Peter M Maurer

A discovery and some theorems

2007-07-30T14:10:42Z

I've been looking for representations of groups, that is sets of matrices over GF(2) or GF(n) that are isomorphic to Sn, the symmetric group of order n.

One well known group is the standard representation, which works in any Field. The standard representation is the collection of matrices that have a single 1 in each row and each column and zeros elsewhere. The collection of all such nxn matrices forms a group of size n! which is isomorphic to Sn.

In a sense this set of matrices is created from the following rows.

1000...
0100...
0010...
0001...
...

Each row contains n-1 zeros and a single 1. Every matrix in the standard representation consists of n rows, each one of which comes from this set. The elements of the standard representation are also called permutation matrices. A permutation matrix is itself a permutation of the n rows given above.

I have discovered that there is another representation that works for any field. To the n rows given above, add an n+1st row consisting of all -1 s.

so now we have

-1 -1 -1 -1 ...
1   0 0 0 ...
0   1 0 0 ...
0   0 1 0 ...
0   0 0 1 ...
...

All permutations of these rows give us n+1 nxn matrices. (We have to leave off the bottom row) These matrices are all non-singular, and form a group which is isomorphic to Sn+1. This also works in GF(2) even though -1=1.

(I don't know if I'm the first to discover this representation or not. I've never seen any reference to it, but it seems odd that something this cool would have escaped everyone's notice but mine.)

Let's call the standard representation Rn and the new representation Qn. Let Tn be the transpose of Qn. Tn is also a faithful representation of Sn+1, sometimes Qn and Tn are conjugate to one another, sometimes they are not.

I'd like to make the following conjectures.

1. Any faithful representation of Sn+1 in nxn matrices over any field is conjugate to Qn or Tn.
2. If k>n+1, there is no faithful representation of Sk in nxn matrices over any field.
     (why? I think such a representation would have to permute n+2 rows. An nxn matrix can represent a
      permutation of n+1 rows, because the omitted row is uniquely determined. For n+2, there is no
      way to uniquely determine the order of the two missing rows.)

3. Any faithful representation of Sn is either
    a. Rn
    b. Qn-1
    c. Tn-1
    d. A direct sum of Rk Qi and/or Tj for k    e. conjugate to a,b,c or d.

Geocaching

2007-07-30T13:39:51Z

Farewell, geocaching. It was fun while it lasted. My reasons for leaving are too complicated to go into here, but let's just say, that life without geocaching has been much more peaceful than life with geocaching. Too bad.

Theorems 1 & 2

2007-07-30T13:35:21Z

It occurs to me, upon reviewing my own work that I've encountered a couple of new theorems in linear algebra. The first is this, which I know to be true, but have never bothered to prove.

GF(2) is the finite field which is also known as the integers modulo 2. We can build matrices from elements of GF(2) and manipulate these matrices just as we can matrices over the real or complex numbers.

In the following, I'm only interested in square matrices.

A matrix over GF(2) is in K form (my terminology) if it has ones on the main diagonal and zeros elsewere, except for one entry which is equal to 1.

The following matrices are in K form.

11000
01000
00100
00010
00001

1001
0100
0010
0001

Let M be an arbitrary matrix over GF(2). Then M can be expressed as:

Q x K1 x K2 x ... x Kn x Qi

where Q is an arbitrary non-singular matrix Qi is the inverse of Q, and K1 through Kn are matrices in K form.

Possible improvements:

1. Insist that Q is a permutation matrix.
2. Eliminate Q entirely.

Another unproven theorem (which I'm certain is correct) is the following.

Suppose f=gM where g is a totally symmetric function and M is a non-singular matrix. Then
f=hN where h is a totally symmetric function and N is a product of K-form matrices.

Anybody have proofs?

On being a revolutionary

2007-07-30T11:56:54Z

This may just be arrogance, but I believe that much of my work is revolutionary in nature. I love inventing new algorithms, exploring new perspectives, inventing new ways of doing things. I have come to believe that the scientific community as a whole discourages revolutionary thinking. Well, not just the scientific community, the world at large hates revolutionary ideas.

