Regression 1 – Normality, and the Chi-square, t, and F distributions

Introduction

(There are more drawings of the distributions under discussion, but they’re at the end of the post. This one, as you might guess from the “NormalDistribution[0,1]”, is a standard normal.)
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Happenings – 2012 Jul 28

The most interesting news this week isn’t really news… it’s rather old stuff as these things go… but it’s new to me. (One of my readers is probably going to say, “You should have learned this 7 years ago from Science News.” He should know… he renews my subscription every Christmas. For which I’m grateful.)

As I said last week, I just ordered 4 books. 2 of them have come in, and one of those was Ian Stewart’s “In Pursuit of the Unknown: 17 Equations That Changed the World”. I wasn’t expecting to learn anything from the book, but I did:

The interplanetary superhighway.

(Also known as the interplanetary highway and the interplanetary transport network. Searches on any of these 3 terms should be quite productive. I’ve provided a handful of links from just the first page of Google results, and from Wikipedia.)

Personally, I think it could better be called the interplanetary hiking trails – it’s a large collection of very low energy, but also very slow, pathways between equilibrium points in the solar system. It means, for example, that if we’re willing to spend more time at it, we can get to Jupiter without using a slingshot – gravitational assist – around Mars.
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Happenings – 2012 Jul 21

I’m almost embarrassed to say it – in view of the terrible massacre in Aurora, Colorado – but my week was pretty good.

I returned to work, but the week off before this let me make good progress on mathematics. The next technical post will be the regression post that I was struggling with before the rings and integral domains posts went out. It’s already through stage V – out on the blog, unpublished, awaiting final edits – although I’m thinking about adding a few more pictures.

After that, I expect to put out one more regression post, and then a bibliography post for regression. And with that I’ll think I’ll stop writing about regression for a while (Yeah? Yeah!)

I’m also making good progress on the post about constructing the final tableau of a linear programming problem – without going through the row operations.

I’m also thinking about a couple of posts on electric circuits… one of which will introduce Laplace transforms… but I was also hoping to say something about how Heaviside’s operational calculus is related to Laplace transforms.

Here is the link that reminded me of it… it turns out I have 2 books that discuss the operational calculus, but neither relates it to Laplace transforms. That link, however, provided a reference… so – what else? – I slipped out to Amazon and ordered the book, Spiegel’s “Applied Differential Equations (3rd Edition)”. I’m hoping it will be a pleasant and informative read.

But I hate to order just one book, so I checked my list of potential purchases, and also ordered

Ian Stewart’s “in pursuit of the unknown: 17 equations that changed the world”…

Rotman’s “introduction to homological algebra”…

Folland’s “quantum field theory –a tourist guide for mathematicians”.

Yeah, just what I needed: another book on quantum field theory, and another book on homological algebra. And that search for books is what delayed this post until after noon.

Meanwhile, my alter ego the kid was looking at Hilbert spaces this morning… those are complete inner product spaces, possibly infinite dimensional, and are a nice intermediate stage between finite dimensional vector spaces and spaces of functions in general (“functional analysis”). They’re also essential to quantum mechanics. Maybe if the kid hadn’t been reading about Hilbert spaces I wouldn’t have ordered the quantum field theory….

Anyway, with a post almost ready to go for Monday, I am free to do whatever I want this afternoon… and I haven’t given it any thought yet today… but I’m sure I’ll think of something. (I have a list… boy, do I have a list!)

Oh, there were no earthquakes in my neighborhood during the past week.

Integral Domains and the failure of unique factorization

Introduction

Recall the Venn diagram illustrating special kinds of integral domains.

I want to look at integral domains in general, but integral domains that are not unique factorization domains (UFDs) in particular. I’m interested in the outer ring of that diagram.
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Happenings – 2012 Jul 15

I can’t believe I completely forgot to put out a diary post yesterday.

I can, however, understand it. I was off work all week, on vacation but staying home. I knew that yesterday was Saturday, but my schedule had been so different during the week… without thinking about it, I guess I extended my vacation frame of mind onto Saturday.

I have made good progress on the next rings post, the next regression post, and the post about finding the final tableau of a linear programming problem (given Mathematica’s rather stark solution of the problem).

As for earthquake record-keeping, there were 4 more earthquakes in my vicinity in the past week, bringing the total to 10 for July.

And, any of you who care about tennis must know that Roger went on to win the championship.

And with that, I’m going back to mathematics. Tomorrow’s post is through stage IV – the lecture, as it were, is written; later this afternoon, I’ll move it out there with pictures and equations.

Happenings – 2012 Jul 8

This is a weekend for distractions from mathematics.

For one thing, I’m watching the men’s singles championship at Wimbledon even as I draft this post. As I’ve said before, I’m a staunch fan of Roger Federer, and I’m excited to see him in a grand slam final again. It’s been a while, what with the younger Nadal and Djokovic each in his prime.

For another thing, this post is a day late. Well, yesterday I visited a friend who was in the hospital. He seems to have recovered from a brief crisis that put him into intensive care, but he remains hospitalized.

As for the blog, last Monday’s abstract algebra post about rings followed in the footsteps of the 1st abstract algebra post a few months ago about groups: it got more than 300 hits, and the blog as a whole set a new record on Monday with 542 hits total.
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Introduction to Rings

Edit 7/12/12. Write Z[(1+\sqrt{-19})/2]\ , near the end. See the second “edit”.
Edit 7/17/12. Write a + (b/2)(1+\sqrt{D})\ near the beginning… so this is the earliest “edit” Life would be so much simpler if all quadratic integer rings looked like a + b \sqrt{D}\ , but they don’t! We’ll get to this in the next post.

The material I’m about to introduce comes from a 1st undergraduate course typically called abstract or modern algebra.

What I really want to talk about are number systems of the form

a + b \sqrt{D}

where D is an integer, negative or positive (and in some cases numbers of the form \frac{a + b \sqrt{D}}{2}\ – oops, edit: make that a + (b/2)(1+\sqrt{D})\ ). These are called quadratic integer rings. What fascinates me about them is that they have many of the properties of the integers – but at least some of them lack the most fundamental property of the integers.

The fundamental theorem of arithmetic is the one that says that any integer can be written essentially uniquely as a product of primes.

Consider, however, the products

(1+2\sqrt{-5})(1-2\sqrt{-5} = 3\cdot7 = 21\ .

That is a big deal – because in the system of numbers of the form

Z[\sqrt{-5}] = a + b \sqrt{-5}\ ,

all 4 of those numbers are irreducible: they themselves cannot be factored.
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