< 1095926399 0 :clog!unknown@unknown.invalid QUIT :ended
< 1095926400 0 :clog!unknown@unknown.invalid JOIN :#esoteric
< 1095956810 0 :rgarro77!~rgarro@200.12.239.1 JOIN :#esoteric
< 1095958704 0 :rgarro77!unknown@unknown.invalid PART #esoteric :?
< 1095961894 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :my laptop is the brokenness now. :(
< 1095961899 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :"was fun while it lasted".
< 1095961928 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :although in retrospect ~1550eur was a bit much for a machine that worked for six days.
< 1095964964 0 :lindi-!unknown@unknown.invalid PRIVMSG #esoteric :uh
< 1095965071 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :that's over 250eur/day, after all.
< 1095968120 0 :lindi-!unknown@unknown.invalid PRIVMSG #esoteric :ZeroOne: btw, while cycling back home i got a new idea regarding that language where one task can be done in only one way
< 1095968147 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :elaborate. clog's back.
< 1095968470 0 :lindi-!unknown@unknown.invalid PRIVMSG #esoteric :so, if every program is presented as a mathematical function, say f(x), then the language could be defined so that only the form of f(x) that is the "most" simplified is legal
< 1095968523 0 :lindi-!unknown@unknown.invalid PRIVMSG #esoteric :fizzie: at least with polynomials it should not be possible to get a function (read: program) that behaves in the same way, unless the functions are identical
< 1095968562 0 :lindi-!unknown@unknown.invalid PRIVMSG #esoteric :so the core idea is that there needs to be some set of simplication rules
< 1095968576 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :wellll, yes, if you limit your mathematical notation enough.
< 1095968712 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :polynomials are safe in that way.
< 1095968777 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :actually I think we proved in one of our maths courses that n-degree polynomials are.. ah, what's the word? n-ulotteisen avaruuden kanta, tavallaan. yksikäsitteiset koordinaatit ja näin.
< 1095968788 0 :lindi-!unknown@unknown.invalid PRIVMSG #esoteric ::)
< 1095968872 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :it's still pretty limited, though, but at least your 'programs' can now take a set of real numbers as input and return a set of real numbers as output.
< 1095968973 0 :ZeroOne!unknown@unknown.invalid PRIVMSG #esoteric :hmm. interesting.
< 1095969002 0 :ZeroOne!unknown@unknown.invalid PRIVMSG #esoteric :fizzie: "radix of n-dimensional space, sort of. unambiguous coordinates etc."
< 1095969017 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :thanks.
< 1095969027 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :it was the 'radix' word I wasn't so sure of.
< 1095969162 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :complex functions make for some really weird plots.
< 1095969193 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :if you happen to have mathematica and/or an equivalent piece of software (perhaps gnuplot would be enough), plot Plot3D[Arg[f2[x, y]], {x, -5, 5}, {y, -5, 5}, PlotPoints -> 100,
< 1095969196 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric : Mesh -> False]
< 1095969200 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :erhm.
< 1095969205 0 :ZeroOne!unknown@unknown.invalid PRIVMSG #esoteric :lindi-: so every program would be a polynomial of degree n, or something?
< 1095969210 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :there's a piece of misplaced copypasting.
< 1095969224 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :since it doesn't include the actual function.
< 1095969276 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :f2[x_, y_] := x^3 + 3x y^2 - 3x + i(y^3 + 3x^2y - 3y), where 'i' is the imaginary unit. when you plot the magnitude of that as a surface, it's as clean and smooth as.. well, as any function you'd like to name, but the argument plot..
< 1095969318 0 :ZeroOne!unknown@unknown.invalid PRIVMSG #esoteric :ok... unfortunately I don't have any math software installed... I guess I should soon get something.
