Five recent book reviews
This book is an introductory course on mathematical analysis, which focuses on the explanation of one of the most fundamental concepts of mathematical analysis, the concept of limits. The author is able to mix both styles relating informal language to mathematical language and giving proofs that are deep but easy to read and follow.
“Limits, limits everywhere” is not a usual popular book on Mathematics nor a traditional analysis textbook of definitions, theorems and proofs. The author is able to mix both styles relating informal language to mathematical language and giving proofs that are deep but easy to read and follow.
The book focuses on the explanation of one of the most fundamental concepts of mathematical analysis, the concept of limits. The book's first part deals with the concepts of integer, prime, rational and real numbers, inequalities, limits, bounded sequences, and infinity series. The second part explains the special numbers e, π, and γ, infinite products, continued fractions, Cantor's different types of infinities, and the constructions of the real numbers. Throughout the book, there are some brief history remarks that explain why and for what these mathematical tools are necessary.
This is a well-written book with a style that is easy to read and follow, which can be recommended for undergraduate students interested in finding out more about Mathematics.
This book gives an algorithm for computing coefficients of modular forms of level one in polynomial time.
After his retirement as a mathematics professor at the University of Boulder (CO) in 2006, Larry Baggett wrote this memoir telling how he managed, being blinded by an unfortunate accident at the age of five, to build up a professional career as a mathematician. It is an optimistic tongue-in-cheek account of how a blind person survives in a world of the sighted. He also displays his passion for mathematics and music, of which some slightly more technical excursions are added as occasional in-text-frames.
Five year old Lawrence W. Baggett accidentally cut himself (he claims it is a design error of Darwinian evolution that arranges bone lengths in a human arm such that if you cut something at eye level, the knife will end up in your own eye). By sympathetic ophthalmia, he lost sight in his other eye as well, which left him totally blind apart from some vague distinction between light and dark.
That was 1944, and in those days, mainstreaming people with some bodily malfunction was not as usual as it is today. However, his mother was very determined and could convince some teachers to take her son along in the class with the other kids. Similar things happened later when he continued his education. He was always lucky to be the first blind person who was accepted in a regular program. He finally got a PhD at the University of Washington in 1966 on unitary representations of compact groups. After that he was hired by the University of Boulder where he had 12 PhD students.
With the technology available to us today, such as computers, audio-books and TeX, we can imagine a blind person writing a paper, but this was not at all obvious forty years ago. At first Braille type-writers were very primitive, and even when they improved, there was no erase button. Hence it required a lot of re-typing. Some books were available in Braille, others were later audio-recorded. For reading a paper he was depending on somebody reading it out for him. He tells us many other things that a blind person has to deal with, like traveling, finding your way in an unfamiliar city, crossing streets with heavy traffic, finding an empty seat in the audience, using a self-service counter, or the use of public toilets, and obviously how to lecture in a theater for a group of sighted students. Baggett takes us along on this journey, embedded in a sauce of gratefulness and a lot of humour. His hilarious evocation for example of what may happen when an unsighted man in a public lavatory is looking for a free toilet booth or when he bumps into the rear of a peer (or is there an 'e' missing) looking for a free urinal. Just a quote to illustrate his tongue-in-cheek phrasing "It is known to people who do research on blindness that most blind men and women can't accurately walk a straight line, which I suppose explains why so many of us get drunk-driving tickets". Until late in life he avoids being exposed as a blind person and acts as sightedly as possible. That may be the reason that he never used a guide dog (at least he never mentions one in this memoir), although he came around to using a cane.
Baggett takes us along the successive stages of his personal life and of general history (e.g. the 'revolutionary sixties', the day JFK was shot, etc.). He tells about his escapades as a student, the "girly-thing" as a teenager, how he travelled to Sweden (partly inspired by the reputation of Swedish girls), how he met his first wife, and how he later remarried with his current wife, and how he got along after being elected as head of the math department in Boulder. It is remarkable how often the reader is almost forgetting that this is a blind person telling this story. He tells about the movies he has watched, the paintings he has seen in a museum, etc. but these words get a slightly different meaning, obviously.
