ChapterZero

Did you know that the 4 equations known as Maxwell’s equations are actually Heaviside’s equations? Maxwell’s equations were actually 20 equations in 20 variables. This is my fourth year as an EE student, and I have never heard this before; this is like having the rug pulled from under my feet— I emailed one of the professors in the EM group to see if he can clarify this for me. Shocking!

I stumbled across that fact while looking for information on Heaviside’s operational calculus, part of my intellectual heritage as an electrical engineer This was a method Heaviside developed during his stint as a telegrapher to solve ordinary and partial differential equations, based upon regarding differentiation as an operator. It has subsequently been displaced by the Laplace transform, even though his unorthodox approach (i.e. regarding division by the differentiation operator as integration!) has been given a solid basis by Bromwich. Here’s an example, taken from “Fourier Analysis and Boundary Value Problems” by Gonzalez-Velasco:

… to solve the simple differential equation for subject to initial conditions , Heaviside would denote the differential operator by — he had used the letter until 1886— and obtain the equation , where denotes the function that vanishes from and has value for [The infamous Heaviside step function]. Hence, if we treat as an algebraic quantity,

.

This cannot be the end of the line, of course. To obtain the actual solution from this operational solution, let us accept that the geometric series expansion is valid for this operator, and then

.

Since represents differentiation, Heaviside regarded as integration from zero to . In this manner,

and

for any positive integer , leading to the actual solution

,

which, perhaps surprisingly, is the correct solution.

I’ll say that’s surprising! Here’s a germane quote: shall I refuse my dinner because I do not fully understand the process of digestion? Well, I guess not.

On a related note, “Fourier Analysis and Boundary Value Problems” is a surprisingly good book— it is tailored more for engineers than mathematicians, I would say, yet it contains rigorous Hilbert space concepts, proofs, and even an introduction to measure theory (in the problems of Chapter 2), although Reimannian integration is used throughout, with a series of proofs given in an appendix to plug all the holes that were sprung from deserting Lebesge world. It doesn’t hurt that the typography is very well done, and there are lots of interesting, as opposed to forced, historical interludes. Too bad I’m borrowing it from a friend, instead of owning it.

I finished reading “Lysergically yours” in about an hour and a half… I was not impressed with almost any aspect of the book. The plot was highly unoriginal: government goons chasing deceptively knowledgable punk rocker with a checkered past, but a heart of gold; the only saving grace was that, instead of stumbling across a new weapon’s technology, he stumbled across a recipe for a new form of LSD which allows the user to make almost preternatural connections. In more than one scene, the protagonist reads incredible amounts of information from the movement of wisps of air. Other than this slightly original twist, the book is cookie-cutter, wanna-be Neal Stephensonesque crap. Well, there is one other point worth mentioning, and probably the only point worth taking from the book: in the end, the protagonist realizes he’ll never be able to run fast enough or far enough, so he releases a virus to spread the information over the Internet. Of course, the author makes a point of how easily and quickly said virus is created and disseminated— there is a lot of techno-savvy elitism in this book—, but what I thought was important about this is the idea that the whole plot in the book couldn’t happen. With access to the Internet, it is impossible to kill people to keep knowledge secret, provided that they have advance warning: all they have to do is post it far and wide. Not even the US government has enough control over the spread of information on the Internet to contain it.

I’m not sure what the moral of the book was, but I’m pretty sure it was intended to have one: either the idea that by restricting drug use, we are limiting our potential for development, or that the coming of age of the Internet destroyed the idea of government repression of sensitive information. The former was indicated on the book blurb, but the latter is the only one that the book even comes close to supporting, realistically.

It’s getting to be too much; everywhere I look, I see the Lebesgue measure/integral being used. Even though most applications are accompanied by ‘oh don’t worry about the details, just imagine you’re dealing with the Riemann integral— 95% of the time, you’ll only be dealing with nice cases’, I still feel like I’m missing out on something crucial.

