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Briefly Noted

December 2005

On pp. 95-96 of T. S. Blyth’s Lattices and Ordered Algebraic Structures we encounter proofs of the following three results: in a distributive lattice every maximal ideal is prime and every proper ideal is the intersection of prime ideals; in a complemented lattice every prime ideal is maximal; and every Boolean algebra is isomorphic to the algebra of clopen subsets of a compact, totally disconnected Hausdorff space. The first two results conspire to show that in a Boolean algebra prime ideals and maximal ideals coincide (a marvelous algebraic result in its own right) even as it is also featured in the derivation of the third fact, a topological representation theorem due to none other than M. H. Stone.

Thus, a raison d’être for the study of lattices and ordered algebraic structures is the facility this subject imparts to demonstrating a certain class of results from algebra and topology. But aside from such contextual connections the subject is of course autonomous and imbued with its own unique characteristics. Consider, for example, the very last result in the book, a structure theorem (due to Janowitz in 1991) with a flavor all its own: the ordered monoid of residuated mappings on a bounded distributive lattice of finite length is regular if and only if the lattice is a vertical sum of lattices of the form M(k) where k≤2 and, by definition, M(k) is “the lattice formed by adding top and bottom elements to the discretely ordered set {1,2,…,k}.” For further explication of the jargon the reader is referred to the book. Suffice it to say that the preceding example is an indication of much of the thrust of this book.

More precisely, the focus of Lattices and Ordered Algebraic Structures falls on such things as Stone and Heyting algebras, ordered groups and ordered semigroups, with the notion of residuated mapping taking central stage much of the time. Indeed Blyth introduces these last-mentioned players already in the Introduction where he likens them to continuous functions in analysis. To wit, just as the inverse image of an open set under a continuous function is open, the inverse image of a principal down-set under a residuated mapping is again a principal down-set (all situated in the respective ordered sets). Again, for further explanation of the jargon, see the book.

Lattices and Ordered Algebraic Structures is extensive and scholarly, dense but accessible. There are a decent number of exercises and a great deal of interesting (if occasionally arcane) material is covered, well beyond what little I have indicated above.

Recommended!

[Michael Berg; posted to MAA Reviews 8/24/2005]

The Carus Monographs of the Mathematical Association of America are rich in memorable associations for a great many of us. I have a fond and vivid memory of scoring a copy of number sixteen in the series, Dedekind Sums, by Hans Rademacher and Emil Grosswald, at a Southern California MAA Section Meeting over ten years ago: even at that time, when I was perhaps not as inclined to reminiscences as I am now, the discovery of such a classic in its familiar deep-blue hard-cover format with the spare printing on the cover done in gold, surrounding the MAA Seal, triggered an irresistible feeling of nostalgia. I seized the book immediately.

The Introduction to Dedekind Sums was written by Ivan Niven, Chairman of the Committee on Publications of the MAA in 1972, the date of the appearance of no. 16, and himself the author of no. 11 in the Carus Monograph Series, Irrational Numbers, the book under present review. Niven first launched this wonderful book in 1956, and now, almost fifty years later, Irrational Numbers reappears in its fifth printing. Recent reprints in the Carus Series no longer sport the deep-blue hard cover, but come out in soft-cover format and varying color schemes: the fifth edition of Irrational Numbers has a cover the color of, well, mud. But all is well: once you crack the spine, “it’s déjà vu all over again.”

We encounter the original material verbatim, starting off with Niven’s elegant and succinct preface (all in the original type-face), carrying the author’s grateful acknowledgment of personal debts owed to Professors Olds and Zuckerman, and tantalizing allusions to Hardy and Wright (i.e. their legendary Introduction to the Theory of Numbers), Koksma, Perron, and Siegel. Accordingly it is crystal-clear that we are in for some very classical stuff, reminiscent (for those of us whose degrees are over twenty years old) of small, informal senior and graduate seminars run by mathematicians who themselves had made contributions to the subjects under consideration. In my own case, my undergraduate school counted Steinberg and Redheffer on its faculty, both of whom figure in Niven’s discussion of Theorem 9.1 (p. 117) on the linear independence of exponentials of distinct algebraic numbers (over the field of algebraic numbers), i.e. the generalized Lindemann Theorem. It is a spring-board for Niven’s proof, in a single paragraph (!) fourteen pages hence, that e and π are transcendental (and he proves a lot more besides!).

