My favourite theorems

[jdhurst]

Mathematics is fun, and Pure and Applied Mathematics is what I did in my University days along with some Applied Physics. There was lots interesting (including studying some very esoteric geometries). And there were several proofs that engaged me, most of which have real world bearing:

1. The Normal Distribution: Taken enough times, every distribution turns Normal. Indeed, the 80:20 rule applies to most anything.

2. Linear equations: Make a system of linear equations large enough in dimension and it becomes unsolvable. Ergo, Numerical Analysis always wins the day for engineers with real problems to solve. Thick bridge decks that vibrate are not easy to figure out and don't solve easily with pure mathematics.

3. Goedel's Incompleteness Theorem: Within any given branch of mathematics, there would always be some propositions that couldn't be proven either true or false using the rules and axioms. This has been used to suggest that computers will never be as smart as humans (because the extent of a computers knowledge is limited by fixed axioms). I relax in the knowledge that, with many years yet to live, it certainly appears that true artificial intelligence will appear long after I am dead, if ever. Look at the sorry state of search engines: Microsoft cannot even find the error messages its operating systems put out.

4. The Cantor-Ternary theorem: Take a closed interval [0:1] and deduct open middle thirds forever. You can prove that what you removed was 1. Therefore the measure of the leftover is 0. However, you deducted a countably infinite number of open sets from an uncountable closed set. Therefore, what is left over is at once: Closed, Bounded, Compact, Perfect, Uncountably Infinite and still measures 0.

This last one is just a mind teaser, and has no real world application I know of.

2007 was a good year here, and is now in the books. 2008 looks like another good year, but we shall see.

Happy New Year to you all. ... JDH

[Trekk69]

I'm all science right here.......that was way over my head!

[Phazer]

Sooooooo...............how 'bout them Ducks?

[mgo]

Mathematics is fun, and Pure and Applied Mathematics is what I did in my University days along with some Applied Physics.

This isn't math, but I've often wondered if some of the dinosaurs (raptor types) may have been equipped with IR sensors, much like today's pit vipers. Seeing that many of the dinosaurs are now thought of as warm blooded or semi-warm blooded IR sensors would seem to be a efficient detector for those predators that needed a lot meat to keep the old engines running...

Also, since modern elephants are able to communicate for long distances using low frequency sounds, perhaps the dinosaurs did that too. In the more dense atmosphere of the times, sounds would have gone pretty good distances.

What better way to detect a dinner than to pick up the low frequency stomach rumblings or communications between herd members.

Sounds might have been better than the shorter wavelength infrared because the sounds would not be stopped by tree growth and other obstacles, as opposed to straight-line IR reception.