I was digging through my stuff last night, looking for something to read right before going to bed, when I came across two sources on Lebesgue integration/measure theory: one, a tutorial by Rich Bass (the link is to a collection of notes he wrote) on measure theory and Lebesgue integration which I printed at least a year ago, and never got around to reading, and the other, the first chapter of a book on the general theory of integration, which I’m sure will give me more than enough detail. So now I can rectify my situation as regard Lebesgue theory.

Although, I just got “Lysergically yours” over interlibrary loan, so I might be spending all my time with that.

I just tried three times to post a comment, and each time it disappeared— didn’t show up in the moderation queue or in the comment list. That explains why I’ve been having a remarkable dearth of spam of late.

I can’t think of anything that might be causing this. My blacklist is clear, and I don’t have any active anti-spamming plugins. So I have another job to do.

Keith Devlin’s latest article is about students’ apparently decreasing ability to deal with the fact that mathematics takes effort. I have to say: isn’t this an obvious trend? There are lots of factors that contribute to this: the rise in graphing calculators, the impression given by popular culture that mathematics is for freaks, and that you can’t be good at it unless you were born a genius (notice, e.g. on Andromeda that Harper is a well-rounded genius: he excells at everything from bioengineering to mathematics; in general, there doesn’t seem to be much of a distinction made between being knowledgable in one science, and being knowledgable in all, or even simply being scientifically literate and being a genius; the choices are Jack O’neill or Samantha Carter). It’s also the case, IMO at least, that mathematics is tied into the more spiritual/artistic pursuits: music, for one, and especially philosophy— for no other reason than these are other pursuits that require dedication and focus to do well in. Clearly the trend is to relegate these pursuits to PhDs— so no one is motivated at a young age to develop their mental capacity in them. There’s a reason all the mathematical geniuses seem to have come way before our time.

I found a Mathematica grammar finally: in an appendix of the Mathematica Book. The section, “Some General Notations and Conventions”, is where the very important table of precedences is, but the other material looks useful also. I did some reading up on the pattern matching system, and was unpleasantly surprised to find that the way I proposed earlier is not nearly powerful enough, as it stands— Mathematica’s pattern matching system can best be described as a regular expression engine that works with mathematica’s data structures. After seeing that a simple integration system can be written using simply the pattern matching system, I’m starting to wonder exactly how much of Mathematica’s internals are written to take advantage specifically of the pattern matcher. I’m going to have so much fun trying to implement this…

Identities and inequalities are the twin bug-a-boos of mathematics. As Neils Bohr’s younger brother said: “Analysts are mathematicians who spend half their lives looking for inequalities they can’t prove”, or something like that. I can’t say I know enough to agree, but my experience in the real analysis course was that a lot of problems I spent hours trying to prove directly, or using simpler inequalities, my prof tackled in about two lines, using relatively complex inequalities. There are books dedicated entirely to inequalites— none of which I’ll ever read, I hope and believe. Identities are just as bad. You never know when knowing may save your life, so to speak. Of course, I realize the necessity of being proficient in handling both… I just don’t like it. That’s part of the reason I’m so interested in CASes: good ones, IMO, can handle all that for you. And, that’s part of the reason I’m mad that it took me so long to figure out that A=B is all about proving certain types of identities automatically. I’ve know about this book for a while, but I didn’t know what it was about… somebody’s got some ’splaining to do.

There are only two problems in this assignment; the first is a give-away. The second is probably easy, but I’m stuck on it right now:

For can be real valued?

This seems to be equivalent to asking if for all , .

Read the broccoli story! I particularly enjoyed the masterful use of a regular expression in this admittedly puerile thread:

[new] Editorial: +1fp, what the fuck? (3.00 / 2) (#22)
by Linux or FreeBSD on Wed Jun 22nd, 2005 at 03:28:15 AM EST

i'm going to have nightmares about this.

[ Reply to This | ]

    * [new] Editorial: s/nightmares/to masturbate/ by Wealthy Foreign Investor, 06/22/2005 05:00:49 AM EST (2.33 / 3)

There’s also the bagel story