To be sure the book is fantastic and remains valuable even fifty years after its first appearance. It certainly qualifies (still) as a wonderful choice for a topics-in-number theory seminar or a tutorial or reading course. Individual chapters of Irrational Numbers already go a long way in this regard all by themselves. For example, the chapter on continued fractions is an outstanding succinct rendition of the subject’s foundations culminating in the theorem that a simple c.f. is periodic iff it represents a quadratic surd. Another example is had in the book’s culminating chapter, giving a complete treatment of the 1934 – 1935 resolution of Hilbert’s Seventh Problem by Gelfond and Schneider (independently). Here Niven notes that the proof he presents is based on Carl Ludwig Siegel’s presentation of the topic in his book (with Richard Bellman), Transcendental Numbers, which I recall very well indeed from a seminar given in the late 1970’s by Basil Gordon, David G. Cantor, and E. G. Straus (requiescat in pace): “the more things change, the more they stay the same…”

All the foregoing having been said, and asking the reader’s indulgence for my nostalgic excesses, I recommend Irrational Numbers in the strongest possible terms. It is the perfect introduction to the indicated subject and serves as a fine precursor to such texts as the aforementioned book by Siegel and Bellman, Alan Baker’s gorgeous but everywhere dense Transcendental Number Theory, and, for instance, the compendium, New Advances in Transcendence Theory (ca. 1988), edited by Baker. Not being up on the latest stuff in this beautiful part of number theory I do not have more recent sources to recommend, but there can be no doubt that, whatever they are, their reader will greatly benefit from a preliminary scrutiny of Niven’s monograph. Beyond this Irrational Numbers is simply a terrific book to read for its own sake.

[Michael Berg; posted to MAA Reviews 10/19/2005]

If I may, I should like to start this review of Volker Runde’s A Taste of Topology with an allusion to my earlier review of A Topological Apéritif, by Stephen Huggett and David Jordan, appearing in Read This!, on 22 April, 2004. In that review I stated that “[the Apéritif] would … play an important role in an introductory course as a supplementary text to one of the established players,” and I went on to single out Munkres’ standard text as a good choice for “the main course.” At the cost of taking word-play to altogether revolting heights, let me note now that if a somewhat more exotic and possibly a little lighter meal is desired the Apéritif might be followed by A Taste of Topology to comprise a very nice introductory course (or two) in topology: both books are easy to digest and leave the reader satisfied. (Having gotten this last pun out of my system, I will now get on with a more proper and specific consideration of Runde’s commendable book.)

A Taste of Topologygrew out of the author’s lecture notes for his 2004 senior-level course at the University of Alberta, in which he evidently tried to solve a pretty sticky pedagogical problem. “There is a very real danger,” says Runde in the preface to his book, “that students come out of a topology course believing that freely juggling with definitions and contrived examples is what mathematics — or at least topology — is all about.” Of course, as Runde goes on to observe, the logistical problem is that introductory topology offers particular difficulties because beginning students invariably lack the background and sophistication needed to fathom topology’s more “natural” (or at least canonical) examples, which often come from subjects not included in the usual undergraduate sequence.

Runde does something very interesting and, I think, very useful, in the book under review: he explicitly plays down what accordingly might to the beginner appear synthetic and artificial, and plays up material which resonates with other, already familiar, mathematical notions from, for example, analysis; and he goes well beyond what is ordinarily found in a first topology course. So it is, for instance, that in the chapter titled, “Systems of Continuous Functions,” a marvelous discussion of Urysohn’s Lemma is immediately followed by a treatment of Stone-Cech compactification and the Stone-Weierstrass Theorem. Runde gives “Silvio Machado’s short and elegant” proof of the complex form of the latter theorem.

Other perhaps unusual topics in the book include the Arzelà-Ascoli Theorem, which Runde characterizes as “the right substitute for the Heine-Borel theorem in spaces of continuous functions,” having shown early on that Heine-Borel breaks down outside Euclidean n-space. (He also appends a nice discussion of the general failure of Heine-Borel in infinite-dimensional spaces.) The highlight of this presentation is the proof of the equivalence of having a normed space be finite dimensional, of having its closed unit ball be compact, and of having each closed and bounded subset be compact. Appearing at the end of the book in the form of an appendix this elegant characterization is a nice encore to what came before.

I should also like to draw attention to my favorite chapter of the book, namely, Chapter 5, “Basic Algebraic Topology.” It is a very thorough treatment everything (appropriate at this level) from homotopy to covering spaces, with Brouwer’s fixed point theorem featured. The chapter (and the book itself) ends with an allusion to a space with a non-abelian fundamental group. Furthermore, this section also contains a truly amusing misprint: Poincaré’s Analysis Situs is given the publication date of 1985 (which may of course be evidence of the great man’s immortality).

