The fields of $p$-adic numbers have long been important in number theory; in particular the use of the $p$-adics allows ``localisation'' to primes $p$, simplifying the study of the ``global'' rational integers. Another important aspect of these fields is that they allow analytical tools to be applied to algebraic areas such as prime numbers and diophantine equations.
Gouvea's new book $p$-adic Numbers offers an interesting approach to the study of these numbers. The areas covered are the ones you would expect to find in a book on this subject which is trying reach advanced undergraduates and honors level students, beginning with congruences modulo $p^n$; then defining absolute values on and completions of fields, leading to the definition of $p$-adic numbers; continuing with some of the ideas of local and global; and finally working through to normed vector spaces and the associated analytic concepts. These topics are carefully presented with basic and easy to follow concepts in the first chapter building up to the more technical aspects in later chapters. The typical section will consist of cycles of three or four paragraphs followed by a couple of exercises inviting the reader to immediately deal with the ideas in the paragraphs. Many concrete examples are scattered throughout. There is an appendix with 50 pages of hints relating to almost every exercise in the text; these hints are not solutions but instead are suggestions which try to get the student started the right direction toward successfully understanding the problem. The book is unusual in that the writing style is casual and much less dry than the normal mathematical text. As a result many students may find it more approachable and indeed more enjoyable to read.
Although the use of the $p$-adics can be quite complicated, the structure of fields of $p$-adic numbers is not, allowing their study to be accessible to students without too much background. However, exposure to rings and fields and some familiarity with infinite series are crucial. The structure of the book, and especially the opportunity for hands-on experience via the exercises, will certainly strengthen understanding of most of those concepts plus many others. Since the $p$-adic numbers touch many different areas of mathematics, their study gives the student an unusual opportunity, especially at such a low level, to see some of the important connections between diverse areas of mathematics. The University of Newcastle
This is the best book on mathematical modelling I have ever come across. I use ``best'' in the sense that, for me, it contains so much useful knowledge but it is explained using an educational style that we all should try to emulate in our lecture courses.
The authors, one of whom passed away in 1992, have been lecturing and researching at the University of Western Australia for over thirty years, and have successfully provided us with a mathematical modelling book which projects their ways of thinking mathematically, physically and intuitively, when confronted with various problems. The problems in this book are well-chosen in that they illustrate most of the relevant features of an applied mathematician's toolbox. Of particular interest are some problems associated with the CSIRO Mathematics-in-Industry Study Group held each year at various venues.
Right from the beginning the reader is introduced to a great summary of modelling, including the importance of mathematics, computation, expewriments and commonsense modelling skills.
The mathematics covered in the book ranges from scaling and dimensional techniques, approximations, dynamics, statistics, stability, applied linear algebra, finite element methods, calculus of variations, variational techniques, differential equations, uniqueness and existence of solutions, similarity solutions, Lie group algebra, the method of images, the boundary integral method, Fourier techniques, inverse problems, integral equations, eigenvalues, resonance, non-linear mathematics, perturbation methods through to shock waves. The techniques used work sometimes, and do not work in other situations. The reader is told why, and what to look for! The authors also suggest when to stop analysing mathematically and start the numerical work. The ``apprenticeship approach'' is used throughout, and students will love it!
The MAPLE computer-algebra package is advocated at various stages, as this is the most common package used in Australian universities. Those who use MATHEMATICA will have no difficulty in modifying the programmes.
I love the style. The reader is led up a particular path by prior knowledge, just as our mind leads us when we are researching a problem or studying a technique for the first time. Then the difficulties are pointed out, and the experience of the authors advises us when to proceed with caution and why. There are lots and lots of useful asides, comments and exercises throughout. The latter are provided with elucidating comments and hints, and would be great for a third year or honours course.
The problems covered include table stability, ship moorings, cooking, the Greenhouse effect, continuous sheet casting, car vibrations and traffic flow.
It is emphasised that approximate models are sometimes better in a practical sense than the exact model, and that some models which appear irrelevant or incorrect at first glance are in fact very revealing when extra effects are shown to be needed in the final model.
The book is good value at $44 (paperback), and is a ``must have'' for every mathematical modeller's bookshelf.
