It’s not yet 10 AM as I begin to draft this, and it’s already been a productive day. I woke up at 3:30 AM… after failing to fall back asleep in an hour, I started correcting a mistake in one of the regression posts… I updated the post at 5:30 AM and went back to bed.
The mistake, in case you’re curious, is in the Toyota post of August 22: I meant to have Mathematica® display the correlation matrix of the data… but I actually asked for the correlation matrix of the parameters… thinking that I was computing the correlation matrix of the design matrix. Sheesh! I know better. I just wasn’t paying attention to the details.
Actually, I feel pretty good about that… over the Thanksgiving holiday I reread all 30 regression posts… and that was the only mistake I found. Of course, that doesn’t mean there aren’t more mistakes I made. I’d be very surprised if there are no more.
I came across some interesting numbers this week or last. Frankly, I don’t actually believe them – having no idea where they came from, but… the IRS (the United States tax collecting agency) has an approval rating of 36%… Paris Hilton has an approval rating of 15%… more surprisingly, British Petroleum had an approval rating of 16% during the Gulf of Mexico oil leak… and –
…and the United States Congress has an approval rating of 9%. (Even my dictation software had 90% as its 1st choice, 9% as its 2nd, when it was trying to transliterate my speech.)
- Maxwell’s equations for electromagnetism and his deducing the existence of the displacement current;
- Euler’s formula for as a special case of
- the quadratic equation and more, but not for degree 5 or higher
- the SVD generalizes the eigendecomposition
- Fermat’s last theorem and the Taniyama-Shimura conjecture
- any possible soccer ball must have 12 pentagons, but may have any number except 1, but including zero, hexagons.
Let me make two more additions to the list.
The normal equations for “ordinary least squares” regression…. Almost every technical student has seen the equations for fitting a line to a set of x-y data. And a lot of them have wondered, how would you fit a parabola or higher polynomial in x… how would you fit y as a function of several polynomials? And the answer is that the normal equations solve all of those cases in one compact form:
where the are the coefficients of the fit. I discussed that early on.
Finally (for today), I think every math geek should know this quotation by the Nobel laureate C. N. Yang. Here’s how he remembered it ten years after saying it:
“There exist only two kinds of modern mathematics books: one which you cannot read beyond the first page and one which you cannot read beyond the first sentence.”
He’s not alone in that feeling. Somewhere, sometime, I came across a quotation from Sir Michael Atiyah, a distinguished mathematician – I knew that even without the “Sir”, which makes it official – who said that it wasn’t fair to ask a mathematician if he had read such-and-such book… mathematicians should get credit for each page they read.
To speak now of pages written instead of read, I do have a post awaiting final edits for this coming Monday. I also have more books on the way from Amazon, and maybe I’ll talk about them after they arrive.