December 8, 1998 
From: Gustavo Aguiar Rocha da Silva
Sent: Wednesday, December 09, 1998 7:21 AM
To: kumar@tri.sbc.com
Subject: golden books 

Dear Mr. Kumar:

    I read your review of Needham´s book and we agree, that's an extraordinary book. I am curious about your list of  "ten gold math books", will you please tell them and why do you think them so good?

   Thank you very much.

Yours truly
Gustavo Silva. 
 

December 9, 1998

Dear Gustavo,

    I had mentioned my choice of "ten golden books" in an off-hand sort of way, not expecting to be ever asked to actually produce a list. However, since you ask, I thought it wouldn't be a bad idea to actually draw up a list. This below is what I can think of right off the top of my head, and I may have missed a few that I should have put in. I'll add those and let you know as and when I think of them. Also my list is not just mathematics, but also the sciences.

    Are you a mathematician? Engineer? I'd very much like to see a similar list from you. Since our tastes match on Needham, maybe they'll match on others too. Do write me your list please.

    In my estimate that more than 98% of books are garbage. They waste your time, and you come away with nothing to show for all that time and effort. And they might even entirely kill your interest in the subject. It is absolutely crucial for people to share lists of books that they actually found useful and worthwhile.Maybe we also need to draw up separate lists of the trashy and useless stuff. 

Regards,
Arun Kumar 

Arun Kumar's list of golden books, Dec 9, 1998. 

• Davenport, "The Higher Arithmetic," Dover. The best first book on number theory that I know of. High-school level, but goes on to tackle pretty tough stuff. Starts nicely with prime factorization, Euclid's algorithm, and Diophantine equations. Gets increasingly more difficult, but never impossible.

• Ivan Niven, "Numbers: Rational and Irrational." High-school level. Beautiful. Majestic. Easy as pi. Niven has written another advanced book on the same subject called "Irrational Numbers" (MAA Carus Monograph?) which is also a very fine book, but I have read only the first two chapters of that so I won't include it in this list.

• Grossman and Magnus, "Groups and Their Graphs," Random House. Elementary. High-school level. Easy. The best introduction to group theory that I know of. Includes a nice little chapter on knot theory. One of the authors teaches high-school. The other college.

• Herstein, "Topics in Algebra," second edition.,Wiley, 1975. The best of the conventional texts on the subject Groups, rings, fields, extension fields, ideals, vector spaces. Concise, complete.

• Halmos, "Finite Dimensional Vector Spaces,"Van Nostrand (in India. May be a different publisher elsewhere). Very easy reading. A very good second book on vector spaces. It may be a little too abstract for someone meeting the topic for the first time. A few pictures would have helped immensely, but there are none in this book. Zip! "And what sort of a book is that that has no pictures!" Alice asked her sister. I should add that Halmos' "Naïve Set Theory" is a book I like to avoid. For set theory and metric spaces my usual reference is Kaplansky, but it is not at a level with the other books in this list. For Banach, Hilbert, and Hardy spaces my usual reference is Young, "An Introduction to Hilbert Space," Cambridge U Press, but I could do with a better book.

• Charles Hadlock, "Field Theory and Its Classical Problems," Carus Monograph, Mathematical Association of America, 1978. You don't need to know any algebra going in, but some familiarity with groups, rings and fields will not hurt. The subject is Galois Theory. Straight-edge and compass constructions. Why cannot a circle be squared? Why do the solutions to a quintic admit of no closed form? Fine fine book.

• Pollard and Diamond, "The Theory of Algebraic Numbers," Carus Monograph, Mathematical Association of America, 1976, Second Edition. For people familiar with the elements of number theory and some algebra, an easy and beautiful little book on algebraic number fields.

• Ogilvy, "Excursions in Geometry," Dover. I think the world of mathematics owes a lot to Dover Publications for their good taste, their fine production, and their easy pricing. Maybe MAA should award Dover a special medal. Harmonic division, the circles of Appolonius, conic sections, projective geometry. Last chapter is on unsolved problems in modern geometry. I think every book should have a chapter on unsolved problems, for us and for our children.