The two authors' wide mathematical knowledge and experience is evident throughout, and their ability to transfer this in terms that are understandable to final year undergraduates is a joy to behold. The students at the University of Western Australia have clearly benefitted from Mahony and Fowkes' efforts for many years. Now we can all benefit. Bond University
This is a fascinating book - well I find it fascinating; but I am a Go player, as well as a mathematician with an interest in the mathematics of games and particularly in John Conway's theory of numbers/games. I rather think you may have to fit into all of those categories to get much out of this work (but then what mathematician could not be intrigued by Conway's astonishing construction of ``everything from nothing'' [2] [3]?).
Go is a game which certainly appeals to mathematicians, and indeed the impetus for the spread of Go in the western world in the last 30 years has been very largely due to its popularity in university mathematics and computer science departments. The appeal of Go to the mathematically oriented is usually explained by the logicality and simplicity of its rules (of which more later!); it has been said that if there are intelligent beings somewhere else in the galaxy who play board games, they may have a game vaguely similar to Chess with quite different pieces and rules, but they probably play a game which is almost exactly the same as Go. The logical nature of the game strongly suggests that at least some aspects of the game may be susceptible to mathematical analysis, deducing correct lines of play from the rules themselves rather than mere exhaustive tree searching. When John von Neumann proposed Chess as one of the basic problems for computer study, he did so in the hope that we would learn something about how the human brain works and also something new about how to play Chess: in the event the recent strong Chess playing programs tell us nothing about the former, and very little about the latter - only in the endgame has there been any contribution to Chess theory. But the endgame in Chess is precisely the part of the game that most resembles Go in the kind of strategic analysis required.
Nevertheless, I did not really expect to learn anything about Go when I picked up this book. I was quite wrong. On page 3, a Go problem is presented to the reader, apparently a simple matter of choosing the better of two possible moves. The particular problem is artificial, in that it would be extremely unlikely to occur in a real game; but the principle is clearly one that does occur very frequently in actual play. No Go player to whom I have shown the problem has chosen the correct solution; most players believe that the two moves must be of exactly equal value, and only choose one of them (the wrong one, for a spurious though plausible reason) because they are told there is a unique answer. The authors' mathematical demonstration of the correct move is utterly convincing (given only a modicum of knowledge of game theory); they then demonstrate that the difference in value of the two moves is actually a true infinitesimal, which means that one player can gain one point!
A one-point gain can mean the difference between winning and losing a game, particularly between Go players at the highest level. Does this mean that all Go professionals are going to have to take up university level mathematics, and in particular Conway's strange ``surreal numbers'', in order to remain competitive? Unfortunately I think not. The kind of position to which the analysis can effectively be applied is very restrictive: essentially it must be an independent (non-interacting) sum of ``simple'' sub-positions, each of which is trivial to analyse completely because for example there is only one move available to each player at every stage of its individual game tree. The only decision to be made therefore is which order to play the sub-positions, and the winner, under best play, is the player who makes the last move of non-zero value. Such positions are then amenable to the kind of analysis used by Conway and Berlekamp [1] for such games as Nim and Hackenbush; but equally they have about as much enduring interest to the serious game player as those mathematically interesting but rather boring-to-play games. Nevertheless these positions do occur at the end of most Go games; in a small percentage of games they are sufficient complex that attention to the analysis given in this book could produce a one point gain over less perfect play; and one point is not infrequently sufficient to win a game. Thus strong Go players would do well to study the book's summary tables of how to play various combinations of endgame positions; the fact that a number of the ad hoc rules provided are quite counter-intuitive may lead them to delve into the mathematical justifications.
Assigning a numeric value to a game position is a basic aim of mathematical and computer analysis of two-player games. A perfect evaluation would enable a perfect minimax strategy; for more complex games which cannot be solved completely, an heuristic evaluation is sought. For Chess a good heuristic clearly weights heavily the number and importance of one's remaining pieces but how to assign a value to positional advantage is not clear. Go has a ``natural'' evaluation function, namely an estimate of the number of points of ``territory'' for each side, and human Go players in fact are ``counting'' throughout the game. As with Chess, the difficulty of the game, the subject of much of Go theory, and the origin of different styles of play, arise from the impossibility of accurately estimating positional advantage, in particular the relative values of safe territory versus potential for expansion and attack. However late in the game territories are well-defined; by the stage of the positions covered in this book, the point value of each move can be made with total accuracy. What the authors show is that an integer value is not adequate to determine the correct order of moves; indeed Conway's very definition of a game can be thought of as an evaluating ``number''.
