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?
posts for september 2007.
yesterday i fell in love. again. with a song, called into cold light, the first track on unholy‘s third album rapture. it catches me. absorbs me. makes me feel like floating, like sitting on a boat, moving on the water with high speed, like flying through space, stars passing by, infinitely, monotone, yet never boring, full of power, psychedelic. somehow reminds me a bit of dream theater‘s constant motion, in particular it’s video, as it looks to me as if everything is in motion, no way to stop, no way to rest, no way to look back. reminds me a bit of life. constantly moving forward in time, not able to stop for a moment, no possibility to rest. making decisions in every moment, some of them good, most of them not. what happened cannot be undone. the music’s intenseness increasing every second, then, guitars hitting in, increasing the pace. hell breaking loose. drop-dead gorgeous. stunning. haunting. so intense, so beautiful. then, fade to black. the stream flows away, leaving me in blackness. again.
“i met a traveller from an antique land
who said: two vast and trunkless legs of stone
stand in the desert. near them on the sand,
half sunk, a shatter’d visage lies, whose frown
and wrinkled lip and sneer of cold command
tell that its sculptor well those passions read
which yet survive, stamp’d on these lifeless things,
the hand that mock’d them and the heart that fed.
and on the pedestal these words appear:
“my name is ozymandias, king of kings:
look on my works, ye mighty, and despair!”
nothing beside remains: round the decay
of that colossal wreck, boundless and bare,
the lone and level sands stretch far away.”
percy bysshe shelley, 1818
i first got in touch with this poem when i was listening to the black league‘s album ichor, which contains the poem in form of a song. a very beautiful one which, in my opinion, makes the poem even more affecting, powerful, than it is already in text form. after some oriental tones, the first nine lines of the poem are read out by taneli jarva‘s baritone voice. then, for the two lines quoting the inscription, the music gets hard and heavy, whilst the two lines are screamed. (i’d say that this is one of the best examples where screaming as a singing style fits perfectly.) after that, the music returns to the slow, oriental sounding style from the beginning, and the rest of the poem is spoken. then, music slowly turning to an end, the two lines of the inscription are whispered in the background over and over.
yesterday i tried to make “real” hot chocolate, resulting in the most intense hot chocolate i ever had. i began with 200 gramms of dark chocolate, half a liter of milk, and a bit of sugar.
using the double boiling method, i.e. putting a smaller pot into a bigger pot filled with hot water, i melted the chocolate. when it was mostly melted, i started adding the milk and stirring. then, i added the sugar, and waited until it got hot again. then, finally, i added some chili powder and a bit of creme de cacao.
one last hint of mine is that you shouldn’t try to drink more than one cup of this in a few hours, it’s simply too much chocolate at once :-)
opeth‘s fifth masterpiece blackwater park contains a gorgeous song called bleak. dark poetry painting pictures of betrayal, of frustration, of anger. luring me into deep thoughts, making me float in sadness. at the same time, making me adoring the music, the interplay between mellow and heavy parts, interwoven into a complex composition. carrying me into it’s world. making me addicted to it.
having listened to it so often, it still touches me, hits me, drags me away, as it did the first time i listened to it. you may want to take a look at a beautiful picture created by rose of sharyn, which captures well some of the emotions i feel when listening to this song.
saturday evening, i watched jim jarmusch‘s movie night on earth (imdb). knowing before only ony movie by him, namely dead man (imdb), one of my favorite movies, i was really anxious to see another movie by him.
night on earth is divided into five stories in five different cities around the earth, namely los angeles, new york, paris, rome, and helsinki. each story being centered around a cab driver, cruising through the city early in the morning. some being funny, some being very sad, some just being daily life, dream life; stories about love and death, about life. some making you laugh, some making you cry. a great movie.
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 18.104.22.168), 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.
well, last night was the first concert in my current schedule: serentiy, machine men, communic and threshold were playing in the z-7 in pratteln. i really enjoyed it. first, serenity, an austrian power prog band was playing. i’ve already seen them at the end of march, playing before sacred steel and morgana lefay, and i still don’t like them too much. it’s not about their technical skills or whatever, i just can’t really stand this symphonic power metal in some cases, and this is one of these cases. and their singer should pose a bit less… ;-) after serenity, a finnish melodic metal act called machine men started to rock off. them, i really enjoyed, and later i even bought one of their albums, circus of fools. after them, the norwegian band communic started playing. i’ve never seen them or heard music from them before and was pretty curious as i read some good things about them. they’re making prog metal and are quite influenced by nevermore, a band i really like. consisting of only three musicians, which is very seldom the case for prog related bands, playing guitars+vocals, bass and drums, they still managed to produce a very intense, beautiful stream of music. if the concert would have endet after them, i would have been perfectly happy (except that they should have played longer in that case). but then, it didn’t. well. they seemed to be very happy with us, their audience, too: after their gig, besides giving us sticks and plectrums, and tor atle, their drummer, even took of one of his cymbals and threw it into the crowd. i’ve never seen that before. unfortunately, it landed somewhere else, so no chance of grabbing it… afterwards i bought their album waves of visual decay, and the stage was prepared for the british prog metal act threshold. threshold seemed to enjoy playing for us, and we enjoyed listening to them, moving to the music, marvelling at their solos, their playing skills. it was simply fantastic. but then, everything’s ending, and after an encore with two very nice songs, i had to leave to catch the last train. curiously, i again met a couple on the way back which i already met after arena’s concert at the end of may; back then, they recommended me to try out ayreon, a project by arjen lucassen which produces fantastic prog operas. again, thanks a lot for this hint, i really like these operas. well, maybe i’ll write something on them later.
a few days ago, kornel wrote something on chili drink chocolate. it’s really easy: take usual chocolate powder, take chili powder, take milk, mix it, drink it, enjoy it. and, well, it is great. if you take a sip, it first tastes as usual drink chocolate, and then, seconds later, the chili starts burning on your tongue. that’s the kind of hot chocolate i like.
today i bought a bottle of creme de menthe, so i’d be finally able to mix one of my favorite cocktails, the grasshopper, myself. additionally to creme de menthe, it consists of creme de cacao, cream and ice, and is very sweet and refreshing.
to mix it, get the ingredients, put them into a shaker. some ice should be in the shaker, too, while the rest of it should be inside the class.
then, shake the shaker for some time, and fill the result into the glass. i don’t have cocktail glasses, so i took something else, happening to be green. then, i finally finished it off by adding a green straw.