Well, I've recently been reading an excellent biography of Albert Einstein, probably the greatest revolutionary the world has ever known. Around 1905-06 he was tired of working for the Swiss patent office and was trying to find a job in academia. He sent out applications everwhere, and included in these applications several of his published papers. His applications were universally turned down. In this packet of papers were three of special note.

1. A paper that established the branch of physics known as "statistical dynamics" and used a purely mathematical approach to predict the phenomenon known as "Brownian Motion." This paper is credited as being the first to prove the existence of the atom. At the time many physicists did not believe in atoms.

2. A paper that explained the photoelectric effect by showing that light consisted of discrete particles known as photons. This paper is credited with establishing quantum theory, because it was the first to show that quantization had a noticable effect in the real world. Until that time quantization was considered to be merely a mathematical device with no physical reality. This paper won Einstein his only Nobel prize.

3. A paper on time and space establishing a theory that eventually came to be known as "The theory of special relativity." A theory which not only revolutionized physics but the whole world.

I feel your pain Albert!

If a guy with this kind of credentials can be universally turned down, what hope do we lesser revolutionarys have?

A new perspective

2007-07-30T11:32:01Z

How things change! This summer I taught a course in VLSI Design Automation. I can see now that some of the work I published in that area was done when I was incredibly depressed. That's not surprising. I published one paper in 2000 that I considered to be "a mere curiosity." However, upon rereading it while presenting it in class I realized it was the starting point for an astounding amount of new research. Why didn't I see that before?

Well, at the time I was battling to try and stay at the University of South Florida. Despite the fact that it was "clearly time to leave" I didn't want to displace my children from the only home they ever knew. Consequently I was battling on every front I could think of to fix the problems and so I could remain with USF. Needless to say, these efforts were pathetically ineffective. I couldn't see it at the time, but I was spiraling into an abyss of depression in which everything I did seemed trivial and pointless.

Reading this paper for the class jarred me back to reality. I decided to do a realistic review of my past work to determine other loose ends that might be dangling. There are quite a few, actually. I think a person could put together a pretty substantial research career tying up these loose ends. Even tying up a few of them could put someone on the map. Metamorphic programming, for example. One could devote an entire career to that concept alone.

I remember leaving South Florida with the feeling that I was finished as a researcher. That I would never have another original idea as long as I lived. The truth is that there is a vast landscape in front of me with virtually unlimited possibilities. What was I thinking?

Be careful what you pray for ...

2006-12-08T11:43:00Z

In many ways it's been an angonizing semester. I was trying to do too many things at once, not exactly by my own choice. I was trying to increase the quality of my two classes by publishing class notes for each lecture. At the same time, I was trying to solve the interminable problems with the Czech student recruiting, I was trying to increase the quality of our graduate recruiting and also trying to put together a proposal for a new joint CS/CE Ph.D. At the same time I was writing and running research software for conjugate symmetries, and writing an NSF proposal. The amazing thing was that I was actually getting everything done.

Then, somebody slipped in a monkey-wrench. The graduate recruiting web pages I created were not exactly to the liking of ITS, so they came to Don Gaitros to explain what they thought was wrong with them. Their comments were only suggestions but Don treated them as commands from on-high and destroyed all of my web pages and substituted a bunch of worthless crap in their place. Don, as chairman of the department, actually has a right to do this sort of thing, but I was led to believe that it was ITS that made the changes, which absolutely infuriated me. ITS DOES NOT have the right to dictate academic matters, but Don apparently didn't want to tell me that HE made the changes and blamed it on ITS. When I went to the Dean about it, he explained to me that I just had to live with this. I've been thinking about relinquishing the job of graduate directory anyway because I detest going to the leadership committee meetings. However, this is the straw that broke the camel's back. Not really a straw actually. I refuse to work in any position where I do not have the support of the people above me. Particularly when I have the rug jerked out from under me. The Dean examined those web pages and enthusiastically approved them. A couple of days later, he gives me no support whatsoever. I refuse to do any job where I don't have the support of the people above me. Second, and probably even more important point. I refuse to work in any position where I'm the bad guy by default. I said we should do X, ITS said we should do Y, it was instantaneously assumed, without any discussion or negotiation that ITS was right and I was wrong. This is a situation that I find utterly, completely, and totally intolerable. The upshot of it is, I resigned as graduate director a couple of days after this happened. With the assumption that I would continue to work the Czech student problem and the Ph.D. proposal. I solved the Czech student problem. The Ph.D. proposal is still under construction.

