eon

Posted by tpc at May 29th, 2010

Picked up a copy of Lewis Carroll in Numberland by Robin Wilson from the library. I must admit that I browsed through it instead of reading it, picking up bits and pieces that I find interesting. For example, it is told of how Queen Victoria was charmed by Alice’s Adventures in Wonderland that she demanded:
Send me the next book Mr Carroll produces …
And the next book that arrived was … An Elementary Treatise on Determinants.

Ok, the story is not true but how cool would it be if it had been. Another fun tidbit is the following game. Starting from the number 1, A and B take turns adding a number from 1 to 10 to the running total. Whoever gets to 100 wins. What is a winning strategy? Working backwards, for B to be certain of winning, he would need to get to 89. That way, A can’t win with one turn but whatever number he picked, B can go for the win. Inductively, to get to 89, B needs to first get to 78, 67, 56, 45, 34, 23, 12 — steps of 11.

Posted by tpc at May 24th, 2010

Martin Gardner passed away last week on 22 May, aged 95. Wikipedia is a good place to read about his contribution in bringing mathematics to the public. My favourite article of Gardner’s is Six Sensational Discoveries that Somehow or Another have Escaped Public Attention, Sci. Amer. 232, 127-131, Apr. 1975. (Also published in Time Travel and Other Mathematical Bewilderments.) Inside, Gardner announces six discoveries among which a counter-example to the four colour theorem. Before you jump off your seat, the article was dated 1st April 1975. Yes, it’s another very clever hoax.

The best among the six is the claim that

is exactly an integer and this fact was found by Ramanujan. The attribution to Ramanujan was clever not because of Ramanujan’s remarkable prowess of calculation but that constant is actually an evaluation of the modular j-invariant

and of course has class number one.

Posted by tpc at May 5th, 2010

is a book by Niels Lauritzen that I just checked out. My initial impression is that it is well written and contains many interesting gems. It certainly looked like a good book to teach from, although the topics covered are a little broad and thus I suspect not in enough details for a student struggling to learn abstract algebra.

One of the gems was how a computational number theorist, Thomas R Nicely, nicely found a flaw in Pentium’s floating point unit. The book provided this link to the discoverer’s site and as usual, wikipedia has a nice coverage.

Other gems include Sam Loyd’s 15-puzzle and of course the last chapter on Grobner Bases.

Posted by tpc at February 28th, 2010

while a rose by any other name may smell as sweet, a mathematician might just be discriminated by his/her name. The perceived ability of a mathematician (as well as most academics) is by publications. There are two norms in listing authorship of joint work. One adopted by mathematics and economics is to list the authors alphabetically, regardless of amount of contribution. On the other hand, for disciplines like psychology and biology, the norm is to list the author in terms of contributions. It’s an interesting phenomenon, because I’ve seen biology papers with many authors. It seems that if you are in a lab which publishes 10 papers, chances are high that you get your name in every one of them, maybe as the 7th author. Perhaps, I can get my name as the 8th author by just walking past the lab! In a sense, as long as the main author’s name is first, he probably wouldn’t mind sharing credit with people, maybe in the hope that the favour is reciprocated.

I must say both practices have their pros and cons. Suppose you are a young person writing with an established guy, people would tend to think the established person did the work, unless you put your name first. On the other hand, a co-authorship might result in A providing an idea and B doing all the work. Now, who should deserve more credit? The one who did the work? Not really. A good idea is usually hard to come by. Plus it can be touchy to always have to establish who should get main credit. I suspect the relationship wouldn’t last or at least be as cordial. Compare that to the Hardy-Littlewood rules for collaboration.

I’m not one to want to upset the established order, so I stick to the alphabetical routine. As my surname is in the last quartile of the alphabet, I’m always the second or third author. I do have this nagging suspicion that this might have impact on how people perceive your contribution. This was confirmed when I read superfreakonomics. That book mentioned the following paper in Economics that concluded that there is some bias towards academics with names arriving later in the alphabet. Here’s the reference if you are interested.

What’s in a Surname? The Effects of Surname Initials on Academic Success, Liran Einav and Leeat Yariv, Journal of Economic Perspectives - Volume 20, Number 1—Winter 2006—Pages 175–188.

One main point is that tenured faculty in the top 5 economics departments in the US have surnames arriving earlier in the alphabet than junior faculty. But the bias disappear as the data is expanded to the top 20 or 35 departments. So perhaps the impact of that bias is not as great as things like race, gender and other forms of discrimination.

