Posted by tpc at January 30th, 2005
… a mathematical problem should be difficult in order to entice us, yet not completely inaccessible lest it mock our efforts. It should be a guidepost on the tortuous path to hidden truths, ultimately rewarding us by the satisfaction of success in its solution.
- David Hilbert (1862-1943).
Posted by tpc at January 26th, 2005
This week, I have to teach predicate calculus to first year computer science students, in addition to the vector calculus for engineering students. I find that I really have nothing to value-add for logic. Perhaps it is because I’ve never really fancied all that logic stuff, but seriously, how do you make things interesting when all that is required is the mundane task of filling up truth-tables?
Posted by tpc at January 24th, 2005
A book by George Boros and Victor Moll. The authors bill the book as a guide to the evaluation of integrals, but it strucked me as a nice guide to old-fashioned (nineteen century) analysis, in the spirit of Gauss and Euler, if I may add. There are many gems in the book including three proofs of
and also this which is not proved
where 
.
Posted by tpc at January 20th, 2005
Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. “Immortality” may be a silly word, but probably a mathematician has the best chance of whatever it may mean.
- G.H. Hardy (1877-1947), A Mathematician’s Apology
Posted by tpc at January 16th, 2005
AIME 1994
Find n for which
.
A fairly straightforward problem that can be worked out quickly once you see the pattern. 313 was my initial answer which was wrong. (The actual answer is 312.) The funny thing is that I first tried to check it with Maple and it returned an answer that was slightly wrong. My guess is bad code resulting in some round off error.
So I checked with Excel and found the answer and my initial error. Athough admitting that I like a microsoft product will destroy whatever “geek” status I that possess, I have to say that there are few things you cannot do with Excel. It is also an excellent (pun not intended) way to investigate patterns in number theory.
Posted by tpc at January 14th, 2005
Taken from Pi and the AGM:
When I was a student, abelian functions were, as an effect of the Jacobian tradition, considered the uncontested summit of mathematics and each of us was ambitious to make progress in this field. And now? The younger generation hardly knows abelian functions.
How did this happen? In mathematics, as in other sciences, the same processes can be observed again and again. First, new questions arise, for internal or external reasons, and draw researchers away from the old questions. And the old questions, just because they have been worked on so much, need ever more comprehensive study for their mastery. This is unpleasant, and so one is glad to turn to problems that have been less developed and therefore require less foreknowledge - even if it is only a matter of axiomatics, or set theory, or some such thing.
Felix Klein (1849-1925), Development of Mathematics in the 19th Century, 1928
Klein succintly described what is the current state of mathematics (at least in Singapore.) In a typical year, you get 30 students learning algebra, 100 students learning coding theory (90% of whom who think a field is some piece of land with grass on it) and 150 students learning graph theory. While the local arts scene has government support and funding, there is none for pure mathematics. It’s really difficult to see where the next generation of pure mathematicians is going to come from.
Posted by tpc at January 11th, 2005
is the number of ways of picking r objects out of n possibilities without order. A simple but elegant identity is this
which appears when you are proving the binomial theorem by induction.
