At this point, we are adding some 30 reviews a month to MAA Reviews. This column represents an idiosyncratic selection of some of the reviews that have appeared there (fairly) recently.
This is a text for a one-year
course in real analysis for graduate students (or for very good
undergraduates). Reviews that give tables of contents can justifiably be
called boring, but someone looking for a text for a course whose content is
not the same at all times and in all places needs to know what it contains:
(Riesz's approach is to use the order integral, measure, rather than measure, integral.)
The treatment is clear and thorough. Everything is here. The author includes exercises in the text that readers are expected to do, or at least think about, as they go along. There are extensive problem lists at the end of chapters, which is a way of guaranteeing that nothing will be left out. For example, after the last chapter there are one hundred and fifteen problems, including Egorov's Theorem, the Hölder and Minkowski inequalities, and a good deal of theoretical probability. This is a very rich book. Serious students would benefit greatly from it.
Traditional reviews start by giving the table of contents and end by noting two or three misprints that the alert reviewer has noticed. Well, I didn't see any misprints, nor any grammatical errors. The book is a pleasure to handle, physically and intellectually.
[Underwood Dudley; posted to MAA Reviews 04/11/2006]
As an avid problem solver with a
strong interest in inequalities, particularly algebraic and analytic
inequalities, I am delighted to supplement my repertoire with the
techniques illustrated in this volume. Geometric extremal problems such as
those treated in this volume are dense in the problem sections of many
journals (e.g., Crux Mathematicorum with Mathematical Mayhem).
The book contains hundreds of problems, classical and modern, all with hints or complete solutions. Solution methods include the use of geometrical transformations, algebraic inequalities (AGM, Cauchy-Schwarz,…), and calculus (Extreme Value Theorem, Fermat’s Theorem,…). The Glossary is very helpful, as is the bibliography of books and journal articles, although I was surprised to find Ivan Niven’s Maxima and Minima Without Calculus missing from the list of references.
Over the years, Titu Andreescu and various collaborators have used their experiences as teachers and as Olympiad coaches to produce a series of excellent problem solving manuals, as a quick MAA Reviews search on "Andreescu" will reveal. The present volume continues that tradition and should appeal to a wide audience ranging from advanced high school students to professional mathematicians. Here’s a sample of four problems that caught my eye:
Cut two nonintersecting circles from a triangle such that the sum of their areas is maximal.
Of all quadrilaterals inscribed in a given half-disk find the one of maximum area.
Consider n2 arbitrary points in a unit square. Show that there exists a broken line with vertices at these points whose length is not greater than 2n.
Show that one can cover a unit square by means of any finite collection of squares of total area 4.
[Problem 26 in D. J. Newman’s A Problem Seminar shows that the covering can be accomplished with a finite collection of total area 3 and that this is optimal.]
[Henry Ricardo; posted to MAA Reviews 03/19/2006]
OK, I'll admit it. I didn't think
I was going to like this book. But it surprised me. It is, in my opinion,
just the sort of thing its intended audience needs, and quite well
executed.
Most American mathematics departments offer a regular course in geometry, usually aimed mostly at future teachers. Given that students now arrive in college with very little geometrical knowledge, these courses have settled on a fairly standard pattern. First, one does a little synthetic geometry, following Euclid as modified by Hilbert, in more or less detail and at varying levels of rigor. Next comes some (still synthetic) non-Euclidean geometry, usually very lightly done. At that point, coordinates, vectors, and transformations can come in, which creates the opportunity to introduce various other kinds of geometry (especially projective) and/or to spend some time considering symmetries of the plane and related topics (Meyer does the latter). From there on, one is free to consider special topics; Meyer chooses to do a little bit of the theory of polyhedra.
All this is fairly standard, as is the provision, made through a web site, of software support (in this case, using Geometer's Sketchpad). What makes Meyer's book stand out are two things. First, he puts to good use his experience in industry (at Grumman Corporation, where he ran a robotics research program) in order to present applications that, while usually simple, seem real. This includes some fairly important (and non-classical) material, such as a discussion of Voronoi digarams.
The second is harder to pin down; I'd describe it as the book's "voice": a humane, intelligent, reflective way of discussing things that is quite interesting to read. Read his discussion of what axioms are, early in the first chapter, to see what I mean. If we can get students to read the book and think about what they read, they'll learn a lot from this book.
So: this may look fairly traditional (especially from the outside), but it's actually quite creative and very well done. Anyone teaching this kind of geometry course should consider adopting this book.
[Fernando Q. Gouvêa; posted to MAA Reviews 03/25/2006]
First held in 1959, the International Mathematical Olympiad, or IMO, is the premier international competition for talented high school mathematics students. Students from all over the world participate. For many of them, the experience is the gateway to a mathematical career.
This book collects statements and solutions of all of the problems ever set in the IMO, together with many of the problems proposed for the contest. Apart from its obvious historical and archival value, it serves as a vast repository of problems at the Olympiad level, useful both to students who want to prepare for the competition and to faculty looking for hard elementary problems. No library will want to be without a copy, nor will anyone involved in mathematics competitions at this level.
[Fernando Q. Gouvêa; posted to MAA Reviews 03/25/2006]
Publication Data
Basic
Real Analysis, by Houshang H. Sohrab. Birkhäuser, 2003. Hardcover,
559 pages, $59.95. ISBN 0-8176-4211-0.
Geometric Problems on Maxima and Minima, by Titu Andreescu, Oleg Mushkarov, and Luchezar Stoyanov. Birkhäuser, 2006. Paperback, 264 pages, $69.95. ISBN 0-8176-3517-3.
Geometry and Its Applications, 2nd edition, by Walter Meyer. Academic Press, 2006. Hardcover, 539 pages, $94.45. ISBN 0123694272.
The IMO Compendium: A Collection of Problems Suggested for the International Mathematical Olympiads: 1959-2004, by Dušan Djukić, Vladimir Janković, Ivan Matić, and Nikola Petrović. Springer Verlag, 2006. Hardcover, 740 pages, $79.95. ISBN 0387242996
Woody Dudley, whose training in real analysis dates back forty-seven years, can no longer do one hundred and fifteen problems but sees no reason why those younger and more vigorous shouldn't be required to.
Henry Ricardo (henry@mec.cuny.edu) is Professor of Mathematics at Medgar Evers college of The City University of New York and Secretary of the Metropolitan NY Section of the MAA. His book, A Modern Introduction to Differential Equations, was published by Houghton Mifflin in 2002.
Fernando Q. Gouvêa teaches at Colby College in Waterville, ME. He is the editor of FOCUS, FOCUS Online, and MAA Reviews. Somehow, he finds time to also review a book or two.
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Read This! is the MAA Online book review column. Contributions are welcome; contact the editor if you'd like to be one of our reviewers. Books for review should be sent to the editor: Fernando Gouv&ecric;a, Dept. of Mathematics, Colby College, Waterville, ME 04901. Publishers, please check our reviews information page.
MAA Online is edited by Fernando Q. Gouvêa (fqgouvea@colby.edu). Last modified: Sat Apr 22 19:30:25 Eastern Daylight Time 2006