• Bressoud, "A Radical Approach to Real Analysis." Assumes some passing familiarity with partial differential equations. Uses the Fourier heat equation for motivation. One of the very few real analysis books that won't bore you to death, or put you instantly to sleep. Contains a wonderful essay on Newton's wrestling match with an infinite series for pi. And much more.

• Needham, "Visual Complex Analysis," OxfordU Press, 1997.

• Feynman , Leighton, and Sands, "Feynman's Lectures in Physics," volumes one through three, Addison-Wesley. The finest introduction to physics. Incomparable. Volume one: mostly mechanics, classical and some statistical, with a pinch of this, that, and the other. If I were to write a book, this is the example I'd wish to emulate. It returns in waves to cover each topic at greater and greater depth. A great chapter on harmonic oscillators. Three great chapters on electromagnetic radiation. Wonderful expositions on kinetic theory of gases, statistical mechanics, thermodynamics. Volume two is all electricity and magnetism. For those who never did completely figure out div and curl in college, Feynman gives you both in less than half a page --- and you'll never ever forget them. Maxwell's equations seen every which way by a person who went beyond to become a principal architect of quantum electrodynamics. Volume three, all quantum mechanics (but I haven't gotten that far).

• Richard Feynman, "QED," Princeton U Press. Quantum electrodynamics explained with no math! Complex numbers representing photon flux pictured as one-handed clocks with elastic needles. Could anything be simpler? Why do we see colors in a film of oil floating on water? Why are peacock feathers iridescent? The modern explanation of reflection and refraction. Forget that old nonsense. Photons don't travel in straight lines --- they go every which way. The Feynman path integral. The strange interaction of light and matter. Wonderful little book.

• Kaufmann, "Universe," fourth edition, Freeman.You need know no physics and only some calculus going in. Provides great motivation for the study of physics. Clear and crisp introductions to the physics underlying astrophysical phenomenon. Astronomical calculations. Beautiful pictures.

• Moore, "Structures: Or Why Things Don't Fall Down." The best book in engineering that I know of. Elementary. Why do nighties cut on the bias cling to their lady? Why do bridges shake and break? Why would a cylinder full of gas rather fracture longitudinally than circumferentially? Beam loading. Crack propagation. Fracture mechanics. Moore invented fiber-reinforced plastics for aviation. He has also written a book called "The New Science of Strong Materials," that I have yet to read.

• Raven and Johnson, "Biology". Most of us people whose tastes run to engineering, mathematics and physics, do not know enough about how Mother Nature designs her machines. And how she does that is amazing. Every little thing she has she uses with immense art and cleverness. Of the books I have tried to learn about this from, this is the best. It is a standard university text, now in its fourth or fifth edition. I got a third edition copy from a second-hand bookstore for $15. A new one costs $90 or $100 I think. I had eyed it in university bookstores and had always been intimidated by the price before I found my cheap copy at Half Price Books. But having read through more than half of it, I think the book is easily worth any kind of money. Starts out with basic biochemistry, proteins, higher-order structures, protein synthesis, lipids, lipid membranes, trans-cellular transport, neucleotides, RNA, DNA, enzyme structure and function, reproduction, the energy cycle, receptor-acceptor signaling. Pretty standard stuff, but all very nicely explained. Well illustrated.

Sent: Wednesday, December 16, 1998, 2:55 PM 

Dear Mr. Kumar:

    You have a very sharp leaning towards Algebra! Here goes my list. I am a mathematician and the oncoming of age (51, almost 52) has made my mathematical tastes a bit more applied.

    We certainly agree on Feynman´s Lectures on Physics ( although I tried QED and gave it up, but some day I'll be back), on Herstein "Topics on Algebra", on Needham's book and on Halmos "Finite Dimensional Vector Spaces", now published by Springer-Verlag. My other choices are 
 

• Richard Courant: Differential and Integral Calculus. I simply refuse to believe there can be any other book so good!

• George Simmons: Introduction to Topology and Modern Analysis. Everything one needs to know about metric spaces, normed spaces and the like, in Simmons delightful explanatory style.

• George Simmons: Differential Equations with Applications and Historical Notes. In content, in philosophy, in form, a book absolutely different from the precedent, but with the superb Simmons Touch.

• Robert Bartle: Elements of Real Analysis. Very, very good! Picks the reader fresh from Calculus and takes him into the realm of Analysis.