< 1095969334 0 :lindi-!unknown@unknown.invalid PRIVMSG #esoteric :ZeroOne: well, if you extend the simplification rules you can support many other types too
< 1095969368 0 :ZeroOne!unknown@unknown.invalid PRIVMSG #esoteric :although I once made a program that was able to draw 2d-diagrams of polynomial functions. in qbasic, even. ;P=
< 1095969379 0 :lindi-!unknown@unknown.invalid PRIVMSG #esoteric :i wrote one in ti86 basic :P
< 1095969418 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :and you need to limit yourself to finite-degree polynomials, otherwise you could have an infinitely-long program which would do the same as 'sin x' (if you write some simplification rules to allow trig. funcs)
< 1095969419 0 :ZeroOne!unknown@unknown.invalid PRIVMSG #esoteric :yeah. but TI features all math functions built-in, qbasic doesn't.
< 1095969447 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :heh, 2d plots of polynomials seems to be a popular project.
< 1095969449 0 :ZeroOne!unknown@unknown.invalid PRIVMSG #esoteric :fizzie: gotta love mr. Taylor.
< 1095969468 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :I think it was an exercise-or-sort-of in my high school.
< 1095969477 0 :lindi-!unknown@unknown.invalid PRIVMSG #esoteric :fizzie: but what if "sin x" is not allowed but only the 'sarjakehitelmä'
< 1095969488 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :mooz at least wrote a.. pretty advanced one, in qbasic.
< 1095969527 0 :lindi-!unknown@unknown.invalid PRIVMSG #esoteric :fizzie: that reminds me, where does mooz irc nowadays? /whois shows no channels but maybe they are secret/private
< 1095969536 0 :ZeroOne!unknown@unknown.invalid PRIVMSG #esoteric :heh. :) we only had some really stupid piece of some software the teacher had made. copied that from paper and got disappointed when it did practically nothing: "oh, no, it shouldn't do anything, it's just the initialization function thingie!" was the teacher's response ;PP
< 1095969599 0 :ZeroOne!unknown@unknown.invalid PRIVMSG #esoteric :who is mooz?
< 1095969605 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :I'm not sure, really. I speak mostly in a query. he's been gradually moving to an apartment in kamppi, probably parted from #da during that time.
< 1095969741 0 :lindi-!unknown@unknown.invalid PRIVMSG #esoteric :ZeroOne: apt-get install maxima and plot3d(x^3 + 3*x*y^2 - 3*x + %I*(y^3 + 3*x^2*y - 3*y), [x,-5,5], [y,-5,5]); if you want to see the function fizzie pasted
< 1095969750 0 :lindi-!unknown@unknown.invalid PRIVMSG #esoteric :
< 1095969811 0 :lindi-!unknown@unknown.invalid PRIVMSG #esoteric :hmm, you can even rotate the plot
< 1095969834 0 :ZeroOne!unknown@unknown.invalid PRIVMSG #esoteric :lindi-: apt-get doesn't work under winblows, you know.
< 1095969897 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :how does maxima plot complex-valued two-variable functions?
< 1095969908 0 :lindi-!unknown@unknown.invalid PRIVMSG #esoteric :ZeroOne: http://prdownloads.sourceforge.net/maxima/maxima-5.9.0.exe?download
< 1095969935 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :with mathematica I need to use either Abs[] or Arg[] (well, or Re[] or Im[]) in front of it.
< 1095969937 0 :ZeroOne!unknown@unknown.invalid PRIVMSG #esoteric :fine... :p
< 1095970032 0 :lindi-!unknown@unknown.invalid PRIVMSG #esoteric :fizzie: iki.fi/lindi/maxima_plot3d.png
< 1095970048 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :ah, but that's just the real part of it.
< 1095970066 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :you want to use abs() or arg() (or corresponding maxima functions) if you want to see the strange-ness I mentioned.
< 1095970119 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :and actually while the magnitude plot with abs() looks smooth, looking at it with the ranges [-2, 2] reveals otherwise.
