Happenings – 2011 New Year’s Eve

Well, I was off work this entire week. I should have a post for this coming Monday… but I still have to write a summary for it. I should even have a post for the following Monday… but in that case I actually have to write almost all of the connecting narrative. The images have already been uploaded and inserted, but I have yet to explain what they are and what I’m doing.

I’ve continued playing with fractional linear transformations, categories, circuit theory, and linear programming. In particular, I might be right on the edge of understanding one last detail in what I want to show you about linear programming.

And, in a blinding flash of insight, it is clear to me why barycentric coordinates are not unique unless we use triangles. I asked that question back in February 2009, toyed with it for a little while, put it down and moved on. And I really did move on, thinking about the question only when something reminded me of it.

Simply put… take a rectangle, cut it into 2 triangles with one of the diagonals, and then draw a point inside one of the triangles. Well, using barycentric coordinates based on the triangle which contains the point, all of its coordinates are between 0 and 1… but using barycentric coordinates based on the other triangle, one of its coordinates is greater than 1 because it’s outside the diagonal edge of that triangle.

In other words, if we want to use rectangles to cut up something (say for finite elements), and if we want to use barycentric coordinates, then within each rectangle we would have to define a triangle to select one specific set of coordinates. We might as well use triangles in the 1st place.

Mathematics, however, is not the only thing I do in this week between Christmas and New Year’s Day. This is when I write an end of the year letter to serve as a Christmas card. I’m still working on it… I don’t have much time for mathematics for the rest of this weekend… possibly not for much of this coming week. That’s one reason I have 2 posts already in stage V… so I don’t have to do very much other than my letter for a while.

And so, for a change, this post closes with: I’m off to do some non-mathematical writing.

Happy New Year.

Happenings – 2011 Christmas Day

Merry Christmas, or happy winter solstice celebration, whichever you prefer, if you have a preference and if you observe a holiday.

This post is a day late because I was very lazy yesterday. I watched the Steelers shut out their opponent, instead of working on this post. Afterwards, I just puttered around the house.

I expect to be having dinner with friends this evening, but I’ve decided to make up for yesterday even though it’s Christmas, and do what I should have done yesterday.

Last weekend and during the week, mathematically…

Happenings – 2011 Dec 17

Let’s see. Last week’s happenings post seems a bit brusque to me. Perhaps I was just impatient to get it out there… perhaps it reflects that my dictating is more leisurely than my typing.

I haven’t said this in a while: I generally use “Dragon Dictate” to create text, rather than trying to type. Last week’s happenings post, however, was completely typed.

Or I might have been a little annoyed that I wasn’t able to talk about either Stokes’ theorem or the Axiom of Choice – and, at best, I will be lucky to write about Stokes theorem today. (It’s another piece of mathematics I think a geek should know.)

We’ll see how the draft develops.

For those of us even remotely interested in particle physics… it appears that the large hadron collider has seen something at 125 GeV. It could be the Higgs particle. Peter Woit’s blog, as usual, has something to say about it and provides a lot of links.

Regression 1: Example 8, Fitting a polynomial

I want to revisit my old 2nd regression example of May 2008. I have more tools available to me today than I did when I first created it – and it was originally done before Regress was replaced by LinearModelFit.

Recap: fitting a quadratic and a cubic

What I had was five observations x, five disturbances u – and an equation defining the true model: y = 2 + x^2 + u. Here they are:

Construct a full data matrix with x, x^2, and y:

Run forward selection… and backward selection…

Happenings – 2011 Dec 10

The blog had passed 140,000 hits since inception at the end of the day Tuesday Dec 6.

I have a post written for this coming Monday; as usual, when I say “written” I mean that it awaits final edits — typically, links and more transition and summary sentences.

It is, as I forecast, a regression post about fitting polynomials – but it’s only one of the two examples I had in mind; the second example is even juicier than I thought, and definitely warrants a post of its own.

My crucial Xmas shopping is done: what had to be shipped to the East Coast has, in fact, been shipped. Everything else is local… oh, there’s one thing I need to order on line.

As for the week that was, I’ve been browsing.

One of the things I came across was a 14-element triangulation of the torus. I have shown that we need at least 14 triangles (instead of the easy triangulation with 18), but I had not exhibited a solution. Here it is, based on Nash & Sen “Topology and Geometry for Physicists” ISBN 012-514081-9.

Happenings – 2011 Dec 3

I know exactly what I’d like to put out this Monday: 2 small examples of multicollinearity when we try to fit polynomials to data. The mathematics is probably done… but I don’t think I’ve actually written a narrative to accompany it… so what I have is, at best, through stage III. Getting a complete draft will take me through stage IV… transferring it to the blog will take me through stage V… and then it will still need final edits. In other words, I have my work cut out for me.

In the background, I’m working on some posts about group theory. My alter ego the undergraduate went through the group theory in Dummit & Foote “Abstract Algebra”… and I want to talk about a few things in it.

He has since worked his way through Pilkey & Pilkey “Mechanics of Solids” and Stilwell “Naïve Lie Theory”, and he has started an elementary book on electric circuits…. I need to start writing up the nifty examples I’ve seen in all these. (Did you notice the alternation of pure and applied?)