Posted by tpc at April 17th, 2009

A classical and computational introduction is a new book by L.J.P. Kilford. New enough that it even has reference to the resolution of Serre’s conjecture. But this book is really an introduction to the classical aspects of the theory of modular forms and it does a great job. I took a few days to read through the book (of course, ignoring the details and proofs) and I would say it is very enjoyable. Kilford adds in lots of funny and quirky anecdotes, most of which I’ve read from different places but it’s nice to have everything collected in one book. For example, he mentioned Lang’s famous foreword:

“It is possible to write endlessly on elliptic curves. (This is not a threat.)”

I remembered being so tickled when I first saw this in Lang’s book.

Back to this book. Even with the subject of modular forms it is the same. He knows he can’t possibly explain everything and so is not afraid to be a little vague at times and cite the various references to where more in depth discussions can be found. Thus, he is able to accomplish much in this modest sized (200+ pages) book. My one small complaint is the title should not include “and computational”. I found the last chapter on computational aspects too brief. He highlighted some history, discussed MAGMA and SAGE, giving some examples of the codes used, but I believe that this is not enough for someone interested in computing modular forms to get started on. And the appendixes on MAGMA and SAGE codes are each one page long with two longish lines of commands.

Posted by tpc at April 11th, 2009

by Yoko Ogawa. I remember watching half a movie on a plane which was called “The Gift of Numbers” (Hakase No Aishita Sushiki) and was based on the 2004 Japanese novel by Ogawa. Since then, I place the book in my to read list and it so happened that two weeks ago, I read a review of the book on the Sunday times. Apparently, the english translation just appeared under this new title and it turned out fortuitously that the library carried this book.

The story of the book is about how the housekeeper was assigned to this number theory professor who had an accident, lost his short term memory and only retained 80 mins of memory. To remind himself, he pinned a note on his clothes that says “I have only 80 minutes of memory.” It’s an interesting proposition to say the least, but what is engaging is the relationship developed between the professor, the housekeeper and her son called Root because he has a flat head. There was only a very brief and subtle exploration into the anguish of the professor at his state.

There are quite some mathematics mentioned by the professor, for example, on amicable numbers, Ruth-Aaron numbers 714 and 715, as well as the perfect number 28 which happened to be the shirt number of the professor’s favourite baseball player Yutaka Enatsu.

For another review see Math fiction.

Posted by tpc at January 24th, 2009

I’ve recently came across several good books that are recently published by PUP and I’m further impressed by that fact that their books are usually cheaper than Springer and significantly cheaper than Oxford. I really liked the Fourier Analysis and the Complex Analysis titles in the four part series by Stein and Shakarchi. Another book that I really liked was Google’s PageRank and Beyond by Langville and Meyer.

They also published quite a number of general maths book. One of which I’ve wrote about. It’s not always hits. One that I didn’t like was A Certain Ambiguity by Suri and Bal which is a fiction. It started out promising, a young Indian studying in stanford saw a reprint of a paper by his grandfather (who was based in India) with the footnote that ideas came to him during his term in prison in the US. So he set out to discover how his grandfather was incarcerated. Then the story unravels in two lines, a transcript of conversations between his grandfather and a judge, plus a parallel line where the protagonist learns mathematics in a class. There is also a third fictional line in the form of journals of famous mathematicians. Much of the action is actually discussion on mathematics focusing on geometry and the 5th postulate, and also ideas on logic, truth, Cantor’s theory of infinity. Somehow, I just don’t find the book engaging at all and I particularly dislike it when authors take liberties in fictionalizing real mathematicians.

Isn’t it funny how I have so much to say about books I don’t like and not much to say about those I recommend.

Posted by tpc at January 17th, 2009

The tour de force by E.T. Whittaker and G.N. Watson. The Math Reviews says that the 1996 reprint of the 1927 fourth edition has 608 citations! It’s certainly a magnificent book and worthwhile to have on your shelf.

The 1996 cambridge version on amazon is listed at USD$94. But I just found a new version from merchant books also listed at amazon for USD$25! The details state that the new version has 568 pages vs the 616 pages of the original. So are these two the same? Obviously if I refer to page 408, it would be a different page altogether and it would be in my mind insane to re-typeset the whole book. Unfortunately, Amazon links the reviews for the 1996 versions to the new version, including the previews. So there’s no way of confirming my suspicions that there are differences.

Posted by tpc at September 28th, 2008

Any rigid body displacement where a point is fixed is equivalent to a rotation. I saw this neat proof from Don Koks’ Explorations in Mathematical Physics.

