ChapterZero

A lattice is a partially ordered set with the special property that for every pair of elements , the supremum (the least element in that is greater than or equal to and ) and infimum (the greatest element in that is less than or equal to ), and are defined. A standard way of denoting the relationship between the elements in a lattice is the Hasse diagram, in which an element is drawn above an element , and the two are connected by a line iff and there is no such that . (See the planetmath page on Hasse diagrams for an example).

Question: given an integer , how many Hasse diagrams are there for a lattice with that many elements? I don’t know the answer, or how difficult the problem is. I do know, however, that for respectively, the answers are .

An almost trivial question: prove that a lattice can be decomposed into a collection of chains such that no two chains are disjoint. Is this a sufficient condition on a collection of chains for it to be a lattice? (A chain is a poset in which every element is comparable, i.e. for all in the poset, either or .)

I made it to school early today, so I was here before they replaced Friday’s edition of the Daily Cougar with today’s. Today’s has an interesting article on the controversy in the Kansas school system on whether ‘intelligent design’ should be taught in tandem with evolution. The author seems to think that there is no problem with this, and that despite the fact that this is mostly being pushed by Christians, it is a move that would benefit all religious groups, as the ‘higher force’ is not specifically named.

What a load of crap! So are we going to be teaching astrology as an alternative to astronomy also? People — Christians are the ones seeming to make a big deal about it — should realize that when they place their kids in a public school, they’re going to be taught science. If you disagree with science, send your kids to a private school. Forget the fact that as far as sane and honest people are concerned, the closest we as human beings will ever get to the truth is via science, there’s an even better reason to teach science in schools, as opposed to pseudo-religious propaganda: science has no objective but to build a body of knowledge, and as such is free of religious and social influences. Well, maybe not completely, but I charge you to find a system as close to impartiality as science.

Here’s a reply I sent to the author:

Hello,

I read your article in the opinion column on the evolution controversy
in Kansas.

There is a very practical reason for not including intelligent design
in the curriculumn: if science is abolished as the metric with which
the truths taught in schools are measured, what’s to prevent our kids
from being taught the validity of astrology or magic the next time a
group decides to protest?

Instead of teaching religious beliefs— however generic, sanitized,
and pseduo-scientifically supported ‘intelligent design’ may be, it
*is* a religious belief—, why not simply emphasize the fact that
evolution is ‘only’ a theory? After all, one of the things any child
should learn is that the ‘truth’ as defined by science is rarely
immutable, and in the past, even ‘Laws’ such as Newton’s have been
shown to be incorrect. Certainly that is a more realistic goal than to
give our kids the idea that either evolution or intelligent design is
the Truth, the whole Truth, and nothing but the Truth.

The agitators should realize that if ‘intelligent design’ was worth
its salt as a scientific theory, it instead of evolution would be
being taught as the leading scientific theory. So if they want to
replace evolution, they should go and do their homework like Darwin
did, instead of trying to fight dirty.

Alex Gittens
EE/Math major

Usually, creating well-illustrated LaTeX documents is a nightmare, and it turns out to be easier to include bitmapped graphics than vector graphics, even when they’re what are really wanted. So I’m recording the steps here that I just used to create a document with high quality vector illustrations:

1. Draw the illustrations in InkScape
2. Select everything in the document and then export to eps, using the “from selection” option (to give a tight bounding box)
3. Include the file in the LaTeX file (for some reason, some of the illustrations seem to be rotated; you might need to use a \rotatebox block)
4. Since pdflatex doesn’t work with eps files, use latex followed by dvipdf

I was going through my “Typesetting/illustration/editing” bookmarks folder, visiting all the sites and adding meaningful descriptions to all the links, when I came across an article on the need for integration between GIMP, Inkscape, and Scribus, the three top OSS graphical packages.

I’ve been an off and on user of GIMP– i.e., when I can’t get to a copy of Photoshop– for years, and have Inkscape installed on my laptop, maybe have used it twice (until reading this article and the comments at OSNews, I would have said Sodipodi was the top vector illustration package). But I’ve never even heard of Scribus before this; I wonder if it too has a Windows port?

Integration isn’t an issue that I’ve ever considered too deeply between graphical OSS, because I don’t take them very seriously. But maybe if they were integrated and had a common interface standard, so I could sit down and learn one of them, and not have to start over from scratch when I moved on to the next, I would. After all, Inkscape does seem pretty nice, and the screenshots of it in use look promising; the only thing that has kept me from more experimenting with it is that the interface is so foreign to Illustrator’s, or even any other Windows program, that I spend too much time figuring out how to do any little thing. As for GIMP, I suppose it has matured and settled enough into it’s interface that if I took the time I could get as comfortable with it as Photoshop. Don’t know anything relevant about Scribus…

One good thing about Inkscape: it can open SVG files, and automatically sets itself up as the SVG handler. I just downloaded a clip art image from openclipart.org and was pleasantly surprised to find that Firescape caused Inkscape to open it, all without a hitch. The world of OSS is coming along nicely.

appropriately enough, via The art of problem solving:

The two great Russian mathemeticians, Lara Antipova and Pasha Gordon, are taking a train ride through the Urals.

