Conference rusure::math

Garrett Birkhoff and Marshall Hall, Jr., editors,
	Computers in Algebra and Number Theory.
	SIAM-AMS Proceedings, volume IV, New York: 1971
	(reprinted 1980).  Softcover. 200pp.
	AMS order code: SIAMS/4. $19.
	ISBN: 0-8218-1323-4

This book consists of articles representing talks given at
a Symposium in Applied Mathematics on Computers in Algebra and
Number Theory, held in New York City in March, 1970.  As such,
the compute-power available to the participants is now 15 years old.
Nevertheless, that does not stop this from being one of the few
technical mathematics books that explain how computers can be used
to help solve theoretical mathematical problems.

Unfortunately, the articles are all at the research level and are
hard reading even for research mathematicians.  Each article has its
own set of references, and there is a master index, unusual
for a set of proceedings.  The list of authors includes an incredibly
large number of very famous mathematicians.

Here is the contents:

Part 1: Algebra in Computation

Garrett Birkoff, The Role of Algebra in Computing

Shmuel Winograd, On Perfomring Group Multiplication by Switching Circuits

Part 2: Number Theory and Combinatorial Theory

H.P.F. Swinnerton-Dyer, Applications of Computers to the Geometry of Numbers

B. J. Birch, Approximation to Cubic Irrationals

Hans Zassenhaus, On the Group of an Equation

J. H. van Lint, Nonexistence Theorems for Perfect Error-Correcting Codes

L. D. Baumert, et. al., A Combinatorial Packing Problem

Part 3: Problems on Finite Groups

Marshall Hall, Jr., Construction of Finite Simple Groups

J. H. Conway, Groups, Lattices, and Quadratic Forms

M. D. Hestenes and D. G. Higman, Groups and Strongly Regular Groups

John J. Cannon, Computing Local Structure of Large Finite Graphs

John McKay, Subgroups and Permutation Characters

J. Neubuser, Computing Moderately Large Groups: Some Methods and Applications

Charles C. Sims, Determining the Conjugacy Classes of a Permutation Group

Donald D. Spencer, Computers in Number Theory. Computer Science Press.
	Rockville Md: 1982.
	softcover, 250pp, $12.95.
	ISBN 0-914894-27-7

This is the kind of book you find at your local Daltons in their
section on computers.  It unfortunately has a goodly number
of trivial sections with titles like "How to Recognize a Computer".
The programming language used is BASIC.  Nevertheless, for a
mathematician who knows little about computers, this would be a better
introduction to computers than many of the other books on the
market.

For an established computer programmer, there is probably nothing
you will learn from this book about computers.  If you know very
little mathematics, you can probably learn a little about things like
the Sieve of Eratosthenes, Perfect Numbers, Amicable numbers, magic
squares, and Fibonacci numbers.  However, if you already know about
these, this book will not help you write programs about them any better.

A typical exercis is: Write a program to find a solution in integers
of 33x+14y=173, or find all Pythagorean triangles with one side equal
to 12.  In most of these exercises, one can find the answer faster
using mathematics than using computers (sigh).

Highly recommended for bright Junior High Schools students.

Contents:				Comments

1. Meeting the Computer
2. BASIC Programming
3. Prime Numbers			Mersenne primes, twin primes
4. Magic Curios				Abundant numbers, perfect nos, etc.
5. Factoring				nothing useful, not even a sieve used
6. Fibonacci Numbers
7. Magic Squares
8. Computers and Numeration Systems
9. Modular Arithmetic
10. Number Theory for Fun

Glossary of Computer terms and an index.

Ulf Grenander, Mathematical Experiments on the Computer.
	Academic Press, New York: 1982.
	hardcover, 525pp
	volume 105 in their Pure and Applied Mathematics series
	ISBN: 0-12-301750-5

Readable by the average mathematician/computer scientist.
By "experiments on the computer" the author means using the
computer to gather data about some mathematical problem from
which a likely result can be conjectured.

He gives 11 detailed case studies of such "experiments"

	1. From Statistics
	2. From Linear Algebra
	3. An Energy Minimization Problem
	4. Neural Networks (Static Problem)
	5. Limit Theorems on Groups
	6. Pattern Restoration
	7. Modeling Language Acquisition
	8. Study of Invariant Curves
	9. Neural Networks (Dynamic Problem)
	10. High-Dimensional Geometry
	11. Strategy of Proofs

The programming language used is APL because of its vector capability
and because it is interactive, so conducive for "trying experiments".
I think this will turn off many mathematicians because of the crypticness
of APL.  This should not bother any DECies.  However, if you don't
know APL, you will not be able to make heads or tails out of these
programs.  There is a section on the APL language, but I'd recommend
reading the DEC manual instead.

