Teddy Bloggie Blog Blogging

I wish I understood enough math to make sense of all the great new discoveries happening in science and mathematics. I never even took a class on number theory... Today's NY Times has me thinking about our endless search for answers in nature - the cryptic way it talks to us. Here's an excerpt:

From Here to Infinity: Obsessing With the Magic of Primes By GEORGE JOHNSON There are practical reasons you would not want to smash atoms in your basement, observing for yourself the peculiar behavior of the fundamental components of matter. But studying primes, those mathematical quarks from which all numbers are made, is a game anyone can play. The only hazard is that it is easy to become obsessed, even for a nonmathematician.

Interesting follow-up on the prime number article posted earlier. For those of you interested in how modern-day encryption works, here's a great article gleaned from ArgMax's blog: A Web-only Primer on Public-key Encryption in the September, 2002 Atlantic Monthly by Charles C. Mann.

A more mathematical explanation is here: How the RSA Cipher Works

Today I tutored Pierce on the definition of a limit. You know, it took me about 2 years to understand what the textbooks were trying to tell me. "Who talks like that?" I thought as I tried to understand the mathematical language. It's really more like some intractable legal jargon than what you would imagine straightforward math talk would be. Is it me or does it seem the more precise a language is, the less understandable it actually becomes?

Ok, so you want to specify an Epsilon so that any Delta you choose from X should have you within Epsilon of L. It still sounds confusing, but I'm glad Pierce got it. They really should find a better way of describing a limit than using the old Epsilon and Delta definition. It's like eveyone knows what a limit is, but they have to confuse and torture you with its definition. Where's the cartoon book of limits? Where's the metaphors and analogies that you could use to teach "limits"?

How do you really talk about:
A function f with domain D in R converges to a limit L as x approaches a number c { closure(D) } if:
given any Epsilon > 0 there exists a Delta > 0 such that
if x {D} and | x - c | < Delta, then | f(x) - L | < Epsilon

My personal definition would be something like...
Hey, do you see a line? Does it go to the exact same vertical position when you're approaching it in both horizontal directions? Well, then what you have there is a limit...

DJ and I've been talking about this for years, it seems like...usually after we've both been drinking. Anyway, it seems official. Gravity propagates at the speed of light. That's right, I can't feel your pull until I can see you... Now, that's something to think about.

Einstein Was Right on Gravity's Velocity

I think this is the coolest thing I've seen in a long, long, time. Nature figured out how to be terminator 2 before we ever thought of it.

Download file

I've always been insulted by De Beers commercials. Their whole objective is to try to persuade the viewer that diamond equates love. How stupid. De Beers has tons of diamonds in warehouses but controls the supply of diamonds in order to regulate their prices. Now, according to Wired Magasine, flawless gem-quality diamonds will soon be made available for $5 a carat. And in the future, Wired predicts, I'll be writing this blog using a computer that will be running on diamonds instead of silicon.

Wired 11.09: The New Diamond Age

Note about physics and other mathy fields:

For half a decade the first equation of quantum theory was there. But nobody knew how to read it.

It is this "what if we took this equation seriously?" factor that is, to my mind at least, the spookiest thing about the unreasonable effectiveness of mathematics in physics. Take the h in Max Planck's equation seriously, and you have the quantum principle--something that was not in Planck's brain when he wrote the equation down. Take seriously the symmetry in Maxwell's equations between the force generated when you move a magnet near a wire and the force and the force generated when you move a wire near a magnet, and you have Special Relativity--something that was not in Maxwell's brain when he wrote down the equation. Take Newton's gravitational force law's equivalence between inertial and gravitational mass seriously and you have General Relativity--something never in Newton's mind. And take the mathematical pathology at r = 2M in the Schwarzchild metric for the space-time metric around a point mass seriously, and you have black holes and event horizons.

- From Probabililty Theory: The Logic of Science by E.T. Jaynes

Many people are fond of saying, "They will never make a machine to replace the human mind, it does many things which no machine could ever do." A beautiful answer to this was given by J. von Neumann in a talk on computers given in Princeton in 1948, which the writer was privileged to attend. In reply to the canonical question from the audience "But of course, a mere machine can't really think, can it?", he said: "You insist that there is something a machine cannot do. If you will tell me precisely what it is that a machine cannot do, then I can always make a machine which will do just that!"

Prior to today, I didn't think the following was possible. What does this have to do with econometrics? I was never that interested in maximum entropy estimation and I don't know much about it, but the following image is maximum entropy applied to image data. Now I'm interested.

Maximum Entropy Data Consultants, http://www.maxent.co.uk/

The following fact by way of Mario Juric:

While most people can see with a resolution of 1?€™, the image on our retina is blurred through a PSF of width as large as 5?€™ due to various effects (the largest being chromatic aberration).

And while we still struggle with finding optimal deconvolution algorithms, the brain happily performs the procedure on a 8500x5400 (43Mpix) image, a few times per second, ~17 hours a day, 365 days a year.
MacKay (2003)
Tidwell (1995), http://www.hitl.washington.edu/publications/tidwell/ch3.html

This Sci-Am article suggests the right kind of encouragement and mindset are important in overcoming challenges for kids and adults. I tend to agree; although, I've seen some hard working students in my class struggle with conceptual problems while others who just "get" it.

At times, the intellectual hurdle seems too high to them and students think the effort required might as well be infinite; and therefore, they give up before they understand a concept or solve the problem. But it's exactly at those times where a little bit of extra effort can see real payoffs. That's something I've forgotten about lately too and could use the reminder. Note to self:

http://www.sciam.com/article.cfm?id=the-secret-to-raising-smart-kids&print=true