ChapterZero

I’m having a problem with an algebra exercise, of all things:

Define, by means of a partition, an e.r. on which has, for each positive integer , exactly one equivalence class with elements. Describe this equivalence relation in the form ‘ iff ’

Update:
Got it! But there has got to be an easier one…

Then is a partition of . Visually, you construct it by defining the 1 element set to contain only 0, then the sets containing odd elements cover the positive integers in increasing order, and the sets containing even elements cover the negative integers. So:

Is there a simpler one? Since I did this assignment a whole two days before it’s due, I have time to pick some smarter people’s brains tomorrow.

Update again;

Turns out I misread the homework assignment: that problem wasn’t assigned after all.

I started studying last night for my Analysis test this Friday covering Chapters 8 and 9 of baby Rudin— ’some special functions’ and ‘functions of several variables’. Both of them were hard chapters for me, in that I didn’t really get several of the proofs.

Most of the material, I have a feeling, will be from Chapter 9— that seems more important—, and that’s what I had the most trouble with, so I got two books from the library that offered alternate approaches than Rudin’s. I wouldn’t be able to do that for Chapter 8, because how many books have comparable sections on at a level useful to me? On the other hand, lots of books had sections on the Implicit, Inverse, and Contraction mapping theorems (ImInvCont), not to mention functions of several variables.

The two books I picked up: “Mathematical Analysis I” by Vladimir A. Zorich and “Applied Mathematics: Body and Soul [Vol. 3]” by K. Eriksson, D. Estep, and C. Johnson turned out to be excellent choices. The first one is very much rigorous, even to the level of Rudin, but his approach is different enough, and he offers more motivtion than Rudin, that I’m going to finish reading his sections on the ImInvCont when I’m done with Rudin. The only thing I am finding a little off putting about this book is the author’s seeming obsession with notation. Sometimes it seems like he’s trying to recast analysis as a branch of logic.

The second book is in a class of itself. I’m considering reading all the volumes, because I love the approach the authors take of using Lipschitz continuity. I’d never seen it used before, but boy does it make all the proofs much shorter and intuitive. Of course, at the same time, you lose some coverage of pathological cases, but that’s fine by me. Between the intuitive approach in this book and Rudin’s rigour (and rigor mortis :)), I think I’ll be able to pull myself together for the test.

Recall the problem I had earlier of trying to define a non-integral number of compositions of a function. Specifically, if is an increasing function that maps onto, what meaning can we assign to , (the composition of with itself times, where )?

I presented some ideas I had come up with earlier this week, and was surprised to see that they were nowhere near the mark.

A little background on the problem: Dr. J. said one of his friends had posed it to him, and he couldn’t figure it out, even over a period of years. Everytime that friend saw him, he would ask him if he had figured it out yet, so eventually Dr. J. asked someone for a hint. Apparently, the hint was enough of an aid for him to figure the problem out.

He passed the hint along to us, really more of a definition— define to be

where is increasing.

You can check that this works out to give you the properties that you would expect: , (the identity), etc. But how do you find the particular , , given ? And are they unique?

It took me a while to get over the outlandishness of this definition, but I can see that it is plausible– is an increasing function from , and is an increasing function to , so their composition determines some increasing function from onto . Only, is in the range of this function?

Apparently the theory this is concerned with is that of ‘differentiable flows’, and is useful in engineering.

There is one complication, but it’s pretty simple to overcome. The problem is this, as the definition lies, cannot be just any increasing function. Look at the definition for a while, and see if you can figure out why not (it has to do with ).

The problem is, by the definition, and is increasing, so if , , while if , . So if crosses the identity function, or even touches it (then ), the definition doesn’t make sense. But this can be overcome by noticing that if you partition into a maximal partition where are the fixed points of and call the restriction of to , then each maps from its domain back onto its domain (because of is increasing). So if you let and , then , so the original definition applies, and

and we can define

so that

Once again, there is a problem here at the fixed points . I don’t think it would be cheating too much to just stipulate that .

Several times mysql has gone down, and I’ve had to restart it by hand. I was hoping this was all just coincidences, but it’s happened twice in the past week, so obviously something is going on. I’m going to try updating, and see if that helps.

God forbid that I should end up living in Houston, much less not having an affiliation with a good library, but it was something I was slightly worried about: after I’m out of school, if I’m not a professor, how will I have access to a good library?

I just found out that UH has a “Library Associates Card” program, for either $75 or $150/yr. I would definitely take the $150/yr. option, and consider it a good deal: a 20 book checkout limit, recall priviledges, on-campus access to all the databases the library carries, and ILL. Sweet.

For anyone who hasn’t already encountered it, Axiom is the best (in my considered opinion) freely available CAS, hands down. The only competition that comes even close is Maxima, but Maxima’s interface leaves a lot to be desired. Axiom has the potential to be better than Mathematica or Maple, and in some ways, I believe it already is— certainly the type system is a stroke of genius.