“I remember you have three sons. How old are they now?” Lara says, to make idle conversation.

“The product of their ages is thirty-six. The sum of their ages is exactly todays date,” replies Pasha, in hopes of making idle conversation difficult.

Lara thinks for a bit, and says, “I’m sorry, but that’s not enough information for me to know their ages.”

“Ah, you’re right. Well, my youngest son has red hair.”

“Oh ok, I know how old your sons are now.”

How does Lara know? And more importantly, what are Pasha’s son’s ages?

I hate problems like these.

Here’s a problem one of my friends was puzzled by on the Math GRE– I have to admit it’s daunting at first sight:

Let be prime, . Find the gcd of where ranges over all applicable primes.

He took the test late last year, and had mentioned it in passing several times, saying he was going to check with the algebra and number theory profs to see how to go about solving it. Today we were waiting outside of a classroom for our Real Analysis test, and the topic came up again, so we sat down and warmed up for the test by solving it together. Although it wasn’t too hard, it still took way more than the 2 and a half minutes you have per problem in the GRE. Admittedly though, the GRE is multiple choice; but in this case, that doesn’t help much, because there was the dreaded “none of the above” option. IMO, this is mathematical competition quality question. Makes me fear the Math GRE

He’s going to let me borrow a prep book he has for it, so I can bone up on my number theory/algebra skills.

So, what’s your answer? Ours is past the fold
(more…)

I’ve been having intermittent problems accessing the site: sometimes when I try to view tangentspace.net/cz, the browser keeps waiting on the site, but nothing comes through. I don’t see anything in the server logs that might explain this. The most drastic thing I did recently was accidentally reinstall perl (I was trying to install perl-doc using apt-get, and it auto installed perl again), but I don’t see why it would cause this problem. I power cycled the VPS just now, and that seemed to clear up the problem, but I don’t know if that’s a permanent solution.

Ok, one more joke…

A great new software announcement!! This memo is to announce the development of a new software system for use in our Mathematics Department.

We are currently building a data center that will contain all files, documents, and date used by our and other Mathematics Departments worldwide. The program is referred to as the “Mathematical Information Access Software System” (MIASS).

Next Monday at 9:00 there will be a department-wide meeting in which I will show MIASS to everyone. We will continue to hold demonstrations throughout the month so that all faculty, staff, and graduate students will have an opportunity to get a good look at MIASS.

As for the status of the implementation of the program, I have not addressed the networking aspects so currently only one person at a time can use MIASS. This restriction will be removed after MIASS expands.

Several people are using the program already and have come to depend on it. Just this morning I walked into a professor’s office and was not surprised to find that he had his nose buried in MIASS. I’ve noticed that some of the less technical personnel are somewhat afraid of MIASS.

Just last week, when asked to enter some information into the program, I had a secretary at the Front Desk say to me “I’m a little nervous, I’ve never put anything in MIASS before.” I volunteered to help her through her first time and when we were through she admitted that it was relatively painless and she was actually looking forward to doing it again. She went so far to say that after using SAP and Oracle, she was ready to kiss MIASS.

I know there are concerns over the virus that was found in MIASS upon initial installation, but I’m pleased to say the virus has been eliminated and we were able to save MIASS. In the future, however, protection will be required prior to entering MIASS.

We planned this database to encompass all information associated with the Department. So as you begin to use the program, feel free to put anything you want into MIASS. As MIASS grows larger, we envision a time when it will be common place to walk by an office and see a professor hand a paper to a grad student and say “Here, stick this in MIASS.”

The program has already demonstrated great benefit to the Department during recent audits by the University Regents. After requesting certain historical data the University representatives were amazed at how quickly we provided information. When asked how numbers could be retrieved so rapidly our Department Chair proudly stated “Simple, I just pulled them out of MIASS.”

Psychologists subject an engineer, a physicist, and a topologist to an experiment: Each of them is locked in a room for a week — hungry, with a single can of tuna fish but without an opener; all they have is pencil and paper.

At the end of the day, the psychologists open the engineer’s room first. Pencil and paper are unused, but the walls of the room are covered with dents. The engineer is sitting on the floor and eating from the open can: He threw it against the walls until it cracked open.

The physicist is next. The paper is covered with formulas, there is one dent in the wall, and the physicist is eating, too. He calculated how exactly to throw the can against the wall, so that it would crack open.

When the psychologists open the topologists’s room, the paper is also full of formulas, the can is still closed, and the mathematician has disappeared. But there are strange noises coming from inside the can… Someone gets an opener and opens the can. The topologist, covered in tuna fish, crawls out. “Dammit! I got a sign wrong again…”