newsletter - New Zealand Mathematical Society

possible, in a friendly manner, has resulted in a text which may be misleading in places.
For instance, the authors discuss in (9.4) the Sturm-Liouville problem. The main result from this
important area is, probably, the completeness of the system of eigenfunctions (Th.9.42). Here the easy
style of the authors brings the unwary reader to the edge of a dangerous swamp by failing to give
sufficient emphasis to the necessity for the function r to be strictly positive on the interval.
The last, but not the least of my remarks is about the list of references, You may find numerous classical
books in it (like Courant and Hilbert's "Methods of Mathematical Physics" or Titchmarsh's
"Eigenfunction expansions") but few of recent vintage. Presumably the recent books by Bartle and Lieb
in the AMS Graduate Studies in Mathematics series and a handful of publications by Kurzweil and others
on the Kurzweil-Henstock integral are too recent to have been included.
All the previous remarks were about the choice of the material and the style. One can also find some
shaky places in the proofs provided. Generally the statements (when proven) are proven accurately. But
the approaches are not systemized, the choice of them is done at random. Sometimes the proofs are too
discursive or too formal. For instance theorem 27.4 gives a condition for boundedness of variation in
terms of absolute continuity. The proof is long and formal. In fact, by defining the absolute continuity as a
uniform estimate , the authors
could use the triangle inequality to estimate the variation V a of the function on the lattice
uniform lattice,
by variation of the function on the product of lattices with the
Still, I guess that creation of a friendly, honest and attractive textbook (or rather a book of problems with
accompanying commentary) for modern students learning classical analysis at Stage 3, is POSSIBLE.
The book in question is a step in this direction but it demonstrates that any attempt to bring students to the
front-line of professional mathematics while avoiding the classical theorem of Analysis is a difficult task,
which needs very accurate placed and exact comments from the tutor, and can't be left entirely to a
written text like one referred to. So I would recommend this book for use by a qualified tutor who could
combine the friendly approach with the production of attractive vistas and exact criticism but I would
hesitate to recommend it to students for independent reading.
Boris Pavlov
The University of Auckland
THE PROBLEM OF INTEGRABLE DISCRETIZATION: HAMILTONIAN APPROACH
by Yuri B. Suris, Birkhäuser-Verlag, 2003, 1092 pp, EU 158.36. ISBN 3-7643-6995-7.
In one of the earliest and still one of the most striking and influential uses of computers in mathematics,
Enrico Fermi and Stanislaw Ulam decided in the early 1950s to use the then brand-new {\sc maniac}
computer at Los Alamos to study problems in physics. They sought a problem which was simple to state
but could not be solved by the existing mechanical computers, and settled on the nonlinear discrete string
(or "lattice'') described by the ODEs
= k(y i + 1 - 2 y i + y i - 1 ) + k a((y i + 1 - y i ) 2 - (y i - y i - 1 ) 2 )
with boundary conditions y 0 (t) = y N (t) = 0 and initial conditions y i (0) = sin(ip/N). Taking up the story in
Ulam's own words,
"Our problem turned out to have been felicitously chosen. The results were entirely
different qualitatively from what even Fermi, with his great knowledge of wave motions,
had expected. The original objective had been to see at what rate the energy of the string,
initially put into a single sine wave (the note was struck as one tone), would gradually
develop higher tones with the harmonics, and how the shape would finally become "a
mess'' both in the form of the string and in the way the energy was distributed among
higher and higher modes. Nothing of the sort happened. To our surprise the string started