But that's not what I wanted to talk about.

I was angry over this whole thing, and I couldn't stop thinking about it. I kept praying to the Lord to take the anger from my soul, and to help me to obey his command to "Love thy neighbor as thyself." It was so hard, and no matter what I did, I just couldn't put all this behind me. Then The Lord answered my prayer in a way that shows He also has a sense of humor. A few Sundays ago I had planned to do some Geocaching, but Brett Caroll (Sawdust) called up and asked if I wanted to go on a motorcycle ride. I hadn't ever been on a group ride, so I said OK. We were up by Corsicana when I took a curve a little too fast. Instead of correcting properly by leaning hard into the turn, I swung wide (as you would in a car). I wasn't really thinking. I swung just wide enough to go off the edge of the road. There was no shoulder, and no easy way to get back on the road or slow down, so I went down into the ditch to slow down and recover. My floorboard hit the edge of the ditch -- I was only going about 15 MPH at the time -- and the bike flipped on its side with my leg underneath it. It was a nice gentle slide with no damage to the bike, but my leg was trapped underneath and as I went flying off, my knee broke with a loud snap. I rode in an ambulance back to Waco and have been on crutches ever since.

And yes, my prayer was answered. I was no longer angry. In the light of this accident I was suddenly forced to focus on things that are considerably more important that a few stupid web pages. Things like life (I'm alive.) Health (will I be healthy again?) Work (I can make it back and forth to class, but it's so exhausting I can barely stand it.) Friends (without friends I'd still be lying in a ditch somewhere near Corsicana.)

Be careful what you pray for.

Well finally.

2006-07-11T11:53:00Z

I finally made it to 2000 caches. I'm surprised at how easy it is to accumulate a bunch in a day. I haven't been caching alone in a long time. I've been over 1900 since April, but I've been busy. Since the doctor gave me the go-ahead to start exercising again, I've been wanting to get out and get a bunch of caches. The only problem is that the heat makes it difficult to manage my blood sugar. I feel dizzy a lot, either from lack of fluids, lack of electrolytes or low blood sugar, and it's difficult to tell which is causing the problem.

Sawdust is too busy trying to get women and trying to advance in the Masons to do any caching. He could probably get to the next 100 without leaving Waco. I guess he's coming to the event though. After all, he's one of the hosts.

Crazy Summer

2006-06-15T12:46:00Z

I'd meant to keep a more accurate record of the summer, but I've hardly had a free moment since the start of summer session I. We're about three weeks in, only two more to go. (Pant pant).

Some of the things I meant to write down ---

I went geocaching a couple of weeks ago, and deliberately chose a strenuous hike to find out if it would kill me. Not too strenuous, just down from Lover's Leap at Cameron Park down to LOE's cache down in the ravine and back up again. I survived. It took me quite a while to get back up, but I made it. I think the real trick is managing blood sugar. Not an easy task when I'm exercising. Too little and the muscles starve (including the heart) too much and it reduces the oxygen-carrying capacity of the blood and the muscles starve for oxygen (including the heart). I could just lie down and rot, but I'll be d***ed if I'll do that. I've just got to figure out the right combination of water, electrolytes and carbs.

Luke (DrDogg) went along with me on this trip. I like to let him just run in rough territory, but he's such a pain. The minute he sees someone, he runs over and jumps all over them. He's great as long as he's on the leash though. I got about five caches. Not much considering my usual weekend count is 30-40. I'll get back into it soon. I've got to get to 2000 eventually. I've been at 19xx forever.