By the hypothesis

Hence is orthogonal and But should vary continuously from the identity transformation, allowing us to conclude So


since we are working in 3D. We are forced to conclude
for some .
Thus has an eigenvector with eigenvalue 1. The transformation is thus the rotation about the direction .

Posted by tpc at April 30th, 2008

I’ve finally wiped the dust off my copy of Ahlfors and started reading from page 1. Previously, I’ve only looked at the chapter on Elliptic functions as a reference. I’ve skimmed through four chapters and decided that it is probably not very suitable for the complex analysis course later in the year. It’s a pity because the international version of Ahlfors is readily available locally and costs less than US$20.

I’ve been looking at several nice books, Stein and Shakachi, Gamelin and Needham.

All seem pretty promising, especially Needham’s Visual Complex Analysis. I have read many good reviews of the book, and now that I’m starting to look at the contents seriously, I agree it’s very good, and unconventional.

Posted by tpc at April 30th, 2008

I have no idea who is Heaviside until I started to teach this course which included Laplace transforms. That got me really interested and I checked out P. Nahin’s biography from the library. Perusing the borrowing slip, the book was last borrowed in Sep 2006, prior to that Mar 1990. 16 long years.

Meanwhile, in the preface a quote attributed to Lazarus Long:

Anyone who cannot cope with mathematics is not fully human. At best he is a tolerable subhuman who has learned to wear shoes, bathe, and not make messes in the house.

Posted by tpc at March 16th, 2008

supposedly used by Russian Peasants as claimed by A. Posamentier and I. Lehmann in chapter 6 of their book The (Fabulous) Fibonacci Numbers.

Suppose you want to multiply 23 to 41. What you do is to write the numbers in two columns. In the first column you successively half (round down) while in the second column you double. By the time you reach 1 in the first column, look at those odd numbers in the first column and add the corresponding numbers of the second to get the answer.

23* x 41
11* x 82
5* x 164
2 x 328
1* x 656

The odd numbers are marked by *, hence the answer is
41 + 82 + 164 + 656 = 943.

The interesting question is why does this work? It boils down to binary numbers.

Posted by tpc at June 19th, 2007

a personal perspective by Terence Tao. This is a new edition of a book which was written by Tao more than 15 years ago, which means when he was only 15! It’s a thin little book that takes a leisurely look at solving some competition type problems. The coverage is not huge, but the author take pains to go through in great detail various strategies one can adopt in solving problems. Quite a nice book but very pricey for 102 pages.

I found exercise 2.1 quite fun.

In a parlour game, the ‘magician’ asks one of the participants to think of a three-digit number . Then the magician asks the participant to add the five numbers and , and reveal their sum. Suppose the sum was 3194. What was ?

My solution is this. If we add all the six permutations, we know that the sum equals

.
So we just need to know the multiples, . Take the smallest multiple larger than the given number, and check by subtracting the difference and summing the digits. You do not have to do it with more than 5 different multiples.

.
But , so incorrect.
now .
And and we found our number.

Posted by tpc at April 15th, 2007

By Keith Ball. Yet another popular maths book, with the usual suspects of Fibonacci numbers and fractal curves. But this book is different in the topics chosen. Influenced by his own tastes, he discusses Stirling’s formula and Pade approximation among other things. The last three chapters are especially interesting for me. I love the way continued fractions keep popping up here, there and everywhere. His discussion on the Fibonacci also carries some depth and goes beyond what is normally done. A section is devoted to showing the for primes p,
, for Lucas numbers.

Posted by tpc at March 22nd, 2007

I spotted this little book at the library and flipped through it. I saw the names Cauchy, Euler and d’Alembert and was immediately thinking “oh differential equations” until I flipped to this lovely nested square root

courtesy of Ramanujan.

This book turns out to be a guide to competition problem solving by Christopher Small. I’m not very big on competition problem solving, probably because I can’t do most of them, but I do like to work on some of them every once in a while. I’ve read most of the book, and quite like it.

Oh, the expression above evaluates to 3. Try it!

Posted by tpc at April 24th, 2006

The following is a quote from Mario Livio’s The Golden Ratio, which in turn quotes Tobias Dantzig’s Number.

A squire was determined to shoot a crow which made its nest in the watch-tower of his estate. Repeatedly he had tried to surprise the bird, but in vain: at the approach of man the crow would leave its nest. From a distant tree it would watchfully wait until the man had left the tower and then return to its nest. One day the squire hit upon a ruse: two men entered the tower, one remained within, the other came out and went on. But the bird was not deceived: it kept away until the man within came out. The experiment was repeated on the succeeding days with two, three, then four men, yet without success. Finally, five men were sent: as before, all entered the tower, and one remained while the other four came out and went away. Here the crow lost count. Unable to distiguish between four and five it promptly returned to its nest.

