ChapterZero

I no longer hate my father; I got over that a while ago. Now I just dislike him with a vague fervor, when I’m not completely indifferent towards him. Needless to say, today was one of those days. I’m no longer shocked, but still hurt, by the things he says sometimes. And the utter lack of trust and respect that he has for me. But, since I have little trust and respect left for him, it doesn’t matter much what esteem he holds me in. It used to… it used to burn me up inside everytime he allowed his utter contempt for me to show. Those fires have burned their way through me, and there’s no fuel left for them; now there’s just ashes, a dull aching. And I’m here in a voluntary emotional void, waiting for my time to leave.

Enough bitterness for one day…

You know you are really over worked when you start to object to the school being closed on holidays. I have a group project that is due in two weeks, and the lab is closed all this week, and none of the circuits my group have been able to find work! We’re trying to make a function generator out of BJTs that will operate in the 20Hz-200KHz range, that outputs square, triangle, and sawtooth waveforms, but the best circuit we have produces only triangle and square waves at frequencies < 130KHz. I need to make an A on this project, in order to make a B in the class. We were going to use the Rice ID of one of our group’s members wife to gain access to their lab, but today the head lab guy is going to be there and he knows everyone, so we can’t show our strange faces. And after today, their labs are going to be closed also.

What pops into your mind when someone mentions a “good”, or equally vague, but more acceptable “well-behaved” function? What are the conditions a function must satisfy to be well-behaved? Depending on the context, it may be any of an uncountable number of things: bounded variation, differentiability, smooth, bounded, continuous, lipschitz continuous, one-to-one, etc. But if you’re like me, if someone shouted out “that’s a good function, buddy!”, you would think of a collage of all of these things.

And if you’re like me, you converged on your personal truth of what makes a function a “good” function through a process of iteration. First, and for a long time, it was continuity; but then, you realized a lot of functions (piecewise defined ones, for example), are continuous, but not nice. Then, and for a longer time, it was differentiability. Why doesn’t this serve as a satisfactory definition of a good function? Well, it could, if you were willing to admit functions whose have non-convergent Taylor series. But, you probably wouldn’t, and there are lots (of course?, since there is one) of functions which are differentiable, but have non-convergent taylor series.

Also, in my intuitive conception of a good function, it is impossible for a good function to be identically zero on an interval, unless that good function is itself the zero function. I thought, until today, that it could be proven that that condition is a necessary result of a function being differentiable. Too bad; that’s not true, at all.

An example of a differentiable function with a non-convergent Taylor series: the Cauchy function, defined piecewise as , is in fact continuous and differentiable at all ; since , the T.S. expansion taken at is identically 0.

“Mollifiers” are examples of differentiable functions that are identically zero on an interval, yet are not identically the zero function. One mollifer is the piecewise function , which is a bump in the first interval of definition, and zero everywhere else.

This all raises a question: is it possible to have a continous differentiable function which has a non-convergent T.S., and/or is zero on an interval yet is not identically zero, which is not piecewise defined?

The search continues for a satisfactory standard for a “good” function. I hear tell of analyticity, but have no idea of what that means yet…

I’m installing the teTeX distribution, which is huge!, on my website right now, so I can conduct some online mathematical experiments I’ve been storing in the back of my mind for a long while. The latest: a web front end to Axiom, or some other free CAS, so math people like myself can indulge ourselves even when we don’t have access to Maple on our desktops.

I really hope this works, and doesn’t piss off my hosts too much. I can see why they don’t have TeX installed- who would use it, for the most part– but I need it to make the max of my site. The things that could potentially piss them off are: the huge and quick downloads that brought the source code to the machine, the long and probably processor cycle intensive compilation process, and the space being taken up. But I figure, the processor usage spike is only going to be this once, and a lot of the space will be freed up once TeX is in its final binary form, and I delete the source.

Anyhow, cross your fingers and hope with me, for a very TeXtual future.

Just remembered: watching TeX compile is reminding me of just about every episode of Alias where Marshal has to corrupt someone’s software, and all you see is a Make/AutoConf script chugging away.

I’m very pleased with the services rendered by my web hosting providers, Imhosted, and I recommend them to anyone who wants a nice package, with great support. I have all the acronyms you could want– PHP, SSH, MSQL– and, they actually fix my problems as soon as I report them! Which isn’t often. Great job guys.

I’ve only been able to find one book on SPICE at our school library, and it’s really old. I think maybe that’s because the people who use SPICE only use the GUI interface, which is straightforward enough that they can avoid having to actually read a manual. Oddly enough, every one of my professors has mentioned SPICE to us in a vaguely recommending manner, yet they don’t seem to use it, or know much about it. Interesting…

So I ordered some books on SPICE through interlibrary-loan, they arrived today, and I just started reading one. Did you know that SPICE originated from UC Berkeley? And that it was created in a class that one professor decided to teach, just so he could build a circuit simulator? That is awesome– some day I hope decide to offer a class because it coincides with what I want to do. As opposed to taking classes that certainly do not coincide with what I want to do!

