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 Posted: Tue Jul 13, 2010 8:37 pm
I've forgotten how to deal with manifolds embedded in Rⁿ. I keep thinking "wait, how are you comparing those two tangent spaces? What do you mean, normal vector?"
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Posted: Tue Jul 13, 2010 9:13 pm
I reject your reality...
I find it so cool that fractals are in a 1.7 dimension.
...and substitute it with my own!
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Posted: Thu Jul 15, 2010 8:30 pm
I reject your reality...
I just hate it when you have a puzzle, a complex problem, a confusing conundrum, that you just can't solve. >.> <.< At least until your father gets home to give you a hint.
Layra-chan...
How do you know so much?
...and substitute it with my own!
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Posted: Fri Jul 16, 2010 6:46 am
Sioga I reject your reality...
Layra-chan...
How do you know so much?
...and substitute it with my own! Years. Years and years and years of solving problems, of devouring textbooks, of forcing myself to go beyond whatever is assigned in class. Years of active, proactive learning and stubborn persistence. Indulge your curiosity, and then when it complains that it wasn't being serious, keep going anyway. Read random books that are beyond your level, and then learn, either on your own or with someone's guidance, what you need to understand those books. When you find something "cool", follow it, ask yourself what it really means, why it's true, what are similar situations, where does it fail? Like the thing about fractals having non-integer dimension. What does it mean for something to have a non-integer dimension? What does it mean for something to have an integer dimension? How does one go about calculating the dimension of something?
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Posted: Fri Jul 16, 2010 6:07 pm
Layra-chan I've forgotten how to deal with manifolds embedded in Rⁿ. I keep thinking "wait, how are you comparing those two tangent spaces? What do you mean, normal vector?" To be fair, that isn't unique to R^n. You could just have embedded submanifolds of codimension 1 in any Riemannian manifold. I remember that was annoying when we got to studying the stress-energy tensor in GR. It's like "Wait, divergence theorem shouldn't work in more than 3 dimensions..." Actually, I've spent the last week working out some fluid mech formulas. I never realized how to prove Reynolds transport theorem for tensors, or that it was possible in generic manifolds. It's been an interesting week, nose deep in Gravitation to figure out the basic principles behind a fundamentally non-relativistic problem.
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Posted: Fri Jul 16, 2010 6:32 pm
Layra-chan It kind of scares me that the most convoluted form of mathematics that I know of is what I basically consider "blind counting" i.e. analytic number theory. Essentially: we don't know how these numbers/functions/algebraic varieties act, how many of them there are, how dense they are, so let's transform them via any of a multitude of convenient groups, stick them in an infinite series, create an analytic continuation, add a point at infinity, and then find the 0s/poles, thus giving us a ridiculous amount of information on the original set of things. This is possibly the only time where my mathematical intuition breaks down, because even though I've seen the mathematics, I still have no idea why the Riemann hypothesis gives us a bound on the growth of primes. Amen. I usually get variables, but when it starts to get overly excessive and you begin finding out all these random numbers from seemingly nothing... It's like inventing answers that somehow magically tie into your previous variables. And a lot of times their right. eek I hate it when I get lost in my calculations and all of the sudden that equation I've been working on for the last hour turns into "Wait, hat the hell did I just do? What the hell am I doing now? Is this even right? Man there's a million places were I might be off, I'ma have to go back and re-check everything..."
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Posted: Sun Jul 18, 2010 12:00 am
Layra-chan Sioga I reject your reality...
Layra-chan...
How do you know so much?
...and substitute it with my own! Years. Years and years and years of solving problems, of devouring textbooks, of forcing myself to go beyond whatever is assigned in class. Years of active, proactive learning and stubborn persistence. Indulge your curiosity, and then when it complains that it wasn't being serious, keep going anyway. Read random books that are beyond your level, and then learn, either on your own or with someone's guidance, what you need to understand those books. When you find something "cool", follow it, ask yourself what it really means, why it's true, what are similar situations, where does it fail? Like the thing about fractals having non-integer dimension. What does it mean for something to have a non-integer dimension? What does it mean for something to have an integer dimension? How does one go about calculating the dimension of something? I reject your reality...
Thank you! I think I will do that, once I finish my current list of books.
...and substitute it with my own!
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Posted: Mon Jul 19, 2010 7:27 pm
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Posted: Wed Jul 28, 2010 4:57 pm
Yeah, so apparently number theory is in fact the craziest thing ever, as confirmed by someone doing number theory. As in, you need to know basically at least some of every kind of math ever in order to do it, because there's very little in the way of elementary machinery; it's all translate into a different branch of mathematics, do stuff there, then translate back.
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Posted: Sun Aug 08, 2010 6:49 pm
Two new numbers: 20, and 1:04.33
The second is the new world record for a single solve of a 5x5x5 Rubik's cube in a competition.
The first is the absolute least upper bound for the number of moves it takes to solve a 3x3x3 Rubik's cube. There are positions that require at least 20 moves, and there are none that require more.
In less spectacular news, I finally got my hands on a 7x7x7.
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Posted: Mon Aug 09, 2010 3:24 pm
Once I realized the simplicity of a Rubik's cube, I lost all interest in it. Figuring out a solution to a puzzle is interesting, but mindlessly repeating the solution is not (to me).
Then again, I never actually figured out the solution to a Rubik's cube. I just looked it up online.
Maybe I'm just lazy?
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Posted: Tue Aug 10, 2010 1:10 am
The solution-wise interest is in optimizing, trying to find an algorithm that strikes a balance between number of moves, amount of memory required, and search length. Every known algorithm takes about two and a half times optimal number of moves, at least, but nobody knows if minimizing the number of moves is actually worth it due to the additional memory and searching required.
The not-so-solution-oriented interest is in figuring out various symmetries and unexpected subgroups.
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Posted: Tue Aug 10, 2010 3:23 pm
I get searching for a better solution. I understand the appeal of trying to find the "best" algorithms. Solving problems is interesting.
I just don't get the appeal of spending hours solving Rubik's cubes just to get faster at it. It seems so mindless.
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Posted: Tue Sep 14, 2010 5:50 am
zz1000zz I get searching for a better solution. I understand the appeal of trying to find the "best" algorithms. Solving problems is interesting. I just don't get the appeal of spending hours solving Rubik's cubes just to get faster at it. It seems so mindless. I think it's a competitive thing. They are quite popular.
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Posted: Mon Dec 20, 2010 10:08 am
I've had too much free time in the last two weeks (and will for the rest of the holiday season). That always is a bad thing for me. You see, when I have too much time, I fill it by learning new things. Unfortunately, when I learn new things, I lose a little more faith in this world. As it happens, I've revisited a topic I mentioned earlier in this thread. I said something here a year ago which at the time seemed reasonable. That was: zz1000zz 2) The planet's sensitivity to doubling CO2 levels from pre-Industrial Era times is somewhat over 1 degree Celsius. Scientists pretty much all agree with this (well, roughly 1 degree, not over it). People who dispute global warming pretty much all agree with this. It seemed everyone agreed on the subject, so I never delved. Now, a year later, I have delved, and I have found out it is bunk. The radiative forcing calculation (resulting in 3.7 W/m^2) used in it is a decade old, and known to have several issues. Worse yet, the radiative forcing is related to surface temperature by a linear relation based upon assumptions made over 40 years ago. There is far more to the issue than what I just said, as I'm starting to suspect the entire concept is meaningless, but I don't want to drone on and on. The basic point is we have lots of people saying global warming is one of the most serious problems Earth has faced, and we have lots of people saying it is a hoax. Yet everybody seems to be agreeing about a calculation which is obviously flawed.
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