Happenings — Feb 20

I have made a few changes: search on “edit”.

It has been a very long time since I put out a diary post. I got a kick out of looking at the last one: written when the very popular Fourier Series & Fourier Transform post was just an idea, and when I had just picked up Cohen’s Color Theory.

Let me try writing a short one. (Hmm. Here’s what it turned out to be….

Color.

I am currently working on a post about 12 metameric gray spectra. Work came to a crashing halt last weekend when I realized I was mixing up some of the calculations. (It happens. Gee, I was only trying to do 48 cases!)

On the other hand, I was also generally having trouble with that post — and that is a strong indication that I need to stop and take a fresh look at it before I post.

I intend to write about getting from XYZ to a spectrum. Frankly, getting to a spectrum is trivial… getting to a physically realizable spectrum — there’s the challenge. In the meantime, there are a few other things I want to, or need to, look at on the way…. I may need to look at numerical integration methods, to try to figure out exactly how other people get their slightly different answers… I have found the “Glassner calculation” elsewhere — so I should stop calling it that! But I haven’t found a proper name….

And I think I understand where the nonlinearities are — more importantly, I understand when linear models are appropriate. Although I need to say this elsewhere, let me say it here too:

There is no valid universal assignment of color to the XYZ tristimulus values, because the human visual system is a nonlinear function of intensity (edit: was “Y”, but then we could do it.).

This leaves us free to make linear transformations in the vector space for which XYZ is one basis. It also leaves us free to make nonlinear transformations if we choose (for example, CIElab).

Logic.

Huh? Where did that come from? Well, my kid picked it up again a few weeks ago and my grown-up stayed with it; I had last looked at it sometime not very long before I started the blog.

The most striking thing is that modern logic and Aristotelian logic differ on one key point. When Aristotle says that all angels have wings, he is also saying that some angels exist. To put that another way, he doesn’t believe in the null set.

Oh, he’s not the only one. Lewis Carroll believed in the empty set, but he still thought that you should not say all angels have wings unless you know some angels exist. That is, he believed that a universal affirmative asserted existence.

Edit: it helps if I add that mathematical logic presumes that a universal affirmative does not assert existence. We happily make assertions about things that do not exist. (The fraction that represents the square root of 2, for example.)

Anyway, I enjoyed looking at Aristotle’s syllogisms from the point of view of modern mathematical logic; and I may write up something about it. Along the way, I may talk a little bit about logic in general and proving things. In addition, Mathematica can do something along these lines, but I’m not sure exactly what.

Oh, work on logic came to a screeching halt last weekend when I realized that Lewis Carroll’s “Symbolic Logic”, Halmos & Givant’s “Logic as Algebra”, and Joseph’s “Introduction to Logic” all agreed that Aristotle presented 19 syllogisms — but they had three distinct lists! They didn’t agree on what the 19 were.

Toward the end of last year, I picked up stepwise regression again, and upgraded my code to version 7 of Mathematica, and added a few refinements. I’m looking forward to writing about this. As usual, sadly, having put it down I am deterred by the mental setup time from picking it up again.

I have made one small change for all of the mathematics I mean to look at now and then. I have assembled a double handful — okay, a triple handful — of books on one small desk. Whenever I feel like it, I pick up one of them and curl up with it.

The only rule seems to be: no more than one book per subject. The second book on the subject can go out there after I finish the first. (OK, in some cases, after I make suitable progress in the first, whatever that turns out to be.)

I think I should put Bloch out there (… done), because I really, really need to summarize chapter 3 so I can move on to curves and surfaces for the blog without feeling guilty. I like the theory of curves and surfaces!

At 15 books, this pile may be just a little too big, but let me see how it works out. They’re either short books which I want to completely read real soon now and be done with, or they’re big books in which I want to make steady progress.

I jump around a lot, and I put things down when I get confused. I mean to get back to them, but then I get distracted by other things. Maybe putting these books in one location will help me get through them.

Most of these books are not in my posted bibliography; I have indicated (biblio) which ones are.

• Aris’ “Mathematical Modeling Techniques”.
• Arnold’s “Mathematical Methods of Classical Mechanics”.
• For Lie groups, Baker’s “Matrix Groups”.
• Bloch’s “A First Course in Geometric Topology and Differential Geometry”. (biblio)
• Primarily for modules, Dean’s “Classical Abstract Algebra”.
• Fulton’s “Algebraic Topology”. (biblio)
• For frequency domain time series, Gardner’s “Statistical Spectral Analysis”.
• Mermin’s “Quantum Computer Science”. (biblio)
• For fiber bundles, Nash & Sen’s “Topology & Geomety for Physicists”.
• Ruskeepää’s “Mathematica Navigator”.
• Sethuraman’s “Rings, Fields, and Vector Spaces”.
• Skogestad & Postlewaite’s “Multivariable Feedback Control”.
• Stillwell’s “The Four Pillars of Geometry”.
• Swallow’s “Exploratory Galois Theory”.
• Thomson’s “Theory of Vibration with Applications”.

Now, breakfast… find out what math my kid wants to do this morning… figure out what math my grown-up should do (logic would be the obvious choice)… and then work on the next color post for the blog.