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 Posted: Wed Oct 01, 2008 12:42 pm
Attempting to make a syntactical algebraic structure for my conlang. The antecedent operator for relative pronouns is really bothersome.
The conjunctive operator isn't much better. I want to smoosh it into the descriptive operator, but that would involve making a fourth case and I don't want to do that. I'm happy with my subject-verb-object set of cases, and a conjunctive case would screw that up.
[Edit] Thanks to some rearranging of things and the introduction of the dual-set of the set of stems, the conjunction operator and the description operator are different enough that I don't feel the need to merge them anymore.
Of course, there's the problem that I don't actually know where the dual-set of the set of stems actually sends the stems, so I can't use it without multiplying (tensoring?) with an actual set, either of stems or sentences.
It's quite problematic that the set of stems doesn't have any structure other than the stuff imposed by the description and conjunction operators.
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Posted: Thu Oct 02, 2008 5:54 pm
Layra-chan Attempting to make a syntactical algebraic structure for my conlang. The antecedent operator for relative pronouns is really bothersome. The conjunctive operator isn't much better. I want to smoosh it into the descriptive operator, but that would involve making a fourth case and I don't want to do that. I'm happy with my subject-verb-object set of cases, and a conjunctive case would screw that up. conlang? *googles*
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Posted: Thu Oct 02, 2008 5:58 pm
Today, out of the blue, I was offered a job as a grader/tutor for the college . ninja ninja
Watch as Trinity's standards plummet over the next few years.
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Posted: Thu Oct 02, 2008 11:39 pm
Ig Nobels! Lots of lots of fun!
Nutrition: The effect of modulating how fresh a chip sounds on its taste
Peace: Switzerland adopting the legal principle that plants have dignity
Economics: The effect of a stripper's menstrual cycle on her nightly tips
Medicine: The effect of price on the effectiveness of fake drugs
Physics: Mathematical proof that heaps of string or hair or almost anything else will inevitably tangle themselves up in knots
Chemistry: Evidence that Coca-Cola can be used as a spermicide, followed by evidence that it's not a very good spermicide
Biology: The jump heights of the fleas on a dog versus those on a cat
Cognitive Science: The abilities of slime molds to solve puzzles
Archeology: The effect of armadillos on archaeological digs
Literature: A narrative on the experience of indignation on the workplace
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Posted: Thu Oct 02, 2008 11:52 pm
Morberticus Layra-chan Attempting to make a syntactical algebraic structure for my conlang. The antecedent operator for relative pronouns is really bothersome. The conjunctive operator isn't much better. I want to smoosh it into the descriptive operator, but that would involve making a fourth case and I don't want to do that. I'm happy with my subject-verb-object set of cases, and a conjunctive case would screw that up. conlang? *googles* Constructed language. I'm trying to make one, but my training as a mathematician keeps interfering; I'm always regularizing and generalizing. At the moment, the entire syntax can be written as three sets, four operators and four equations (and maybe a pushforward). As far as I can tell, it's complete, modulo semantics.
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Posted: Tue Oct 07, 2008 10:03 pm
I spent three hours today trying to figure out an anomaly, only to realize it was due to a difference in the MATLAB and R filtfilt functions. On the upside, the filtfilt.m file actually contained the information needed to discover this. On the downside, it is one of the few functions I have seen which is not pre-compiled, so I was rather lucky.
The company does a great job, but I still prefer open source.
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Posted: Wed Oct 15, 2008 11:48 am
Wow, the exotic 7-spheres are really simple to write down. That's ridiculous!
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Posted: Thu Oct 16, 2008 2:39 pm
Yay! I missed the Cambridge application and fellowship deadlines! As well as most of the other overseas scholarship deadlines. Bye-bye dreams of graduate study abroad! Although I'm starting to doubt whether or not I'm cut out for academia at all, even just grad school, so it balances out.
Math GRE on Saturday. If this is all it takes to get into grad school, what math/science undergrads *can't* get in?
What are the exotic 7-spheres? I heard one of my professors talk about generating them in 8 complex dimensions, but he didn't go into detail.
Source-free Maxwell equations (as well as many simpler systems of PDEs) are Lie remarkable. That's awesome, but in the jet space formulation that just makes a certain manifold unique. Is there any classification/uniqueness theorem for charting such manifolds, at least under the condition that the charts be algebraic, homogeneous, other possible variable relations?
Curse you, algebraic geometry, and the fact that I know none of you.
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Posted: Thu Oct 16, 2008 5:27 pm
Graduate studies abroad was a bad idea anyway; rootbeer is hard to find here.
So... vim or emacs... Which should I pledge my loyalty to?