A game is defined to be an ordered pair $\{L | R\}$ of sets of games $L$ and $R$ , which may be empty. The interpretation is that if it is player Left's turn to move, she may move to any game in the set $L$ , while if it is player Right who is to move, he may move to any game in the set $R$ ; a player loses if they cannot move, ie, their set of options is empty. Never mind the irregularity of the definition (which is deliberate - to produce rabbits out of hats, magicians need some suspension of disbelief), there certainly exists a game $\{ \; | \; \}$ with no moves for either player. This game is denoted 0 ; it is a loss for the first player. (Note that the definition of a game does not include which player Left or Right has the move.) There is also a game $\{0 | \; \}$ which is denoted 1 , which is a win for Left, no matter who plays first (if Right has first move, he loses immediately as there no available move; if Left has the first move, she moves to game 0 which is loss for the player who has to move). Similarly there are games $2 = \{1 |\; \}$, ${1 \over 2} = \{0 | 1\}$ etc. which are wins for Left, and games $-1 = \{ \; | 0\}$, $-17{1\over4} = \{-17{1\over2} | -17\}$ etc which are wins for Right.
The analogy with Dedekind cuts is unmistakeable: since the sets $L$ and $R$ may be infinite, we have no trouble defining games corresponding to any real number. Moreover Conway also defined equivalence, comparison, addition and subtraction of games so that we may indeed think of equivalence classes of games as numbers:
But there other kinds of games than just real numbers, since $L$ and $R$ are unrestricted. Of course it easy to define infinities and infinitesimals using the set of all positive games. A more bizarre ``number'' is $\star = \{0 | 0\}$ (the number between 0 and 0 ?); as a game it is of course quite easy to understand: either player playing first has a single move to the game 0 . This game actually occurs as the value of many of the Go positions described in this book; $\star$ is in fact not comparable with 0 ; it is a win for the first player. Stranger still is ``up'' $\uparrow = \{0 | \star\}$ ; this game is comparable with 0 \- it is positive (a win for Left) but is infinitesimal, that is, less than every positive real number! Other surreal values which occur are double up, tinies and minies.
A typical value for the simple positions studied in the book is $1+\uparrow$ , which is infinitesimally greater than 1 ; if Left plays first, she may convert the position to value 2, whereas Right can convert the position to value 1 . Positions which are sums of such simple positions are then easily analysed using a minimax strategy in the arithmetic of these Conway numbers. A further technique, called ``chilling'', is introduced to include more complex positions. This notion would no doubt be claimed by the authors as their major mathematical contribution to mathematical game theory. Unfortunately while a rigorous mathematical definition of chilling is given (in the appendix), the term is used in a somewhat vague sense throughout the book. Perhaps this was deemed necessary for the non-mathematical reader; but I am still rather mystified as to its exact significance in producing the ``simplification'' claimed - the temptation I suppose is to read this as a Go book and not put in the study required for a thorough understanding of the mathematics.
Fully a quarter of the book is given over to an exposition of various sets of rules for Go and their mathematical modelling; it is interesting to speculate which of the authors wrote this section, as the style is dramatically different and perhaps more familiar to the mathematical reader. Here again there is much interest for both Go players and mathematicians. There is a sense of course in which any game can be defined by rules in which the last player to move wins, a basic requirement for the application of the Conway-Berlekamp analysis; but this is not usually a helpful comment (although the ancient Chinese rules for Go come very close to using this rule to define the winner). Go is admired for the simplicity of its rules; but in fact there is a real difficulty in defining the end of the game (in a manner satisfactory for use in computer programs for example), and in resolving the definition of territory in certain ambiguous positions. The Chinese rules differ slightly from the traditional Japanese rules. The Taiwan, American and New Zealand Go Associations have also proposed rule sets; the New Zealand rules are particularly pithy and elegant, being written by mathematicians! However most players throughout the world supposedly follow the rules of the Japanese professionals; the old Japanese rules required dozens of pages, were commonly regarded as unsatisfactory and incomplete (!) and were out of print for decades. They have recently been revised, but the new regulations are quite controversial. In fact Berlekamp and Wolfe show that alone among the above sets of rules, the Japanese version cannot be modelled in a mathematically consistent way!
There is a short chapter on possible future extensions of the analysis to more complicated parts of the game, including kos (which confound the concept of independent positions), life and death, and approximate results. There is also a mention of NP-hardness results. A chapter of Go problems of the type analysed in the main section of book is also provided; it is claimed that some of these problems could not be solved by top professional Go players. A computer program is available which implements the analysis to solve these problems.