• Edwards: Advanced Calculus of Several Variables (Dover books): Everything from Curves in Euclidean Spaces to Stoke´s Theorem on Manifolds, with lots of attention to some difficult topics as the Implicit Function Theorem and Lagrange Multipliers.

• Edwards (not the same): Differential Forms - a complementary course in Several Variables Calculus (the title certainly is wrong, please excuse me, but the author is Edwards and the publisher is Birkhäuser). An unusual book, an unusual text but a riveting one. After reading the first page you are hooked. It grips you like a good political novel.

• S. Berberian: Hilbert Space. Published by Chelsea Books, it seems to be out of print. Try it in your favorite public library.

• Morris Hirsch and Steve Smale: Differential Equations, Dynamical Systems and Linear Algebra. This is probably the best introductory book on Dynamical Systems I have ever seen.

• Vladimir Arnold: Ordinary Differential Equations. Certainly THE best second book on Dynamical Systems available. The new Springer-Verlag edition is simply outstanding!

• Vladimir Arnold: Mathematical Methods of Classical Mechanics. Well, this is the book to end all books on Classical Mechanics.

• Courant and Robbins: What is Mathematics? An ageless book, first published in the thirties and still going strong. 

    I wish you a very happy 1999.

Yours,
Gustavo Silva. 
 

December 17, 1999

Dear Gustavo,

    It will be very difficult for me to address you as "Dear Gustavo" if you keep up with your "Dear Mr. Kumar". I am very happy to hear back from you. I am not too far behind you, as age goes, at 45.

    I am happy to find a bunch of real analysis books in your list. I am a practical sort of person myself. Have to work for a living. I work on (wavelet) transforms, the design of algorithms for image and video compression, and the design and testing of communication systems. I often run into notions from real analysis, functional analysis, and harmonic analysis that leave me perplexed as to their intent and purpose. So I have some motivation to learn about these things. But most often I like to enjoy my mathematics in the manner of music and poetry and good food --- not because of its utility, but for its plain pleasure.

    Though I have discovered on occasion that even abstract stuff turns out to have its uses. For example about ten years ago I used a theorem by Lagrange on the coset decompositions of a group to design the memory for a chip designed to display an arbitrary 2-D slice through an n-D image. I was surprised to find a connection between two such different worlds. This was for an application in radiology.

    One problem that's been bothering me through the last few weeks is an attempted shortcut proof of a theorem by Claude Shannon that has to do with the capacity of a communication channel contaminated by Gaussian noise. Why I mention this you will soon see. I tried to model Shannon's theorem ("Shannon's Third" I call it) as a Lagrange optimization problem. The variable of optimization,  f, is a probability density function. My problem is that I am a complete stranger to the calculus of variations. When I run through my proof treating partials with respect to f mechanically and formally, things work out, more or less; but my two of my friends, Matt Stafford and Si-Jian Lin --- bona fide mathematicians, down the hallway from me --- start changing colors at the methods I use. But they too are strangers to the calculus of variations, and have not been much help fixing that proof. Now it seems to me that since you have been out drinking with Hilbert and Courant and company, maybe you could point me someplace where I could learn calculus of variations in a day or two. Perhaps even a little longer. Remember that I am just a poor engineer with a very moth-eaten mathematical background. Do you suppose that the book by Courant, number one in your list, is a good place to go to? Or would you recommend Edwards at number five?

    I tried Fox's "An Introductionto the Calculus of Variations" (Dover) but that already looks like a lost cause. Wim Sweldens, in the very beginning of the Introduction to his dissertation, put in a quote from Paul Halmos. It reads, "The beginner... should not be discouraged if ... he finds that he does not have the prerequisites for reading the prerequisites," but I am not that stout of heart. I like to follow stuff all the way out of the gate from page i and all the way in to the finish at page 1998.

    Please give me a lead if you have the time. If you like I could write you the objective function in latex. But if you are busy, please don't bother. Its not such a big deal.

    I will certainly be looking up the books in your list. One at a time. For this list I thank you "from the bottom of my hart" as my daughter likes to write. Maybe I'll start with Courant and Robbins. I checked at www.amazon.com and found that this book was once reviewed also by Albert Einstein. Einstein wrote: "A lucid representation of the fundamental concepts and methods of the whole field of mathematics ... Easily understandable."