Besides mathematics, music has been another lifelong passion of his. He has played in several bands as a youngster and continued to do so during his later career. Both of these passions are illustrated not only by his story, but there are several framed text blocks inserted that elaborate on some of the topics that he mentions. These are not really advanced, but discuss things like the number of possibilities in a set of 6-6 domino tiles (inspired by the 163 possibilities formed by a 6 by 2 dot matrix of the Braille alphabet), or the comma of Pythagoras, the sequence of musical chords, the limit of a sequence of numbers, the irrationality of √2, mathematical induction, etc. Sometimes he just challenges the reader giving a question of an IQ test he had to do: what is the next number in the sequence 6, 42, 7, 12, 48, 16, 18.
This account is indeed 'on the sunny side' (and this probably refers to the jazz standard song `On the sunny side of the street' performed by many of his much admired jazz heros). Apart from his accident as a child, Baggett may have been lucky at many other instances of his life, nevertheless this book testifies that, even 'in the dark' it is possible by using one's creativity and perseverance to achieve remarkable things in life.
This is a revised edition of previous version (2010) with adaptations for the new release of Python 3.0 (2008). It treads the classical subjects of an introductory course on numerical methods for engineering students with chunks of Python code implementing the algorithms. Python is just a working tool subordinate to the numerical techniques. So this should not be mistaken as a programming or a Python course.
Python is a general purpose programming language supporting object-oriented and structured programming. It is free, simple to use and implement, and well structured, and equally useful for non-numerical as for numerical applications. For these reasons it is sometimes chosen as a language for a first programming course. It is clear that in such case, a first course on numerical methods prefers using Python as a tool for implementing and testing the algorithms over an alternative such as MATLAB, or scilab, the open source MATLAB clone.
The present book is written in line with the previous observations. It is assumed that the student/reader is familiar with some programming language that is preferably, but not necessarily, Python. For those who do not know Python yet, a short introduction is given, and it as assumed that their programming skills suffice to learn the details along the way. Thus the book should not be considered as a course on Python. Only part of the language is used. For example, the only object that is used is an array. On the other hand, modules with many predefined functionality beyond the Python core such as numpy and matplotlib.pyplot are essential. Source code for the algorithms in the book are available for download at the CUP website.
After the introductory chapter on Python, the book follows the classical structure and items of a first numerical analysis course: Solution of linear systems, interpolation and curve fitting, roots of equations, numerical differentiation and integration, (ordinary) differential equations (initial value and boundary value problems), (symmetric) eigenvalue problems and (elementary) optimization. It is a true hands-on course where formal proofs are almost always omitted or just replaced by some motivating examples. There is an ample number of exercises, mostly numerical (i.e., not asking for a proof or a generalization or the derivation of a formula) and they are often taken from engineering applications.
Some peculiarities that struck me are the following.
+ There is no list of references. With proofs and more formal aspects missing, some students may be interested in further reading or a more advanced approach. They are left on their own, although these references hould be readily available in any library or even on the Internet.
+ There is no treatment of the mechanism of rounding errors in floating point computations and neither is error analysis or the numerical stability of a method formally included (except for stability and stiffness of ODE solvers). Remarks on rounding errors, and hence on stability, are downgraded to remarks sometimes with computing in double precision as a possible cure.
+ Polynomial interpolation is included in extension, even rational and spline interpolation are discussed, but not the choice of the interpolation points like adaptive interpolation or the use of Chebyshev points for a finite interval. A simple idea that may influence the error and the convergence considerably.
+ Speed of convergence for iterative methods is mentioned but not formally discussed.
+ Multivariate optimization is very concise as compared to the extensive discussion of less important issues. Discussion includes only a method of M. Powell and the simplex method (not to be confused with the simplex method of linear programming which is not included).
So we may conclude that this is a practical introduction, pushing the theory as far in the background as possible. There is a lot of material, probably far too much for a single course. Some sections that could be deleted for a shorter course are marked with an asterisk.