There are a LARGE number of good useful programs in this book.
In fact almost half the book is devoted to algorithms.
Someone should go through and type in the more useful ones.
Although none of the code is commented, each routine is preceded
by a comprehensive routine header describing the inputs and
outputs and there is extensive documentation of the algorithm used.
If you know APL, you will have no trouble if you want to translate
the programs to some other language.  However, the APL is standard, and
should therefore run under VAX APL.

Contents:

1. Definition of the Subject
2. History of the Project
3. Case Studies
4. Other Areas
5. Software Requirements
6. Hardware Requirements
7. Practical Advice
8. Language for Mathematical Experiments
9. Mathematical Objects - Data Structures
10. Functions, APL Statements
11. APL Operators
12. APL Programs - Defined Functions
13. Designing the Experiment
14. An Experiment in Heuristic Asymptotics
15. Organizing a Program Library
16. Algebra
17. Analysis
18. Arithmetic
19. Asymptotics
20. Geometry
21. Graphs
22. Probability
23. Special Functions
24. Statistics
25. Utilities

refernces, solutions to exercises and index.

There are over 100 useful routines provided.  Here are the names
of a selected few of them to give you an idea of what's available:

INVERSE
JACOBI
POLFACT
CONFORMAL
DETERMINANT
KUTTA
NEWTON
SPLINE
CONFRAC
GCD
PRIMGEN
FIT
AREA
CHRISTOFFEL
HULL
BERN
BETA
GAMMA
MARKOV
MULTINOM
HERMITE
LAGUERRE
LEGENDRE
HISTOSPLINE
KOLMOG

No analysis of running time is given for these algorithms although
in many cases, references are given for where the algorithm comes from.

                        Computability and Unsolvability
            Martin Davis, dept. of Mathematics and Computer Science,
         	 Courant Institute of Mathematical Sciences NYU
                                        1958, 
   Dover 1982, adding Matiyasevic's proof that Hilbert's tenth problem is
                               	recursively unsolvable.
                                     248 pages
                                       $6.50
I had wanted just to see the symbols mathematicians use to discuss turing 
machines.  Popular accounts, such as the Scientific American article,
eschew the notation.  I read the first couple 
chapters as well as I could for the rudiments, and now feel that I understand 
the basic idea of a turing machine in a mathematicians's terms.

Here is an example of a machine, from page 12:

	Let a(1)=q(1)(m(1),m(2))=q(1)m(1)Bm(2).  The m's are represented
	in a special way, by m+1 ones.
	The machine, Z consists of the following quadruples:

		q(1)1Bq(1)      in state one if you see a "1" on the tape, 
				print a blank state 1.
		q(1)BRq(2)	in state one if you see a blank, move the
				head right and go to state 2
		q(2)1Rq(2)      In state 2 if you see a 1, move right, goto 2.
		q(2)BRq(3)	In 2, if you see a blank, go right, goto 3
		q(3)1Bq(3)	In 3, if a one, print a blank, goto 3.

The machine starts in state "one" and at the left most "1."

If the tape starts like this:

     11111111111111111 111111111111111111111
     ^
It ends like this:

      1111111111111111  11111111111111111111
                       ^

When there is no instruction for the current instantaneous description of the 
machine, the machine halts.  *The number n on the tape is represented by a
string of (n+1) ones at the start, but only (n) ones at the end.*
This is an adding machine.
		
The rest of the book covers consequences in detail.  I only skimmed that, not 
having the mathematical knowledge or stamina.  It is better for people with 
knowledge of number theory, although there is a 5-page review of number theory 
in an appendix.

It's pretty heady stuff, but there are timely references to the origins and
purposes of the techniques.

2 Prefaces
Glossary of Symbols
Introduction
Part I
	Computable Functions
	Operations on Computable Functions
	Recursive Functions
	Turing Machines Slef- Applied
	Unsolvable Decision Problems

Part 2 Applications of the General Theory
	Combinatorial Problems
	Diophantine Equations
	Mathematical Logic
Part 3
Further development of the general theory
The Kleene Hierarchy
Computable Functionals
The Classification of Unsolvable Decision Problems
Appendices
References
Index

Tom

This one doesn't quite fit, but we don't have a general note for 
recommended books, so here goes... I recently ran across the 1985 edition 
of HANDBOOK OF MATHEMATICS by Bronshtein and Semendyeyev, published by Van 
Nostrand Reinhold Co. - it's a reissue of the original Russian->German
->English first published (in German) in 1957. It has been significantly
expanded with new chapters on functional analysis, set theory, measure 
theory, tensor calculus, operations research methods, numerical methods, 
computational techniques, etc..... a lot of chapters rewritten and revised.
It has just about everything you might want to have in a handbook, except 
possibly number theory!

Anyhow, I was delighted to find it. I bought mine for about $38 at 
Wordsworth's in Harvard Square.