    Below this comment by Einstein is an Amazon Editor's note. "Hmmm ..." it says. I wouldn't have guessed that the Amazon editors could be that witty! 

Regards,
Arun 
 

December 18, 1998

Dear Arun:

    Many thanks for your message. I'll disappoint you right away: I had a term in Calculus of Variations, but it was some 26 years ago and I do not remember anything from it. But there is an introductory chapter in Courant´s Differential and Integral Calculus, volume two. Now, let me try do undo the fuss I made with the Edwards guys: C.H. Edwards is the author of the (excellent!) "Advanced Calculus: a differential forms approach", published by Birkhäuser. He is a teacher at the Courant Institute, New York. The other Edwards is C.H. Edwards, a teacher at the University of Georgia and his book, "Advanced Calculus of Several Variables" is published by Dover books. A very good book, indeed.

    I forgot to include a precious gem in my list: it is Louis Pennisi's "Complex Variables" , unfortunately out of print. Try it in any library and I hope you'll join me in pestering the Dover guys so that put this book in print again. I am an Engineeringdrop out and I earn my keep as a teacher at Rio de Janeiro State University, ...

    Do not blame yourself if you do not like Real or Functional Analysis. Analysis is intrinsically hard, because it deals with those properties that make Calculus tick, so the student of Analysis must be a neurotic guy always interested in the rules as well as in the exceptions and very happy when he can press anything between two inequality signs. This has even turned up as a definition of an analist: a guy who is happy only when he is manipulating some sort of inequality. 

    You are THE ONLY LIVING PERSON I have ever heard about who succeeded in applying Algebra to anything of practical value, believe me! This and this alone should suffice to earn you a prize, but the sponsors do not think so, I suppose. 

    Well, I have never seen anything about wavelets before. Would you please suggest me something to read??? Poor of me, I'd like very much to be a drinking partner of Courant and Hilbert, but Life wasn't so generous. But I read them, I always read them, very slowly, by the way, but I am stubborn enough to keep trying.

    Hope to hear about you soon.I wish you a nice NEW YEAR. 

Yours truly, 
 

December 23, 1998

Dear Gustavo,

    About your request for wavelet literature, I have bad news and good news.

    The bad news is that there is no book or paper that I know of that I might recommend to a person seeking a first introduction to the subject. There is a bunch of gushy literature but almost entirely void of substance. There are some very fine papers, but too technical for those new to the subject, or too out of date. There is a whole host of books which assume either a good grounding in specialized areas of mathematics, or a good grounding in signal processing, and which all (it seems to me) get mired too quickly in dull detail and fail to communicate the grand design.

    The good news is that I can explain wavelets to you perhaps as well as anyone else. This is not because I claim to be a big expert in the subject. I do not. However, I understand the substance of most of the ideas in the field, and have had many years of familiarity with them. And what I understand I can explain in the simplest possible way. And there is nothing like an informal discussion between friends to thrash out problems and questions.

    Let me know.

    I wish you a Merry Christmas and a very happy 1999. I hope to get to some of the books on your list in 1999. I enjoyed your pun on "analyst". That wasn't unintended, was it? Good one! And how true! 

Regards,
Arun
 

January 4, 1999

Dear Arun: 

    Many thanks for your message. I wish you and your family a very happy 1999. As for wavelets, thanks a lot! I'd like very much to talk and be talked into wavelets by you, and if you find it more comfortable to write to me, here is my postal address: Gustavo Silva ... Rio de Janeiro 22431-040RJ BRASIL.

    As for the puns on Analysis and analysts: Sterling Berberian, a very good analyst in his own right, wrote (on his excellent Functional Analysis) that "an analyst is someone who is always in the company of the real and complex numbers", and I wholeheartedly agree with him. Now, please take a look on volume five of Spivak's "A comprehensive Introduction to Differential geometry" and you'll find two very funny remarks about analists, give me two days and I'll tell you the pages. By the way, do you have ready access to any mathematical library? I am always telling you to "look at this and that" but I really do not know whether or not this is possible or feasible. Let's keep in touch and talk about wavelets. 