A Taste of Topology is also rich in exercises of varying degrees of difficulty and contains excellent historical material, primarily contained in the sections labeled “Remarks” at the ends of all chapters. One particularly noteworthy historical aside occurs on p. 59 where Runde takes on Bourbaki. To wit, earlier, on p. 47, Bourbaki’s Mittag-Leffler theorem (as a prelude to Baire’s theorem that the intersection of a sequence of dense open sets in a complete metric space is again dense) is given an altogether accessible form: “Suppose that {X(n), d(n)} [n = 1, 2, 3, …] is a sequence of complete metric spaces, and let [each] f(n): X(n) → X(n-1) … be continuous with dense range. Then [the intersection of all the iterates] f(1)f(2)…f(n)(X(n)) [for all n] is dense in X(0).” Twelve pages later Runde notes that ‘[f]or good reason, our theorem … is somewhat less general than the result from Bourbaki. As Jean Esterle remarks … ‘Incidentally, the reader interested in the French way of writing a result as clear as [this one] in a form almost inaccessible to the [human] mind is referred to the [original] statement by Bourbaki …’” Manifestly Runde possesses the gifts of understatement and a light touch which contributes substantially to the book’s readability.

Coming back to my erstwhile review of A Topological Apéritif, I also mentioned there that I should certainly choose that book “at the very least as a supplementary text” the next time I am asked to teach topology. I will now amplify this by noting that I intend to use Runde’s A Taste of Topology as the main textbook for that course. I highly recommend the book.

Introductory courses in statistics range from the course dealing with conceptual understandings of statistics without the use of any formulas or derivations, to the most abstract mathematical statistics course. In the first type, students get very little understanding of the basis of statistics, while in the second type they get very little sense of applications of statistical theory. This book falls in the middle of these two extremes.

The book consists of two parts. The first part (217 pages) is a standard introduction to descriptive statistics, random variables and statistical inference. The second part (323 pages) takes up the linear model with one, two and several independent variables. There is much more material than can be comfortable covered in one semester, and had the work been presented in two volumes, students coming from a variety of introductory courses could have used the second part nicely alone.

The first test of a textbook is to examine the exercises. Here there are many, but several commit the cardinal sin of asking the students to compute something. Period. There is no final question about the meaning of what has been computed. The second test is to see whether the authors are up on current trends in statistical education. The StatEd section of the American Statistical Association is very active, and the international conference on statistical education held every four years has produced excellent proceedings. The book seems to fail the second test. For one thing, students now are much more visually stimulated by MTV and other videos, and the book would have benefited from a much livelier layout where definitions would stand out in boxes and uses of a variety of fonts, maybe even a dash of color here and there.

There are too many irritating moments. For example, why bother with the pdf of the chi-square distribution since that formula does not seem to be ever used anywhere, why not the F distribution also? The study of observational data is not an experiment. Using state data to regress the number of births on population size does not give a regression coefficient that tells us anything about what would happen if we changed the population of a state. Instead, the coefficient tells us that if we chose two states that differ by one population unit, then their births will on the average differ by the size of the regression coefficient. Why use Greek letters to denote population mean and variance and not for the population regression intercept and slope? Why not say that a significance level tells us how often we would get data from a population where the null hypothesis is true, instead of the probability of rejecting a true null hypothesis. “The sample mean is really a proportion…” (p. 159). I thought it was the other way around, with the proportion being a special case of the mean. Why not say that the way to interpret a confidence interval is if the study was repeated a large number of times, then 95% of the intervals would contain the true parameter value. Whether our particular interval belongs to the large set of many intervals that contain the parameter or the small set of intervals that do not contain the true parameter value is anyone’s guess. That way, students may actually have a chance to understand what is going on instead of turning the interval around and saying that we are 95% confident that our one interval contains the true population value. Whatever that means?

The second half is more enjoyable than the first. Others have written better introductions to statistics, while the second half is a very solid introduction to linear models. It includes nice discussions of regression diagnostics and model building, together with optional sections on the use of linear algebra. But why miss the opportunity of representing the three sums of squares by a right triangle with the length of the hypotenuse as the square root of the total sum of square and the lengths of the other sides as the square roots of the regression and the residual sums of squares and thereby the correlation coefficient as the cosine of an angle?

[Gudmund R. Iversen; posted to MAA Reviews on 10/19/2005]

Publication Data

Irrational Numbers, by Ivan Niven. Carus Mathematical Monographs 11. Mathematical Association of America, 2005. Paperback, 164 pp. $39.95. ISBN 0-88385-038-9.

A Taste of Topology, by Volker Runde. Universitext Series. Springer-Verlag, 2005. Paperback, 176 pp. $39.95. ISBN 0-387-25790-X.

Introduction to Linear Models and Statistical Inference, by Steven J. Janke and Frederick C. Tinsley. Wiley, 2005. Hardcover, 584 pp., $89.95. ISBN 0-471-66259-3.

Michael Berg is Professor of Mathematics at Loyola Marymount University in California.

Gudmund R. Iversen holds a PhD in statistics from Harvard University and is Professor Emeritus of Statistics at Swarthmore College where he taught statistics for many years.

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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.