I did pass 300 hides though, that's something.

Last weekend I went down to Temple to pick up a few caches, and again I took DrDogg with me. He's not that great a companion unless the finds are well away from traffic. Otherwise he takes so much watching that it's hard to do any looking.

The chiggers are wicked this summer. Just a short walk through the grass and you've got dozens of bites. Enough to drive you nuts.

My main accomplishment is the maurer family website, http://www.TheMaurerFamily.org. I wanted to make it "the web site of the century" and I think I've come pretty close. I registered the domain on a Linux/PHP server with a year's worth of add-free hosting for only $35. I uploaded and activated a number of open source scripts so now we've got a forum, blogging, real-time chatting, picture and file uploads, Wiki, Web mail, an appointment calendar, SMS messaging, a Family tree program, and a pretty cool guest book. We also have the ability to dynamically replace news blurbs about individual family members. I'm experimenting with a few other scripts, but I don't expect too many changes. I don't use the tools myself, I just like getting them to work.

The situation with the foreign students were were trying to admit has resolved itself. We got an exception to admit one, the other's GRE was just too low, but he's going to retake it. Maybe he'll make it the second time around. Class is now at mid-term, that's why I have time for blogging. I get a day off today, because I'm giving an exam, and don't have to prepare a lecture. Preparing lectures for freshmen is a hugely time-consuming process. You have to keep reminding yourself that they don't know anything, you have to explain everything.

Maybe I'll do a little more geocaching tomorrow.

The CS graduate program

2006-06-01T23:47:00Z

We had a long talk with Ben Kelly about the graduate program today. The problems we've been having with recruitment and how we're going to deal with these problems. He's 100% supportive of our efforts and enthusiastic about some of the initiatives we're taking. His support is absolutely necessary for proceeding to the next step.

CSI 1430

2006-06-01T23:44:00Z

This course is consuming most of my time right now. It takes a long time to prepare, having never done this one before, and summer pace is incredibly fast. Today we covered variable declarations. We got through the integers and will probably cover scoping, floating point, character constants, and basic arithmetic tomorrow. I need to make up some more programming assignments and more lab assignments. I've used up the ones I've done so far.

CSI 1430

2006-05-31T18:28:00Z

Today we covered "everything you ever wanted to know about "cout." How to make string constants, other constants, key words and the basic framework for a C++ console program. Tomorrow: variables!

Geowoodstock IV

2006-05-31T18:10:00Z

Seems like the Austin geocachers had a lot of fun at Geowoodstock IV, but of course there's a fly in the ointment. Some of the many temporary caches that were created for the event, and some of the caches that were taken from their usual hiding spot to the event were forceably archived by certain reviewers. I don't know of any in Texas that were treated this way, but some from other states were. One guy even transfered his logs to another, archived cache by the same owner, only to have his logs deleted by the reviewer! Of course, now all the Austin cachers are upset. There's even this guy who maintains a "hall of shame" list for event temp caches and for caches that are taken to events, and (of course) he's trying to harrass the appropriate gc.com people into forceably archiving these caches.

The appropriate thing to do is, of course, to ignore the whole thing. Replace the archived caches with new caches in the same spot (no need to replace the log or container) and let everybody get a new smiley for an old cache. But people will fret about such things, so I offered a prize of a fully stocked ammo can to the two Austin cachers who can 1. Get the most caches on the "hall of shame" list, 2. Get the most logs deleted by the reviewer. I don't think anyone will take me up on it, but at least everyone has quit bitching about it. I think finding temp caches at an event is fun, if they're hidden like normal caches, and the same is true for temporarilly relocated caches. I'm not sure I would ever be the hider for such a cache, however. Once I place a cache, I want it to stay put. I'm certainly not going to go get it unless I plan to archive it. The same goes for temporary caches. I want my hides to be permanent.