Posted by tpc at April 9th, 2006

Honestly, I know next to nothing about statistics. It’s no wonder that I have not heard of this book, and most of the actors in the stories within. This is a good book that tells the tales of the origin of statistical studies and should be a must-read for all statistics majors. One major flaw was that he was writing too much for a layman, and most of the details are hidden behind some general explanation. While this is alright for a brief history of the subject, I sincerely believe that anyone who is willing to read a 300 page book on statistics, would like to see a deeper discussion. Another gripe is that the author seem to come across as one who is very critical of those who are only interested in pure theory/mathematics. That aside, the book is filled with gems.

I particularly like this delightful quote attributed to R. A. Fisher

A scientific career is peculiar in some ways. Its raison d’etre is the increase of natural knowledge. Occasionally, therefore, an increase of natural knowledge occurs. But this is tactless, and feelings are hurt. For in some small degree it is inevitable that views previously expounded are shown to be either obsolete or false. Most people, I think, can recognize this and take it in good part if what they have been teaching for ten years or so comes to need a little revision; but some undoubtedly take it hard, as a blow to their amour propre, or even as an invasion of the territory they have come to think of as exclusively their own, and they must react with the same ferocity as we can see in the robins and chaffinches these spring days when they resent an intrusion into their little territories. I do not think anything can be done about it. It is inherent in the nature of our profession; but a young scientist may be warned and advised that when he has a jewel to offer for the enrichment of mankind some certainly will wish to turn and rend him.

Posted by tpc at March 23rd, 2006

by Steven D Levitt and Stephen Dubner. A local bestseller that attempts to find the hidden answers from (not necessarily economic) data. The answers presented are controversial to say the least, but yet not quite far fetched. That’s I guess the main selling point of the book.

The following passage is from the book. It has nothing much to do with the theme/thesis proposed by the authors. But when I read it, I began thinking whether or not it was mathematically consistent.

The ECLS project surveyed roughly one thousand schools, taking samples of twenty children from each. In 35 percent of those schools, not a single black child was included in the sample. The typical white child in the ECLs study attends a school that is only 6 percent black; the typical black child, meanwhile, attends a school that is about 60 percent black.

Another completely unrelated thing. Simpson’s paradox. A google search will reveal several good write-ups.

Posted by tpc at February 13th, 2006

By John Stillwell. I picked this up from the new arrivals counter at the library. The hyperbolic tessellation on the front cover was invitation enough for me.

The book is about teaching geometry to undergrads from four different perspectives: 1) axiomatic a la Euclid, 2) linear algebra, 3) Projective and 4) transformations. It’s quite interesting and comes with lots of illustrations, although personally, I would prefer a book that delves more deeply into each aspect.

One main focus of the book is the cross-ratio which was discussed at length in the second half of the book. There is this wonderful quote.

At this point I can hear someone asking, “What is the geometric significance of the cross-ratio?” Although I first encountered cross-ratios as a senior in high school, and have dealt with them many times since then, I must say frankly that I cannot visualize a cross-ratio geometrically. If you like, it is magic. Here is this algebraic quantity whose significance is impossible to understand, and yet it turns out to do something very useful. It works. You might say it was a triumph of algebra to invent this quantity that turns out to be so valuable and could not be imagined geometrically. Or if you are a geometer at heart, you may say it is an invention of the devil and hate it all your life.
- Robin Hartshorne, Geometry: Euclid and Beyond p 341.

Posted by tpc at November 2nd, 2005

By Denis Guedj. I read it (the English version of course) years ago, and picked it up again recently when I ran out of light reading. It has a wonderful plot although the ending fizzled out. Read Simon Singh’s review here.

The book can really be thought of as a walk through the history of mathematics, and despite knowing most of the stories already, it’s still fun to re-read them again. Actually, if it was up to me to decide, I really wouldn’t mind using this book to teach a course on the history of mathematics at my faculty. Murder, mayhem, pistol duels, crazy genius, there are enough stuff to whet the appetites of undergraduates, no matter what their mathematical inclination. I can even throw in Fibonacci and the Da Vinci connection.

A more scholarly textbook could be The Adventure of Numbers by Gilles Godefroy. It’s interesting that both books are all originally French.