ChapterZero

I’m noticing that this semester, while I’m still attempting to read more books that physically possible at one time, for a change, it is _not_ due to the fact that the courses I’m taking do not live up to my expectations.

I’m taking Statistics, Thermodynamics, Antenna Design, Introductory DSP, Electromechanical Energy Conversion, and DLD. DLD seems like it will be fun, and I enjoy the way the prof teaches– very mathematically oriented, even though he is sometimes incorrect when he talks about the field itself. What’s more, it seems like I will learn genuinely useful information in there, that I can apply to designing useful logic circuits. Not that I have much call for logic circuits yet, but something tells me I will in the future. Along with that, I joined the Theory-Edge discussion group, and have found that, for some reason, binary functions may have revolutionary applications, which keeps me alert in class, trying to figure out why.

The conversion class will be the most challenging, I can tell already. But that’s ok– that’s why I took it. I feel as an EE, I should be as well-versed in the magnetic aspects of engineering, as the electrical; however, it seems even in the EM courses, we avoid talking about magnetic fields, except as they arise from, or can be used to derive, the electric field. The focus is always on electricity. But in this class, the focus is on the magnetic field. It’s a tough class, no doubt, one I’ll have to work hard in to score an A. Just yesterday, I spent four hours doing three very basic magnetic circuit problems!

Antenna Design is ok– not much theory, just plugging in formulas, but it seems like a very practical engineering class, which I appreciate. And I’m not sure I could handle the derivations DSP is challenging also, and potentially very useful, given my interests. And stats is getting more and more interesting everyday– now I see how people can actually major in statistics. Fascinating stuff, and it requires subtle thought.

This is a state of the Alex announcement. I noticed that Google has my algebra page at the top of the results list for the search “computer algebra notes”– I wish I could say it was deserved. But of course it isn’t, or why would I be making that particular search? Sad to say I haven’t done any work on that portion of the site lately; in fact, I haven’t done much math lately, just thought about it. Seems like I never have the time. I’m still learning Scheme, averaging about 10 pages of SICP per week, which then promptly disappears from my mind with disuse. I’m a apprehensive of using Scheme to tackle any programming projects– still unsure of how to use functional programming to accomplish significant tasks. I’m semiseriously contemplating a test project of using Scheme to implement a Boolean expression simplifier; only thing that stopped me from jumping right in is that it would require symbol or string manipulation, which I’m far from knowledgable about. I’m contemplating removing the dates on these postings, to make the site less cliched– I’m tired of the whole blogscene. I write only for myself, and whoever chances by. The record here is solely for me. In fact, might rip up the current “design” and start over, with something more personal, if I ever have the time. Am highly intrigued by algorithms and data structures, once again, but no time to study up on them.

I saw these interesting comments on sci.math

> I know mathematics is a maze.
> But it is a maze without end.
> It is a maze where goals shift
> And directions change.
> It is almost impossible,
> If it is not impossible,
> To climb upon its walls
> And view everything whole.

I would add that mathematical proof is

A maze of infinite complexity,
of infinite exits and entrances
(each valid),
and of infinite dead-ends
(each invalid),
a maze of opaqueness and transparency,
of obviousness and mystery,
of unappreciated beauty
only seen as it finally becomes known
to esoteric eyes...

But seriously,… When I sometimes try to prove a math result, I will often get sidetracked, following other paths, paths not originally intended, to some other related results instead. (I might as well prove *something*!…)

Does anyone else have an excellent example of this?

(I know the maze-metaphor applies to Wiles’ {et al} proof Fermats Last theorem, since it turned out that a once-abandoned path was actually the way to Wiles’ {et al} ultimate goal.)

And of course, in mathematics, if mathematicians fail to solve something as originally planned, still their work need not be completely for nothing, since other related results can still come from their partial successes.

And more ambitiously, as far as the maze metaphor is concerned, can viewing mathematics and proofs as a maze or as a graph (one result leads to another leads to another), lead to any new ways to solve math-problems in general?

(I bet the ‘right-hand-rule’ algorithm for solving mazes has no obvious analog that can be applied to proving math results, as an example, but you never know…)

Leroy Quet

This was posted to the UH Math Enthusiasts mailing list:

Moday in Adv MV Calculus, Dr. Wagner mentioned this poem.  I thought
others might enjoy it.

There's A Delta For Every Epsilon Lyrics

There's a delta for every epsilon,
It's a fact that you can always count upon.
There's a delta for every epsilon
And now and again,
There's also an N.

But one condition I must give:
The epsilon must be positive
A lonely life all the others live,
In no theorem
A delta for them.

How sad, how cruel, how tragic,
How pitiful, and other adjec-
Tives that I might mention.
The matter merits our attention.
If an epsilon is a hero,
Just because it is greater than zero,
It must be mighty discouragin'
To lie to the left of the origin.

This rank discrimination is not for us,
We must fight for an enlightened calculus,
Where epsilons all, both minus and plus,
Have deltas
To call their own.

Show , using the quadratic equation. Nice huh! Maybe that’s why I only hear of continued fractions, and not continued radicals… because they are equivalent. Of course that only proves a simple case, where the radical/fraction is periodic. What if it isn’t; then can you find a way to convert a given continued radical into a continued fraction? If you can, then it would make perfect sense not to have a theory of continued radicals, otherwise…

Another interesting question is, what is ? From the equation above, I would think -1, but matlab disagrees. When I use it to approximate the sequence, the numbers fluctuate between 0.5+.866j and 0.5-.866j. Unfortunately, I forgot so much that I can’t remember how to plot with MatLab, so I can’t see what’s going on with the sequence of approximations.