Special matrices of mathematical physics: Stochastic, circulant and Bell matrices (Q2755075)

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. NEWLINENEWLINENEWLINEIndeed, 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. NEWLINENEWLINENEWLINEAlthough 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 matrices''). NEWLINENEWLINENEWLINEThroughout 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. NEWLINENEWLINENEWLINEOne 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., in order to understand what I mean).