My goal is to learn the SPICE netlist format, and how to read SPICE’s output, so I don’t have to rely on the display capabilities of commercial Windows SPICEs, and I can do circuit simulations from the convenience and safety of UNIX.

My hands are very tired. I turned in an EM project this Monday that I spent about 12 hours on the night and morning before, pretty much just typing (mostly exporting data from Maple to Gnuplot, to Acrobat, to Illustrator– this is a process I find myself going through way too often– I will pay cash to anyone who can show me how to automate this [ I thought of using DDE or some similar programmed solution, but I don’t have prodigious amounts of time to waste on learning Microsoft’s crappy technology] ). I better get an A for effort, at the least. I hate those projects; it seems they exist solely because most people score low on the tests in that class. Since I have no problem with my test scores, I just find them tedious and boring; even if I did need to boost my test scores, I don’t think I’d find those very useful, because they are too damn tedious for you to do well on! If you’ve ever worked a contiguous 4 hour period on one assignment, knowing that you’re not even half way done yet, and it’s due tomorrow, you know what I mean.

Back to the topic of the moment: the calculus of variations. And what a cool topic; if you’ve never heard of it before, I’m shocked. If you’ve heard of it before, and actually have an idea of what it is, and you’re not a hardcore mathematician, mechanical engineer, or physicist (or the appropriate type of student), I’d be shocked and pleased. I personally have found it to be one of those topics, like measures, that everyone (read I) have encountered, but never found a completely satisfactory explanation of. Either it is mentioned in a context which assumes you know what it is already (” and this is a standard result from the calculus of variation”, … ), or the explanation attempts to subsume too many technical details into itself (like, explaining measures as tools used in constructing generalized integrals– what the hell does that tell you, unless you know that measures are generalized volume metrics?)

So here’s an excellent introduction to the subject (caveat: this is only in my meaningless opinion, since I haven’t read more than 10 pages on the topic, much of which I’m not sure I understand) :

The calculus of variations is concerned with finding extrema and, in this sense, it can be considered a branch of optimization. The problems and techniques in this branch, however, differ markedly from those involving the extrema of functions of several variables owing to the nature of the domain on the quantity to be optimized. A functional is a mapping from a set of functions to the real numbers. The calculus of variations deals with finding extrema for functionals as opposed to functions. The candidates in the competition for an extremum are thus functions as opposed to vectors in R^n, and this gives the subject a distinct character. The functionals are generally defined by definite integrals; the set of functions are often defined by boundary conditions and smoothness requirements, which arise in the formulation of the problem/model.

Nice.. it introduces the calculus of variations, and also the concept of a functional. Now I have a ready answer for those people ( non-electrical engineers, of course — we use it for just about everything, so we better not bite the hand that convolves for us) who proclaim the δ function is not a function, so is mathematically insensible. It is a function, just a function that maps from the space of continuous functions to the real numbers in the following fashion: . With this definition, the δ(t) relation has to be a function. It would be interesting to see how the usual properties of the δ function follow from this.

Now I wonder, are there any interesting uses for functionals that operate solely on other functionals? What would a functional have to be like to be considered continuous? I can’t wait for function theory! On that note, I go back to minding my undergrad p’s and q’s.

I just discovered another Jerry Springer like show, “Cheaters.” Just now, they staged a meeting between a man, one of his hos, and another ho he got pregnant. So the pregnant ho and the dirty ho were all over each other, not even really fighting about him, just calling each other dirty bitches and hood rats. Of course, this was in a public parking lot in front of a store, at night, so there was a lot of traffic. After that obvious ploy for some ho-fighting action shots, as the crew climbed back into their SUVs and rolled out, the host, Joey Greco, was trying to console the aggrieved pregnant ho.

I just got back from a UH Math Enthusiasts (aka math club) meeting, and I have to express my dissatisfaction. At both meetings that I’ve gone to (out of three), all the questions raised were either trivial or unclear. Here is an example of the best one raised today:

Let and be two consecutive odd primes. Prove that can be written as the product of at least three primes.

And even that one is not satisfactory; it is precise and sweet, but not very substantial. The majority of the problems we had were of the visualization sort, where the problem is fuzzy and there are as many interpretations of where we’re going as there are people in the room. In general, I like those fuzzy questions, but I don’t think they’re fit for a group of people to work on, unless you really make an effort to limit the problem and specify exactly what you’re looking for.

So I guess my complaint is that we lack the discipline to pose well-defined problems. Other than that, I had fun. Even though the problems were fuzzy, they were still fun to work on, until it became frustratingly clear that we would never agree, not because anyone was approaching it incorrectly, but because the problem was too ill-defined.

And I got a resolution to one of my struggles with the δ function, although not the one I wanted.