< 1095970343 0 :lindi-!unknown@unknown.invalid PRIVMSG #esoteric :fizzie: plot3d(cabs(x^3 + 3*x*y^2 - 3*x + %I*(y^3 + 3*x^2*y - 3*y)), [x,-5,5], [y,-5,5]); looks like iki.fi/lindi/maxima_plot3d_2.png
< 1095970393 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :yes, doesn't it look pretty smooth?
< 1095970555 0 :lindi-!unknown@unknown.invalid PRIVMSG #esoteric :yep
< 1095970574 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :well, what about either a) the range [-2, 2] or b) the argument of the result?
< 1095970588 0 :lindi-!unknown@unknown.invalid PRIVMSG #esoteric :argument?
< 1095970594 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :also called 'angle'.
< 1095970599 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :I don't know which maxima function it is.
< 1095970715 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :extra credit for proving in which points of C that function is a) complex differentiable b) analytic. (the b part was relatively easy, but I'm not sure how I'm supposed to do that a part.)
< 1095970861 0 :lindi-!unknown@unknown.invalid PRIVMSG #esoteric :fizzie: now it does not look that smooth either indeed, iki.fi/lindi/maxima_plot3d_3.png
< 1095970929 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :uh-huh.
< 1095971337 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :eh, scratch that, now I'm not quite sure of the b part either. :p
< 1095971353 0 :lindi-!unknown@unknown.invalid PRIVMSG #esoteric :no problem, i'm even more unsure :)
< 1095971367 0 :lindi-!unknown@unknown.invalid PRIVMSG #esoteric :fizzie: were you in L1 or C1 math btw?
< 1095971392 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :L1.
< 1095971473 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :I thought the fact the function satisfies the cauchy-riemann equations only in the points of the imaginary and real axis, and therefore not in any real region of C, would be enough in stating it's not analytical. it still _might_ be, but..
< 1095971568 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :the canonical definition would be that a complex function is analytical in point z0 if a derivative f' exists for all points z in a neighbourhood of z0, but in order to use that would require me solving the a) question.
< 1095971582 0 :lindi-!unknown@unknown.invalid PRIVMSG #esoteric :hrm
< 1095971664 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :some days I'm not quite sure why I'm in the L* series of maths courses anyway.
< 1095971764 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :ah, right. in order for f to be analytical in a region G it would have the partial derivatives of its component functions satisfying cauchy-riemann, and since that isn't the case it can't be analytical. (I think.)
< 1095971860 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :I still need some way to solve that 'a' part. I'm not sure how I'm supposed to prove that lim h->0 (f(z+h)-f(z))/h exists when h is a complex number and can approach 0 from any direction in the complex plane.
< 1095971874 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :(and this for any z.)
< 1095971915 0 :lindi-!unknown@unknown.invalid PRIVMSG #esoteric :don't know much about handling those (yet)
< 1095972002 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :neither do I, but I'm sort-of supposed to. there could be a better way to do this though.
< 1095972158 0 :lindi-!unknown@unknown.invalid PRIVMSG #esoteric :but at least i've learned something this week. when i cycled back home today i could also do some partical fraction decompositions in my head :P
< 1095972172 0 :lindi-!unknown@unknown.invalid PRIVMSG #esoteric :s/partical/partial/
< 1095972333 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :I freely admit I don't remember ~anything about how to do a LU-decomposition of a matrix.
< 1095973386 0 :lindi-!unknown@unknown.invalid PRIVMSG #esoteric :anyway, need to sleep so that i can be awake on the math lecture
< 1095973391 0 :lindi-!unknown@unknown.invalid PRIVMSG #esoteric :s/on/during/ :P
< 1095973399 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :hnu.
< 1095973403 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :g'night.
< 1095974783 0 :Taaus!unknown@unknown.invalid PRIVMSG #esoteric :Oh, yippee... Complex analysis.
< 1095974948 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :loads of fun.
< 1095975009 0 :Taaus!unknown@unknown.invalid PRIVMSG #esoteric :Actually it's not too bad. I'm taking a course on it this semester.
