
Special matrices of mathematical physics: stochastic, circulant and Bell matrices.
Singapore: World Scientific, (ISBN 981-02-4708-7/hbk). xv, 323 p. (2001).
Primary Classification:
15A90Applications of matrix theory to physics
Secondary Classification:
15A51Stochastic matrices
37-01Instructional exposition textbooks, tutorial papers, etc.
81-01Instructional exposition textbooks, tutorial papers, etc.
81S30Phase space methods including Wigner distributions, etc.
82-01Instructional exposition textbooks, tutorial papers, etc.
53Z05Applications to physics

Despite its zoo-reminding title the book is aimed at a physicist.
Its material is centered around the Markov-chain models of glass
(part ``stochastic matrices"), around the Wigner-type quantization
on a finite lattice (part ``circulant matrices") and around
certain closed formulae emerging in the theory of gases (part
``Bell matrices"). The readership is even free to pick up any
third for a separate study. Indeed, the three thirds of the text
offer a more or less independent presentation of the material
concerning, roughly speaking, the overall questions of an
evolution in time, the underlying differential geometric
structures and some available combinatorial tools of analysis,
respectively. Even the style of the three amalgamated stories is
significantly adapted (e.g., via the relevance and frequency of
citations) to the specific needs of the separate sub-subjects. As
a result, the text as a whole proves extremely inspiring, both by
the enthusiasm of the author himself (especially in the middle
part concerning the differential geometry in the medium of the
discrete quantum mechanics) and by the maximally updated list of
the recent references.

Although the unifying principle of this triplet (= ``special
matrices of mathematical physics") is a bit formal (the thirds are
fairly independent, like the Julius Caesar's ancient Gallia), its
parts still stick together via the sharing of the underlying
principles of physics (especially statistical mechanics) as well
as via the mathematical language using linear algebra as a certain
``lingua franca" . Non-negligible unifying role is also played by
the contemporary changes in mathematical techniques. In this
setting one might emphasize, e.g., the quickly growing feasibility
of the computerized symbolic manipulations (part ``Bell matrices")
as well as of the numerical Fourier transformations (part
``circulant matrices") or iterative algorithms (part ``stochastic

Throughout all the text the author often indicates the existence
of a broader mathematical framework of many questions discussed
via their technical simplification. Thus, for example, it is a
pleasure to read compact comments concerning the deep role of the
braid group and Hopf algebras in mathematics (duality) as well as
in many branches of physics (a bunch of examples), with the reader
being offered just an absolute minimum of the necessary details.

One may summarize that the style forms one of the most gratifying
features of this book. Its user-friendly character is achieved via
a hierarchy of abstracts, step-by-step explanations and
pre-processing summaries (plus, do not forget, a compact
``formulary" at the end). All this makes the book easily
accessible precisely to its target group of readers (for proof and
contrast, you may pick up, say, the case of quantum groups and try
to read some original literature, e.g., ref. [55] by Drinfeld, in
order to understand what I mean).
Remarks to the editors:

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