Happenings — Mar 27

I can’t believe how late it is as I start this draft… but I am determined to put out a diary post today.

Not much is happening, and that’s the problem.

Most of the time, I have a project… I am focused on something… I am posting mostly about whatever it is. And minor, interesting things happen and they are noteworthy largely because they add spice to whatever I’m working on.

Right now, however, I have no mathematics project in hand. Yes, I expect to put out more posts on logic. Yes, I can put out a few posts on regression (ordinary least squares).

I know the mathematics for both of those subjects… more importantly, I know the specific topics I wish to address… I just have to crank out the posts themselves.

This is not a bad state of being. In fact, it is a very good one: I am under no immediate pressure to find topics for the blog. I like that.

I just can’t decide which of a zillion mathematical things to do. (Interplanetary orbits are a very likely candidate, but we’ll see. And some small topics — how about the Fast Fourier Transform? — might show up.)

Books added: more logic & proof

The following books have just been added to the bibliography.

Copi, Irving M. & Cohen, Carl. Introduction to Logic.

Gensler, Harry J. Introduction to Logic.

Tao, Terence. Analysis I.

Logic: truth tables

edit: 29 Mar, the compact proof truth table for modus ponens needed “2”, not “1” under column 7.

I think this will be the first of three posts on logic. In this one, I will look at truth tables and at using them to prove tautologies (valid logical propositions).

(If I had known how much typographic trouble this post would cause…. Well, after a little practice with the new symbols, it wasn’t so bad.)

I expect that the second post will deal with “quantifiers “, namely “there exists” and “for all”, and their classical or linguistic counterparts, “some”, and “all”.

And the third post should deal with Aristotle’s syllogisms. They started it all — for me in particular, and for the world in general. All I wanted to do when I started was review the syllogisms given what little I knew of modern logic. It turns out there’s a major difference between Aristotle and modern logic, largely motivated I think by the explicit idea of the empty set. We’ll see the difference in principle in the second post, in practice in the third.

As usual, I’m not trying to write an introductory text here, just picking out a few things that interest me.

Introduction

Happenings Mar 20

Well, it’s already past noon as I begin to draft this post. Maybe I’ll keep it short so I can get going on the mathematics and technical blogging I should be doing.

In many ways last weekend was very frustrating, and things didn’t get better when I grabbed some time during the week.

Color. I’m still trying to figure out how to design residuals which, when added to a sometimes negative fundamental spectrum, will give me a spectrum that is everywhere positive (actually, of course, everywhere nonnegative). Nothing I tried last week worked — but I do have another idea. That’s encouraging.

In addition, I really should have a go at working out the nonlinear characteristics of my computer screen.

Orbits.

Happenings — Mar 13

Well, it’s a little later than I would like on a Saturday morning, but that’s the way things go. (And yet, as it happens, it looks like this will go out like most of these posts, shortly after noon.)

My kid has already had his time playing. Right now he’s reading through Jänich’s “Topology”. No, of course that’s not on the small desk — the kid gets to rampage through my entire library, and he gets to grab anything he wants — well, anything of mathematics or applied mathematics.

I hope to speak more about Jänich’s “Topology”… but, for now, let me just say that it is one of several delightful “Undergraduate Texts in Mathematics” from Springer. Many of the books in that series seem to be standard textbooks — but the ones that especially delight me are those that either take one small topic, or take a stroll through a subject.

Color: from XYZ to spectrum

Introduction.

In this post, I will go from XYZ coordinates to a spectrum.

I have to ask: what would somebody do with this? Especially, what would they do that couldn’t have been done using the XYZ tristimulus values directly? I don’t know – but let’s just solve the problem. I do not yet always have a satisfactory solution, and I will illustrate both satisfactory and unsatisfactory solutions.

Yes, we have done this before – but not as our primary purpose, but rather as part of another computation. It is worthwhile to tackle this specific problem, because there is one subtlety.

In general terms, the solution is simple: the XYZ tri-stimulus values are proportional to the components (with respect to the dual basis) of a (fundamental) spectrum. Find the dual basis, then get the linear combination defined by those components.

Happenings — Mar 6

“Live as if you were to die tomorrow. Learn as if you were to live forever.” Mohandas (“Mahatma”) Gandhi.

(That’s one of the inspirational quotations on the wall of my study.)

Regardless of what else I might choose to talk about this week, the dominant subject was, in fact, particle physics.

I went to the annual “Oppenheimer lecture” at Cal last Monday in the late afternoon. Frank Wilczek, a co-winner of the Nobel Prize in Physics 2004,”for the discovery of asymptotic freedom in the theory of the strong interaction”, was the speaker.

Books Added – Logic

I think my kid picked logic up again late in January. In particular, he wanted to look again at two recent books intended to help students make the transition to abstract mathematics — i.e. to having to prove things.

Those two books are Exner and Hummel. They were highly recommended for that, out on the sci.math newsgroup, and, therefore, I immediately bought them.

In addition to those two books, I ended up looking at Aristotle (indirectly), Lewis Carroll — yes, for logic! — Paul Halmos on logic as algebra, and a few books on symbolic logic.

Speaking first of the two textbooks about how to prove things….

Color: more on color-primary transformations

introduction

(Notation: It has become my custom to use, for example, N’ to denote the transpose of N.)

This post has two purposes.

1. To show you Giorgianni & Madden’s (see bibliography) version of the calculation I first found in Glassner (see bibliography; and there is a link below).

2. To show you what happens when we change the white point — only the white point.

I want to show you Giorgianni & Madden’s version of the calculation because they do it slightly differently — they compute a transition matrix. As always, the surest way to avoid misunderstanding is to show you a calculation. You may not know why I did something, but there should be no doubt about what I did.

And I want to show you what happens when we change the white point. I talked about this at the end of the “color primary” post (okay, what I still think of as the “Glassner” post); the link is below.

I can spare you some suspense, but I’m going to do the calculation anyway.

One disadvantage of having Mathematica at my fingertips is that I can compute before I think. (There are advantages to walking around the block every so often, and they’re not just physical; it gives me a chance to think, stuck in circumstances where I cannot compute.)

Let us first review the calculation as Glassner presented it.