Yours truly
Gustavo Silva

January 6, 1999

Dear Gustavo,

    Today is a day to mark with a white stone because today I feel like a pig that's just had a bath. This morning, in about an hour's time, my colleague Yongdong Zhao, down the hallway from me, taught me the calculus of variations, using by way of example the design of the trajectory of a missile. And we discussed also how to use the variational method to obtain a Lagrange multiplier proof of Shannon's Third. So everything looks good today, and I celebrate the end of a painful month's worth of ignorance. I'll write down a water-tight proof today to go sailing in. That even my local mathematicians will like the smell of.

    I'm glad to hear that you feel kindly towards a wavelet discussion. I look forward to that too. We will have to find a good way to read and write math. Can you read a latex file on your machine? Do you write latex? I can read and write latex at home.Here too.

    I can also convert from latex to postscript, if that is more convenient. And freeware to read postscript is easily available, and easily installed on a PC.

    The Microsoft equation editor stinks. I think it is written by dipsomaniacs or cobblers or people like that. The sort of people out of whose heads the pigeons have all flown away. So if either latex or postscript won't work, we could try either html, with equations written as images, if you have a browser handy; or just plain old handwriting in snailmail. Let me know.

    Sadly, I do not have ready access to a reasonable library. My company library has fewer books thanI have at home. (And that is almost true.) And most of even those are only good for use as toilet paper for the smarter managers. And more mathematical literature is usually found on the Dewey decimal stickers on book spines than inside them. We do subscribe to a reasonable collection of IEEE journals.And that is about all this library is useful for.

    The closest reasonable library is a good half-an-hour's drive at the University of Texas. But there it is (a) hard to find free parking, or (b) I have to pay $2 per hour for parking and walk 15 minutes to the library. Also the company I work for, not being very heavy in the upper story, does not like employees to go visiting libraries when they could be gainfully-employed attending those delightful meetings we are known for out here. I very rarely breathe library dust these days. So when you say look up this and that, I can only look up at the ceiling above me and say a sad little prayer. Next incarnation maybe. 

Regards,
Arun 
 

January 7, 1999

Dear Arun: 

    I share your joy and I sincerely envy you because a colleague could teach something mathematical to you using a very concrete example, and this DOES NOT happen with mathematicians.I hope you write the water-tight proof you want and, if there is some time to spare, please send me the proof by snail-mail. As for the "look-up" nonsense, please excuse me. I'll put all the quotations in full form and let's begin right now!

    The book is Michael Spivak'smagnificent "A Comprehensive Introduction to Differential Geometry", volume five (yes, sir!), Chapter 10, about Partial Differential Equations and on page 115 one reads: "The proof is carried out in detail in Courant-Lax [1]. It involves a series of estimates that only an analyst could love and there doesn't seem to be much point in reproducing it here, since the paper is readily accessible, and the sane differential geometer would probably skip it anyway." 

    On page 140, same volume, onereads: "Moreover, the proof has one true beautiful feature - there isn't a single inequality in it. "

    I prefer to discuss wavelets by snail-mail. I think I have already given you my home address, but here it is, anyway:

    Gustavo Silva ... Rio de Janeiro22431-040RJ BRASIL

    I must stop: my students arealready waiting in classroom.

Gustavo
 

January 4, 1999

Dear Gustavo,

    Handwriting it is for wavelets.And I'll also send you a proof of Shannon's Third, rather of a lemma that is almost all of the Third.

    And what that lemma says this: If X and Y are two random variables, and if X is Gaussian, then the differential entropy of X is greater than or equal to the differential entropy of Y.

    The term "differential entropy" may not be familiar to you. However, if you are familiar with the entropy of a discrete memoryless source, then differential entropy is readily understood as a similar measure of information (but not quite) associated with a non-discrete or continuous source. I'll state this better when I write you the proof.

    Shannon defined the capacity of a communication channel as that maximum amount of information that we may deduce about an input random process (at the transmitting end) from an observation of the output random process (at the receiving end), given that the output random process consists of a sum of the input random process and a noise random process. Very nice definition that. But these are just hurried remarks to be explained better sometime later. Right now I got to rush home. 

Regards,
Arun