Archive for the ‘Math’ Category.

16th February 2002, 10:58 pm

I’ve played a lot with
MetaPost today — I’ve
been making fractals. It’s a lot of fun to make graphics with
MetaPost, and making
the fractals was no exception.

I’ve made three kinds of fractals: Koch
Curves, Hilbert
Curves and finally the one shown at
the right, Sierpinski’s Sieve.
The last one is also know as Sierpinski’s Triangle, for obvious reasons.

You can download the sourcecode and a document here:
`/downloads/metafractals/`

. If you’re trying
to remake the document, then please note that the images are called
`filenameN.mps`

instead of just `filename.N`

. This is because PDFLaTeX
works best with those filenames. So you’ll have to rename the files
generated by MetaPost
before running LaTeX/PDFLaTeX. If you’re
running Zsh, then you could use this very little
shell-script:

```
#!/bin/zsh
for basename in $*; do
mpost $basename
for file in ${basename%.mp}.<0->; do
mv $file ${file/./}.mps
done
done
```

7th February 2002, 12:38 pm

Yesterday was a rather long day — or to put it more precisely, it
*became* a long day, when Manitou (Jérémy) and I decided to attend a
lecture about Models and Logic (the course is called
dModLog at
DAIMI). We had started our day at 8 O’clock and
was finished five hours later. But then we decided to go to the lecture —
it lasted for another three hours.

The lecture was really interesting, we heard about Finite Automata, which I
believe is also called Finite State Machines. We heard about languages and
how to define a Finite Automata in terms of five parameters: Q, Σ,
δ, q_{0}, and F:

Q is a finite set of states. The states are the memory of the machine,
and in any given moment, the machine will be in exactly one of the states.

Σ is a finite input alphabet. It could be binary digits, the ASCII
characters or another finite set of symbols.

δ is a transition function δ: Q×Σ → Q. δ
is the function that makes the machine act on it’s input.

q_{0} ∈ Q is the start state of the machine, and

F ⊆ Q is the set of final, accepting states. If the machine is in a state
in F, then it will answer “Yes”, if not, then it will answer “No”.

You can take a look at the slides that were
used if you want to know more. I
hope to be able to follow the lectures loosly, as they’re going to prove
Gödel’s incompleteness theorem later. Ever since I first heard about
it, I’ve wanted to understand what it really says — I’ve only heard the
informal explaination of it, namely that it says, that there are things in
every closed system that cannot be proved, although they’re known to be
true. That sounds like a very fundamental theorem, and I’m really looking
forward to learning about it — if not this year, then definitely next
year, where dModLog will be part of our
mandatory courses.

9th January 2002, 07:05 pm

As you might have guessed, then I’m studying for my upcoming exam in linear
algebra, also know as Mat10.
The exam is the 15th so there’s still plenty of time. I’ve now worked by
way through all the subjects we will be tested in, and have made a
(Danish) document in
LaTeX with all the theorems and proofs.
I’ve now written something for every subject — there’s even a proof of
the nasty Cayley-Hamilton Theorem! At first I thought it was *very*
confusing, but then I found a proof on the net (sorry — I didn’t save
the link) that took the time to explain things properly, and that helped.

The exam is without preparation, so I’ll probably have to rehearse some
more… I’ve never tried this kind of exam so I don’t know what to expect.
It sounds a bit scary, that I have to be able to talk about a random
subject — just like that. My last exam in math was with 20 minutes of
time for preparations — plenty of time to look things up in the books
and so on… But on the other hand, we wont get a grade for this exam —
we either pass or fail. That should make things much easier.

7th January 2002, 06:12 pm

I’ve placed my notes for the forthcoming exam in Mat10 (linear algebra) on the web.
You’ll find them here. (The notes are in Danish)

1st February 2001, 05:07 pm

Last Monday I finished by big Danish paper on Mersenne
primes, or *Mersenne primtal*
as it’s called in Danish.

I made it to my school only ten minutes before the deadline, because it
took over an hour to print the darn thing! I had underestimated the
printing, because when I printed it during the week, I only printed the
important pages, and skipped the title page, the table of contents, the
appendix etc. But with all those pages the report is 36 pages long, and
since I have to print at “Best” quality on my Deskjet 970Cxi it took a
loong time to print… But I made it — with time to spare :-)

I’ve put the paper up in the Danish section of my
site, so that you can take a look. You’ll find several different formats
to choose from, so everybody should have a chance. But if your browser
can’t display the image above, in PNG format, then don’t bother trying
the HTML versions. Due to patent problems, all the images are in
PNG format.