Recently I added an unstable repository to my list of Debian mirrors; that has given me a lot of goodies to mess around with.

Most spectacularly, after dreaming about having access to Axiom, I just installed in on this server! Which means that I can run it over a text display only, since this is a rented server, but I’m working on that… nothing seems to be stopping me from installing X Window and tunneling a connection except for the fact that I don’t know how to configure X Window properly over SSH.

Hopefully sometime soon I will have Axiom hooked up so I can run calculations through it, have Axiom convert the results to TeX, and then pipe that back here. I’m not really that familiar with Axiom (I read most of the Axiom book a year or so ago, but never had the opportunity to play with a working version), but I could probably do this, seeing as how that looks like what the AxiomInterface does.

Axiom ports are available for Windows and Macs. Luckily I brought the laptop to school today, so I’m going to download Axiom for Windows and give it a spin. And order the Axiom book from the library again.

TexMacs– a beautiful, albeit rather mispurposed program– can be used as a GUI for Axiom, under both Windows and Unix.

Chapter 9 of Principles of Mathematical Analysis is on functions of several variables, from a vector space approach. My major problem with it– the same problem I have with almost every chapter in this book– is that Rudin seems to go out of his way to make his statements obscure. Is that what I have to look forward to in graduate literature?

Here’s a typical example:

Suppose maps an open set into , and is differentiable at a point . Then the partial derivatives exist, and

Oh, okay… I see that is explained in the form I was about to suggest later on. I was going to complain that he didn’t simply state

but he does end up doing that, after the end of the proof. I say he should have put that at the start, because that is the form that a student who took, say, Multivariable Calculus would have the most familiarity with.

In general, there are a lot of places in this chapter where at least mentioning the matrix notation equivalent of the purely algebraic statements he makes would save a lot of time in understanding what he’s trying to say. And let’s not even start on the topic of illustrations.

Maybe someone should make a project of making illustrations for this horrible book, since so many schools seem to use it. Then we could hand out a little booklet to students that has illustrations for each chapter, with references.

A function is homogeneous of degree if . I’ve seen this definition before, in the proof of some inequality about integrals (a really famous one that I can’t recall the name of), but since I didn’t understand the proof, never really got interested in homogeneity until today.

I came across it again in a problem in a book: if is homogeneous of degree , then . I still haven’t figured out how to prove this, but it caused me to examine the idea of homogeneity. Like, does this problem assume that is continuous, or is that a result of the fact it’s homogeneous?

First I tried looking for individual examples of homogeneous functions, to get an overview of what they look like, and I came up with some: , , etc. But then I realized from the first example, that a metric has to be homogeneous of degree 1, by definition— or I thought I did, until I realized this holds true only if .

Then I tried generalizing my thought process to higher dim spaces, and realized that because of homogeneity, has only to be defined on the unit ball, and then for any , for some in the unit ball.

This gives us a way to construct arbitrary homogeneous functions, given just their values on the unit ball. Of course, you need to be careful that for each , . So, if and odd (even), then has to be odd (even) to be homogenous of degree , for example.

I realized from this ability to construct arbitrary homogeneous functions that homogeneity does not imply continuity, so the problem that got me started isn’t well-stated.

All of these observations can be made directly from the definition of homogeneity, but I think the thought process that lead to them in this particular instance was pretty neat.

I got a ticket last November, and I’ve been putting off taking the driver’s safety course. The certificate is due on Tuesday, along with my driver’s record, and I’m now taking the course. I opted for the overnight option, but I don’t really expect the certificate to be here tomorrow (because it is a weekend and I’m not going to finish the course until around 7 tonight). Also, ordered the driver’s record with the 5-7 day option last Tuesday, so it’s a toss up as to whether that’ll get to me in time. So I might end up missing my March 22, 2005 deadline… I wonder what happens then? I suppose even if they issue a warrant for my arrest, the most that will happen if I go in a day late is they will refuse to recognize the course, and make me pay the ticket. What a drag! Since that first ticket (it was actually two: one for cutting through a parking lot, the other for not stopping when exiting from said lot), I’ve spent > $300 on this, and I’d like to get it over with, so I can start having money again.

I’m posting this while taking the test on another computer (somehow, the software can detect when IE loses focus, and stops the clock that times the amount of time you spend on each page, so I have to use 2 computers). The test is set up so that I have to spend a certain amount of time on each page– like I really need 9 minutes on one screenful of information. Also, the questions are slightly retarded: what color robe is the judge wearing in the introductory video? And I signed up for the course using my sister’s car’s license plate number, because I don’t remember mine, and out of nowhere in the middle of the course, a question popped up, saying I must answer in 45 seconds, what is the number at the beginning of the street address where the car is registered to? I almost had an apoplexy, because I didn’t see our address on there, then I realized that the address of the apartment we used to live at was a choice.

5+ more hours of joy to go. Oh wait… it just forced me to take a 15 minute break. This just gets better and better.