In general, I think one should make light of other's bad behavior. If you let it get to you, you're the one who suffers, not the person doing the evil. Ignore it, make a game of it, but don't get upset.

Intro to CS

2006-05-30T22:42:00Z

First class today. I covered all of chapter 1, and moved the class into the lab. Chapter 1 mostly contains pointless nonsense, so it didn't take too long to cover. I used Bill Booth's overheads for the chapter. (Which HE stole too.) Now what am I going to do tomorrow?

COVER ARITHMETIC OF COURSE!

Yeah!

Geocaching

2006-05-30T22:38:00Z

I can't believe it. Here it is May 30th and I've found a total of one (1) cache this month. I did manage to pass 300 hides though. (I was at 292 at the beginning of the month.) I think I've been "resting" too much.

Class BLOG

2006-05-27T23:21:00Z

Well, the class blog for CSI 1430 seems to be working. I had somebody E-Mail me about submitting the first assignment already, and the class hasn't even started yet. I posted the assignment in the blog, so he must be subscribed. Cool!

Quick Blog Problems

2006-05-27T17:56:40Z

Ah yes, we go round and round. I reported three more bugs. This time, it didn't quite get to the developer. Some muddle-headed intermediary decided that I must be setting the parameters wrong somehow, and emailed back with a bunch of questions. Of course I E-Mailed back with a very annoyed letter recreating the incident. The Mobile Blogging feature is so totally screwed up that they've GOT to fix it. Some how I've got to get this idea through their thick heads. Idiots.

Mike's Graduation

2006-05-27T17:51:55Z

Mike got graduated from High School last night. A good time was had by all. Gramma Erbe came to visit, and so did Aunt Becky and Aunt Kathy. Laura is visiting from Florida. We had a great time. Later, Mike went to an "All Night" party at the YMCA. He has gotten friendly with the Valedictorian, and she was going to the party, so of course he had to go too. We took a bunch of pictures and posted them on line at http://www.neighborhoodnuts.org. Everybody liked them.

Blackboard

2006-05-26T13:50:00Z

At the behest of (I think) my one and only subscriber I have begun experimenting with blackboard. It provides a lot of services and is less cumbersome than updating my own site. Of course, I don't have any discussion boards on my site, so that's also a nice feature. Not quite like being able to E-Mail entries into your BLOG and have them (more or less) instantly appear in your students' RSS feeds, but cool.

HOT Geocaching website

2006-05-25T23:54:00Z

We have a modest beginning, such as it is. Mainly a page of links. I've gotten some helpful input from M-T-P.

So, we have a couple of puzzle cache confirmation pages, some links to discussion boards, and some blog links, including a link to this one.

Well, for a while anyway.

Travel bugs & geocaching

2006-05-25T20:00:00Z

I've got ten new travel bugs. Some of the last set are moving out. I'd like to get a few of the new ones out, but I haven't got goals for them yet. I've also got over a hundred geocache containers waiting to be hidden. Gee, if it weren't for working for a living, having health problems, and ministering to people whose kids have been hit by cars I could really be cranking those numbers! And then there's this weekend! My son picked a very inconvenient time to graduate from High School! Doesn't he know this is prime geocaching season?

That reminds me of a story. I once went out to dinner with my brother and a bunch of his big-shot friends. This was at Ruth's Chris Steakhouse in Baltimore and one fellow, a multi-millionaire, was buying dinner for the whole party. Several $100+ bottles of wine were consumed and everyone was talking about "rich guy" stuff: expensive cruises, skiing in Aspen at the height of the season, European tours, ... you know the stuff. A slank, elegant, beautiful, lady sitting to my left was telling me about the cruise she was going on with some other folks, and how much fun it was going to be. And then she said, "You know the only bad thing is, I'm going to have to miss my daughter's high-school graduation."

I never did catch the lady's name. Or perhaps I just didn't bother to remember it.

I'm pleased to say that when my son receives his high-school diploma, I won't be on a cruise. I won't be geocaching. I will be applauding the best son a father could ask for, and be loving every minute of it.