i finally started another project: a math blog. the aim of this one is to write about mathematical things which interest me, for example things related to my research. the formulae will be rendered with latex; mathml is simply unuseable so far.
an example post shows a feature with i added to my wp-latex enhancer plugin: (primitive) environments for definitions, theorems, proofs, etc., including a very basic labeling system allowing hyperlinks which jump to the right environment; for example, here’s a link to a lemma in the post. the post features my favorite proof of the fundamental theorem of algebra, using complex analysis.
posts about math. (page 1.)
i was thinking on using latex in maybe some blog entries, maybe here or maybe somewhere else. so i decided to see what existing plugins there are. after a bit of searching, i stumbled over wp-latex, which is apparently also used by wordpress.com. unfortunately, it has a kind of clumsy syntax (“$latex formula$” instead of simply “$formula$”). and it has no support for display style formulae, i.e. something centered in its own line, as opposed to inline formulae which try to fit neatly into the text.
so i tried to fix that. and it worked out, and i can still use a “normal” $ by appending a blackslash in front of it. well, euler's identity is , as simple as that. if you want to see something more complicated:
let
be a number field or an algebraic function field. then, we have the following commutative diagram with exact rows and columns:
here,
simply denotes the cokernel of the map
which assigns to every unit
its principal divisor
; in particular,
. finally,
denotes the cokernel of the degree map
, where in the number field case,
, and in the function field case,
.
this is written as follows:
1let \$K\$ be a number field or an algebraic function field. then, 2we have the following commutative diagram with exact rows and 3columns: 4\$\$\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] & \\ 0 \ar[r] & 5\calO^* / k^* \ar[r] \ar[d] & \Div^0_\infty(K) \ar[r] \ar[d] & 6T \ar[r] \ar[d] & 0 \\ 0 \ar[r] & K^* / k^* \ar[r] \ar[d] & 7\Div^0(K) \ar[r] \ar[d] & \Pic^0(K) \ar[r] \ar[d] & 0 \\ 0 \ar[r] 8& K^* / \calO^* \ar[r] \ar[d] & \Id(\calO) \ar[r] \ar[d] & 9\Pic(\calO) \ar[r] \ar[d] & 0 \\ & 0 & H \ar@{=}[r] \ar[d] & H 10\ar[d] & \\ & & 0 & 0 & }\$\$ 11here, \$T\$ simply denotes the cokernel of the map \$\calO^* \to 12\Div^0_\infty(K)\$ which assigns to every unit \$\varepsilon \in 13\calO^*\$ its principal divisor \$(\varepsilon)\$; in particular, 14\$T \cong \Div^0_\infty(K) / (\Princ(K) \cap \Div^0_\infty(K))\$. 15finally, \$H\$ denotes the cokernel of the degree map \$\Div(K) \to 16\G\$, where in the number field case, \$\G = \R\$, and in the 17function field case, \$\G = \Z\$.
note that this example also shows a problem: namely, the vertical alignment of the inline formulae sucks bigtime. let's see how to fix this...
this week, i was attending the ants. that’s not a conference on ant colony optimization and swarm intelligence, but the eighth instance of the algorithmic number theory symphosium, held in banff, canada.
banff is situated in the canadian part of the rocky mountains, whence one has a pretty neat sight:
here are some impressions from the conference center facilities. the first is a shortcut to the dining hall, the second the professional develop center, and the third one a pathway inbetween:
the next two photos show a view from inside the professional develop center, where we were housed, and me together with my poster:
this conference has been really great. i had lots of fun, learned a lot, met a lot of nice folks (again). and some more.
mathematics is nothing but the search for structures. beautiful structures. i myself tend to think i understood something if i manage to have a picture of it in my head, some geometric interpretation. sometimes it’s more easy to understand something if one can somehow draw it, by hand, or using a computer, which can sometimes give insights which would not be possible without it; the best example maybe is the beautiful mandelbrot set. most visualizations are by no means as impressive as the mandelbrot set, unless the viewer is acquainted with the underlying mathematics. like the following one, produced by the programs i wrote yesterdays and today, whose beauty probably won’t unfold if you don’t know what it depicts:
do you know the feeling when, while doing something completely unrelated, like taking a shower, you get an idea? an idea which is somehow cool, which will change life completely, in some sense? with such ideas, usually lots of endorphines and adrenaline is set free, making you unable to sleep, to rest, to think clearly. then, after spending the day working the idea out, you finally notice a tiny flaw, something that didn’t worked as smoothly as you thought it would or should. such tiny, but yet so fatal, killing the whole idea, throwing you back to the ground of the pit, where you continue to lurk around, in search for new ideas. but then, it’s not as bad as it sounds. usually you learn something. you get a better understanding of something, the object in question has turned into something which you can see better with your mind, from a bit farer away, not as near as before, so near that you can see every bump but have no clue what these bumps belong to. like a new stack of jigsaw pieces, given to you to plug it into your big world puzzle, to complete it a bit more, to widen your glance on the truth. the universe got a bit smoother.
