Mathematics

Literature references and annotations by Dick Grune, dick@dickgrune.com.
Last update: Fri Dec 16 16:53:02 2022.

These references and annotations were originally intended for personal use and are presented here only in the hope that they may be useful to others. There is no claim to completeness or even correctness. Each annotation represents my understanding of the text at the moment I wrote the annotation.
No guarantees given; comments and content criticism welcome.

* Mark Ronan, Symmetry and the Monster, Oxford University Press, Oxford, 2006, pp. 255.
This is a history book about the history of research into group theory and the discovery of the "Monster", not a book about that Monster. The math has been simplified beyond recognition, and even after reading up on the subject in the Wikipedia and with a PhD in computer science, I could not make head or tail of it.
     The first problem is that the author does not make clear what he means by "a symmetry". We learn that the "zillions of symmetries" of the Rubik cube are "generated by 90 degree turns", which in the lines above are compared to "symmetry operators". This suggests that the 24 turns (4 on each of the 6 sides) are the operators and that the positions that can be achieved are the symmetries. But operators in a (mathematical) group have the property that the combination of two operators is again an operator in that group, so any configuration can be achieved with a single (compound) operator. So are all these operators "symmetries"? I find it confusing. Symmetries are also explained as permutations, but the relationship remains vague.
     A second problem is that the level of explanation is very uneven: the root sign is explained, but the j-function is written out without any explanation.
     We learn a lot about the people around the Monster but next to nothing about the Monster itself, except that it is 196,884-dimensional, but that's already on the cover. Does it have a geometric representation, like a cube? Or is it just a network of symbols? (Does a network of symbols have symmetries?) If it can be geometric,it must have sides. Are all sides the same length like in a cube or a dodecahedron? How big is it if the length of the shortest side is 1 unit? Answers to such questions would have made the Monster much more accessible.
     Perhaps the subject is too complicated to allow a popularized treatment, in which case sticking to just the history is OK. But it would have been nice to see an example or two of representatives of the simpler symmetry groups. Some examples are given, but they are not assigned to groups. And it would have been nice to be told to what position in the periodic table of symmetries Rubik's cube occupies, probably the most complicated symmetric object any of us can relate to.

* Marcus Du Sautoy, The Music of the Primes: Why an Unsolved Problem in Mathematics Matters, Harpercollins, 2003, pp. 335.
Mostly about the people involved in attacks on the Riemann hypothesis, and indeed supplying interesting biographies of them. The application of primes in cryptography is emphasized, justifying the second half of the title. The math is exceptionally shallow; modulo arithmetic is called "clock arithmetic".

* Julian Havil, Gamma -- Exploring Euler's Constant, Princeton Science Library, Princeton, 2003, pp. 266.
"Fun with Series" would probably be a better title, but within that realm the book indeed focuses on γ, the Gamma function, the harmonic series, etc., in 14 chapters. The book closes with two chapters on the distribution of primes and the Riemann zeta function. Two appendices, about Taylor expansions and Complex Function Theory, provide handy refresher courses on the subjects.
     Most chapters start in low gear but soon speed up; not all explanations are as clear as I'd hoped. The material is covered in quite reasonable depth, the most difficult results sketched only.

* M. Copi Irving, Carl Cohen, Introduction to Logic, Prentice Hall, Upper Saddle River, NJ, 1998, pp. 714.
Thorough, interesting, readable, good.

* Samuel D. Guttenplan, The Language of Logic, Basil Blackwell, Oxford, UK, 1987, pp. 336.
Pleasing introduction.

* William H. Press, Brian P. Flannery, Saul A. Teukolsky, William T. Vetterling, Numerical Recipes -- The Art of Scientific Computing, Cambridge Univ. Press, Cambrigde, England, 1986, pp. 818.
A much more amusing and easy-going account than one would expect, given the subject. Chapters on: linear algebraic equations, interpolation and extrapolation, integration of functions, evaluation fo functions, special functions (Gamma, Bessel, Jacobi, etc.), random numbers, sorting(!), root finding and non-linear sets of equations, minimization or maximization of functions, eigensystems, Fourier transform spectral functions, statistical description of data, modeling of data, integration of ordinary differential equations, two-point boundary-value problems, and partial differential equations.
With programs and program diskettes in Fortran and Pascal.

* H. M. Edwards, Riemann's Zeta Function, Dover, Mineola, NY., 1974, pp. 315.
Of considerable depth. The first chapter explains Riemann's famous 1859 paper "On the Number of Primes Below a Given Size", and the subsequent 11 chapters cover many famous papers and theorems based on Riemann's paper. Requires serious study.