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Posted: Sun Oct 19, 2008 2:51 pm
Swordmaster Dragon What are the exotic 7-spheres? I heard one of my professors talk about generating them in 8 complex dimensions, but he didn't go into detail. He didn't go into detail? But it takes like five seconds to write them down! For a fixed k between 1 and 28 (inclusive), take R^10 ~ C^5 and consider the following two manifolds: S^9 and the zero-locus of z_1^2+z_2^2+z_3^2+z_4^3+z_5^(6k+1). The intersection of the two manifolds is a 7 dimensional manifold homeomorphic to a 7-sphere, but only for one k between 1 and 28 is the intersection diffeomorphic to the standard 7-sphere. Isn't that ridiculously simple?
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Posted: Sun Oct 19, 2008 3:51 pm
Layra-chan Swordmaster Dragon What are the exotic 7-spheres? I heard one of my professors talk about generating them in 8 complex dimensions, but he didn't go into detail. He didn't go into detail? But it takes like five seconds to write them down! For a fixed k between 1 and 28 (inclusive), take R^10 ~ C^5 and consider the following two manifolds: S^9 and the zero-locus of z_1^2+z_2^2+z_3^2+z_4^3+z_5^(6k+1). The intersection of the two manifolds is a 7 dimensional manifold homeomorphic to a 7-sphere, but only for one k between 1 and 28 is the intersection diffeomorphic to the standard 7-sphere. Isn't that ridiculously simple? The talk he was giving was trying to describe some of his work to pre-math majors (sophomores), so he didn't go into detail. But he stated that equation and the result. (For some reason I thought it was in a different ambient space.) And only the statement is simple; I have no idea how you would show that each of those is a 7-sphere, much less prove that those are the only ones. Morberticus: You misunderstand quite how much I want to leave the States. Though I do like root beer...
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Posted: Tue Oct 21, 2008 9:29 pm
No, Professor Georgi, we don't understand the adjoint representation. sad
Stupid Cartan subalgebras.
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Posted: Wed Oct 22, 2008 5:30 pm
Are you taking a course in Lie Groups or Lie Algebras?
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Posted: Thu Oct 23, 2008 1:58 am
Lie algebras in Particle Physics, based off the book of the same name, written by Professor Georgi.
He doesn't mention nearly enough about the adjoint representation. For example, he expects us to figure out on our own how to find how a subalgebra acts on the adjoint representation without giving us a workable definition of the adjoint representation that doesn't involve manipulating (n^2-1)-dimensional matrices. He just says "commute" or "diagonalize." I'm not diagonalizing that monster! Sure, a computer can do it; I'm not a computer. I want something prettier than that.
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Posted: Thu Oct 23, 2008 8:17 am
Layra-chan Lie algebras in Particle Physics, based off the book of the same name, written by Professor Georgi. He doesn't mention nearly enough about the adjoint representation. For example, he expects us to figure out on our own how to find how a subalgebra acts on the adjoint representation without giving us a workable definition of the adjoint representation that doesn't involve manipulating (n^2-1)-dimensional matrices. He just says "commute" or "diagonalize." I'm not diagonalizing that monster! Sure, a computer can do it; I'm not a computer. I want something prettier than that. Perhaps I'm misremembering.... but isn't the adjoint action of the Lie algebra upon itself the bracket? Hence X * Y = [X,Y] If X and Y are realized as matrices then [X,Y] = XY - YX, but that definition of the bracket is really the Lie Algebra equivalent of embedding a manifold, or choosing a coordinate system -- ie it's not an intrinsic/invariant description of the operation. What you'll find is that the action of the Cartan subalgebra, h, decomposes your Lie algebra into weight spaces.... where a "weight" is a linear functional in h*.... so if alpha is a linear functional on h, and X is an element of the corresponding "weight space", then [h,X] = alpha(h)X. (Note: for most linear functionals on h, the corresponding weight space is just 0) Think of a weight space as a generalization of eigenvalues and eigenvectors-- a weight space for alpha is really just an eigenspace for every element of the Cartan subalgebra h, but the "eigenvalue", alpha isn't a constant... instead it's a linear functional on h. As long as your Lie algebra is well behaved (as, for instance, complex semisimple Lie algebras are), then the decomposition is complete AND (this part is cool), if X is in the weight space for alpha, and Y is in the weight space for beta, then [X,Y] is in the weight space for alpha+beta (which may be 0)... if you think about it... it's a direct consequence of the Jacobi identity. Now THAT'S pretty. It gets better... There is a beautiful set of symmetries (the Weyl group) associated to the weights, which allows all the weights to be deduced from a few fundamental weights... this gives rise to root systems, and through examination of a reduced root system, the entire structure of the Lie algebra can be represented in a simple diagram called a Dynkin diagram. Even better, because of the necessary constraints on the symmetries... ALL complex semisimple Lie algebras can be classified in this fashion! Now how cool is that? I recommend JP Serre's Complex Semisimple Lie Algebras. It's small, concise, and the perfect reference for basic abstract theory. PS... Cartan subalgebras are NOT unique, but once they are chosen everything else falls into place.... and they ARE all isomorphic to each other (at least in nicely behaved Lie algebras)
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