no, this post is not about the area of abstract algebra in mathematics, but about the experimental metal project abstrakt algebra founded after candlemass‘ dismanteling by leif edling. they released one album and started with a second, but ran out of money, and leif ended up escaping bankrupcy by re-using some of the second albums material to create another candlemass album, dactylis glomerata (named after a flower he’s allergic to). the 2006 reissue of dactylis glomerata contains as a bonus disk the unfinished version of the second abstrakt algebra album, entitled abstrakt algebra ii (now guess how the first one was called). well. the arrival of that one kicked me out of bed this morning, when the mailman rang the doorbell. but it was worth getting up for it, and even if it is not a very good album, it is not good, it is interesting, and it is one of the albums every mathematician who has a similar strange taste of music than me should have in his cd shelf. of course, together with their first album, which i just ordered. ;-)
yesterday i started reading the xkcd archives (maybe you’ve already noticed that yesterday). if you don’t know xkcd, it’s a webcomic “of romance, sarcasm, math, and language”. here are the ones i found particularly interesting, funny, or whatever:
- pi equals. reminds me of the classic “help, i’m trapped in a fortune cookie factory!” joke.
- what if.
- barrel – part 3. i wouldn’t say “wow!” in that situation, though.
- fourier. poor cat.
- secrets.
- useless. standard approachs suck for love.
- su doku. even i can solve these.
- national language.
- binary heart. if you check the parity of the read ones in every column, you’ll notice most of them are even. is this a coincidence? (and don’t ask why i stumbled about that…)
- laser scope. sometimes the primitive word jokes are the best.
- riemann-zeta. about love & primes. this one is not only for number theorists.
- nihilism. squirrels! cool!
- alice and bob. the real story of eve.
- matrix transform. if i’d ever had to solve linear algebra exercises again, i’d try to turn this one in.
- valentine’s day. nothing to add.
- cat proximity. yay, how true. meow!
- code talkers. another one on cryptography.
- fixed width. nerd talk on irc. i don’t know what scares me more, laughing about this one or thinking that i’m probably knowing people who would really do this.
- exploits of a mom. well. if you don’t know what an sql injection attack is, read about it here.
consider the field of all numbers. meaning, of course, the complex numbers. we say that a number is representable if we can describe it by a text (for example, by a binary coded string of 0′s and 1′s of finite length). the set of all representable numbers is countable, as there’s a surjection of the set of binary strings (which is countable) onto it. moreover, it is a field, as if a and b are representable numbers, we have that a + b, a − b, a ⋅ b and a / b are represented by strings as “sum of (description of a) and (description of b)”. obviously, every algebraic number is representable, so our field of representable numbers contains the algebrically closed field of the algebraic numbers. but then, our field also contains euler’s number e and archimedes’ constant π, so it’s strictly larger. this opens the question: how does it’s algebraic closure looks like? not too surprisingly, it turns out to be already algebraically closed: every element in its closure can be represented by “root of polynomial with coefficients (description of coefficients)”, as all coefficients are representable. hence, our field, being countable, is strictly larger than the smallest algebraically closed subfield of the complex numbers, but still countable. and it contains lots of transcendental numbers. isn’t that cool?
i’ve always thought that mathml is a great idea, allowing you to write formulae in html which look good afterwards. or so i thought. then, today, i saw this. at least in my browser (firefox 2.0.0.6), i have the feeling that word by default produces nicer formulae than this. then, i started searching the web for other examples, maybe it is possible to produce better output. but then, even when looking at (seemingly) standard examples as this and this (the latter supposed to display that 44.997 is an element of the reals), i’m shocked how bad this looks, if it works after all (the first did, more or less, and the latter didn’t). i mean, i know that it might be illusive to expect that the formulae look as good as in latex, but i would have at least expected them to be rendered correctly…
well, i’d guess the only way is to stick to rendering formulae with latex and including them as images, as it is standard practice on basically all sites displaying good looking formulae on the web, as wikipedia, matroids matheplanet, matheraum, etc.
or simply don’t use any formulae in html files, as i’m doing so far. too bad.
sometimes i wonder why mathematics is working so smooth as it is. for me, this is one of the most intriguing things around, next to the questions on why do we live and on whats the question to the answer 42… based on a small set of axioms, nobody knowing whether they are free of contradiction, an enormous building of constructions and proofs has been built. so huge, that no one can learn about all of it in one lifetime. and inside this building, between lots of dirty corners with ugly computations and hard work, there sit so many beautiful small and big results, sometimes showing surprising connections to completely different parts of mathematics. this connectivity is what surprises me most. every time again. that’s what i like about mathematics, and that’s what’s keeping me doing math. and makes me even more curious about the question on why it is working so smooth.