Inevitably the book tends to fall between two stools - from the mathematician's point of view, the mathematical content and indeed the contribution to mathematics is perhaps insufficient to justify a book of this size (although it contains all the necessary definitions and statements of theorems), and at the same time it is rather too abstruse for most non-mathematically inclined Go players to follow and considers Go problems of only marginal significance. Personally however I think it's great!
$f(z)$ is an interactive graphing program designed to aid in the study of functions of a complex variable. It allows one to display the domain and range of a complex function in separate windows, and then explore the function by studying the images in the second window of given configurations in the first window. A pleasing feature is that this can be done in ``animation'' mode: one can actually watch a line turn into a circle, or an arbitrary squiggle turn into its image. In addition to being able to draw on the complex plane $f(z)$ can draw on the Riemann sphere, and also form fairly general two-dimensional projections of the (four-dimensional) graph of a complex function.
The program recognises the elementary functions and the beta, gamma and Riemann zeta functions, and can form the myriad combinations of these via the algebraic operations and composition. There are also facilities for constructing Julia sets and Mandelbrot sets, which include the feature ``zoom during animation''.
The earliest review I could find of $f(z)$ was in the July/August 1989 issue of the Notices of the American Mathematical Society . There was another review in the October 1992 issue. I refer the interested reader to both of these for descriptions and some critical comments of a specialised nature. The 6th version of $f(z)$, which I was given, boasts ``many enhancements and improvements'' over earlier ones, but as far as I can discern, does not incorporate any significantly new ideas.
Unlike many users of $f(z)$ who might be proficient at Windows but not so proficient at Complex Analysis, I am a novice at Windows. I know how to use e-mail, have had some experience with Maple, knew how to write a Fortran program 20 years ago, am fairly computer-illiterate; but have worked in and taught Complex Analysis for about 20 years. A positive thing $f(z)$ did for me was to provide a wonderful opportunity to get acquainted with Windows, a superb tool so enormously useful in so many other situations!
So what can we say about this example of ``Sesame Street'' Mathematics? After we've had our bit of fun, not much. Here is why.
Complex Analysis proper begins with the Cauchy Integral Formula (CIF); what precedes it is comparatively straightforward. (Even in the pre-CIF stages of the subject care ought to be taken over the definitions of the elementary functions, but rarely is: the damage has already been done in first year with a sleight-of-hand treatment of the circular functions.) There is a beautiful logical coherence in the development of the subject, and the consequences of the CIF are themselves of remarkable elegance and far-reaching significance in both Complex Analysis itself and in several related areas of both pure and applied mathematics.
Sadly, students are, on the whole, unprepared for the intellectual sophistication of the subject. School teachers and, to a lesser extent, university teachers have betrayed their calling to present mathematics as an intellectual pursuit, in favour of quick results and misleading anxiety about ``relevance''. Even at the more basic level of calculation and routine algebraic manipulation, instead of solid foundation we often find a quicksand of carelessness and shoddiness.
$f(z)$, the program being reviewed, does nothing to remedy the situation. If anything it makes it worse. The overwhelming impression being given is ``OK you can't think, so visualize''. There is no mention anywhere of the Cauchy Integral Formula, making the title The Complex Variables Program the grounds for a possible complaint of misleading advertising. In fairness, I should say that there is mention of the Cauchy Theorem and we are invited to check visually that the image of an indefinite integral is a simple closed curve beginning at the origin and ending somewhere near it! (We are asked to allow for the fact that the integrals are only approximated by Riemann sums; but in any case all the curves that appear look as though they have been drawn by someone with advanced symptoms of delirium tremens.) It may not be entirely appropriate to quibble here about assertions made in the User's Manual, so I confine my attention to ``A series is an infinite sum where the terms are described by some simple pattern''.
How simple? What about the MacLaurin series of $\tan$? What about the MacLaurin series with $a(n)= n$ th digit in the decimal expansion of $\pi$? At least the next sentence is reassuring: ``$f(z)$ will add finitely many terms''. Sophistication?
By contrast Maple is of never-ending usefulness, but then of course the aims are different. Even for Julia and Mandelbrot sets the $f(z)$ User's Manual freely admits the superiority of other programs.
I suppose $f(z)$ offers lots of fun. But for the student who uses it as an aid in Complex Analysis it is likely to be little more than a distraction. One day is useful, more than one may lead to addiction. Architects are more likely to enjoy the animated rotations of axes in the graph projections. University of New England