For those who are familiar with the second edition, the changes (apart from the necessary adaptation of the code to Python 3.0) include: an introduction to matplotlib.pyplot in chapter 1; an interpolating polynomial plot routine is added to the interpolation chapter; in the chapter on differential equations, the Taylor series method is dropped in favor of Euler's method; an improved implementation is given for the Jacobi method for eigenvalues as well as for the Runge-Kutta method for differential equations; and some new problems are added.
Mainly relying on two-dimensional transformations, the book is a mixture of proper mathematics explaining some aspects of Euclidean and non-Euclidean geometry, philosophical discussions on aesthetics, and all of this amply illustrated with many examples from mainly two-dimensional pieces of art, and one chapter on Bach's music. These examples illustrate many forms of symmetry, tessellations, hyperbolic and projective geometry, the use of perspective, other non Euclidean geometries, etc.
The cover states that this book grew out of a liberal arts course. Felipe Cucker is Chair Professor of Mathematics at the City University of Hong Kong. The result is a collection of chapters, some of which are just plain mathematics, others analyse the philosophical and psychological aspects of aesthetics, and many discuss a wide diversity of works of art and how the mathematics are recognized in their features (like symmetry or translation invariance for example) or how mathematics have influenced the techniques available to the artists (like perspective or hyperbolic geometry).
Almost all examples are two-dimensional, which is related to the mathematics that are covered. These are all geometric, treating transformations in the plane or different kind of projections and non-Euclidean geometry. So most art examples are graphical like paintings and drawings but also carpets, and occasionally poetry and dancing is mentioned. One exception is a complete chapter devoted to Bach's canons, but architecture and sculpture, which is clearly three-dimensional, is almost completely absent.
After an appetizing introduction showing symmetry and structure in work by Simone Martini (painting), John Milton (poetry) and Johan Sebastian Bach (music), the first chapter introduces geometry and its history from Euclid to Descartes and this is followed by a mathematical treatment of plane transforms: translation, rotation, reflection, glide, isometry, completely with definitions and proofs. Artistic examples illustrate the mathematics in another chapter where it is shown that there are exactly 7 friezes (translation invariant pattern in one direction) and 17 wallpapers (translation invariant pattern for two independent vectors). Pieces of art with planar symmetry are easily found. Tessellations and patterns in carpets from Central Asia, Chinese lattices and of course Escher's work.
Much more philosophical is an analysis of George D. Birkhoff's attempt to define a measure of aesthetics and Ernst H. Gombrich's sense of order. We often see symmetry when it is not really there. Da Vinci's Vitruvian man is not perfectly symmetric. A sense of beauty is raised by a balance disorder and boredom. This is illustrated with several examples from op-art, and for example repetitive work by Andy Warhol, but also from ballet performances and the rhyme and rhythm of poetry. Mathematics re-enter with homothecies, similarities, shears, strains and affinities and conics, which is illustrated by the use of the ellipse (a circle in perspective) in the Renaissance. More patterns are illustrated with musical canons, in particular the ones in J.S. Bach's Musical Offering.
The introduction of perspective in European paintings triggers some more mathematics introducing projective geometry and projections. This allows to produce proper representation of a reflection in a sphere, but also optical illusions based on false perspective. The rules of perspective are left with the start of cubism and modern art. The parallel in mathematics is that Euclidean geometry is left to introduce alternatives based on axiomatic systems and formal languages. In a final short chapter, Cucker ponders briefly on the geometry to describe our universe, but this requires to leave the two-dimensional world he has been discussing so far.
Rule-driven creation is moved to an appendix. Literature does not have the same geometrical basis as the other examples according to Cucker, yet he describes some patterns of constrained writing like anagrams, palindromes and other word plays.
This wonderful survey shows that, even though the author has restricted his approach mainly to two-dimensional geometry and transformations of the plane, it should be clear by now that this is still a very broad area when this is related to visual (and aural) art. From Euclidean geometry to Gödel's completeness theorem, from stone age artifacts to modern dance theater, from short biographies to quantitative aesthetics, the scope is enormous, forcing the author to be selective. The illustrations used are not always the ones that are best known though. So there is certainly something new to be discovered for every reader. The book grew out of a course, and so it is obviously possible to extract some interesting lectures from the material that is presented.