< 1095975468 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :oh? then you can tell me what's the easiest way to show where the derivative f'(z) for a complex function f(z) exists.
< 1095975520 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :looking at the definition of f' (that is, f'(z) = lim h->0 (f(z+h)-f(z)/h) is awfully untrivial since h can approach 0 from any direction and I need to look at all points z.
< 1095975667 0 :Taaus!unknown@unknown.invalid PRIVMSG #esoteric :Well... "Easiest" depends on the situation. But the Cauchy-Riemann equations are a safe bet.
< 1095975948 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :but are CR enough? I mean, they definitely are enough to prove whether the function is analytical anywhere, but f'(z) could still exist in some isolated points without it being analytical there.
< 1095976178 0 :Taaus!unknown@unknown.invalid PRIVMSG #esoteric :A function f=u+iv from an open subset of the complex plane to the complex plane itself is holomorphic if u and v are differentiable, and if the partial derivatives satisfy the relations du/dx = dv/dy and du/dy = -dv/dx (where the ds are "soft ds". I.e. partial differentiation.)
< 1095976211 0 :Taaus!unknown@unknown.invalid PRIVMSG #esoteric :s/holomorphic if/holomorphic if and only if/
< 1095976296 0 :Taaus!unknown@unknown.invalid PRIVMSG #esoteric :It can be applied to single points of the function as well. To prove differentiability in, say, (x_0,y_0).
< 1095976452 0 :Taaus!unknown@unknown.invalid PRIVMSG #esoteric :I'm not sure what you mean by f'(z) existing in isolated points without being analytic in said points.
< 1095976875 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :my definition of an analytic function says a function is analytic at point z0 if f'(z) exists for all z in a neighbourhood of z0, and it has a side remark saying "existance of f'(z) in a single point is not enough"
< 1095976880 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :this in my lecture notes.
< 1095977073 0 :Taaus!unknown@unknown.invalid PRIVMSG #esoteric :Hm. My notes don't mention analyticity in a single point. In gives the usual limit definition of differentiability and says a function is holomorphic if it is differentiable in all points. I suppose it makes sense that there needs to be a neighbourhood around z_0. You need more than a single point for an open subset.
< 1095977185 0 :Taaus!unknown@unknown.invalid PRIVMSG #esoteric :If you can read Danish, you can take a look yourself. They're online at http://math.ku.dk/noter/ The course is called 2KF.
< 1095978184 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :well, maybe my notes try to say that it needs to analytic in the open subset, but simply having f'(z) (that is, the limit existing) at a single point doesn't yet make it analytical.
< 1095978195 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :s/al.$/./
< 1095978211 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :who knows, I was probably at least half-asleep when I wrote that.
< 1095978279 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :but our exercise question gives us a function and asks "in what subset of C the function f: z -> ... is (a) differentiable (b) analytic".
< 1095978335 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :and I can easily say it's analytic nowhere in C, because it satisfies CR only at z=0, but I'm not quite sure about the differentiability thing.
< 1095978379 0 :Taaus!unknown@unknown.invalid PRIVMSG #esoteric :Well... It's differentiable in the points where the limit exists... I guess you'll have to do it the hard way.
< 1095978485 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :agh. I tried writing open "f(z+h)-f(z)" (with z=x+iy and h=hx+ihy) and ran out of paper width. granted, it's only a c5 envelope I'm writing on, since all proper paper is several meters right and I can't get myself off the chair
< 1095978512 0 :Taaus!unknown@unknown.invalid PRIVMSG #esoteric :What's the function?
< 1095978577 0 :fizzie!unknown@unknown.invalid PRIVMSG #esoteric :f: z=x+iy -> x^3 + 3xy^2 - 3x + i(y^3 + 3x^2y - 3y). I'm tempted to abuse the component functions u(z) and v(z) somehow here, since they are more humane.