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    Algebraic Topology: SO(3)/A5

    I was watching this video on Abstract Algebra and the professor was discussing how at one point a few mathematicians conjectured the special orthogonal group in ##\mathbb{R}^3## mod the symmetries of an icosahedron described the shape of the universe (near the end of the video). My question is...
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    Trying To Learn Category Theory

    So I cleaned my solution up a little based on the comments but I also worked through mathwonks solution which is much nicer. It used some mathematics that were a little beyond where I'm at currently but I managed to get a handle on those mechanics by "reading ahead." Anyway, if anyone is...
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    Trying To Learn Category Theory

    Thanks mathwonk and fresh_42 for the great comments. I've read them and will rework this problem shortly. Seems like it needs a lot of work though so it might take a day or two.
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    Trying To Learn Category Theory

    I'm trying to learn Category Theory; this isn't homework or anything. I've attached a problem from the text "Basic Homological Algebra" by Osborne and I show my attempt at a solution. My solution doesn't seem exactly correct and I state why in the attachment as well. Can someone take a look...
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    Tensor Product Functor

    That's along the lines of what I was guessing. I suppose an idea for a paper would be to carry this line of thought out and make it precise. I suppose I would have to learn some GR which seems rather daunting.
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    Tensor Product Functor

    At the risk of sounding ignorant I'd like to propose a question to someone well versed in Homological Algebra and General Relativity. I'm starting to study the tensor product functor in the context of category theory because I'm interested in possibly doing a paper on TQFT for a directed...
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    Abstract Algebra: Automorphisms

    OK, thank you both for your reply. It's a little more clear now.
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    Abstract Algebra: Automorphisms

    I have a question about Automorphisms. Please check the following statement for validity.... An automorphism of a group should map generators to generators. Suppose it didn't, well then the group structure wouldn't be preserved and since automorphisms are homomorphisms this would be a...
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    Modern Algebra: Stabilizers and Conjugacy Classes of Dodec

    We were studying the Icosahedral (or dodecahedral which ever you prefer) group or equivalently the rotational symmetries of the Icosahedron. I should elaborate on my question with a little more specifics using an example. In one step he claimed all elements of order 3 are conjugate and to...
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    Modern Algebra: Stabilizers and Conjugacy Classes of Dodec

    My professor was proving that the Dodecahedron is isomorphic to ##A_5## and in the process utilized the stabilizer (which one can intuit ) of an edge, vertex or face to determine the conjugacy class (which is hard to intuit) of elements of the same order. This seems like a valuable skill but I...
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    Abstract Algebra: Dummit and Foote Exercise

    This isn't homework, I'm just trying to refresh my memory on cyclic groups. My question is, in this problem solution, how does ##{\sigma_i}^m=1## follow from ##\sigma_i## being disjoint?
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    Interesting Subspaces of ##L^p## Spaces

    He said it can be anything to do with ##L^p## spaces as long as I find it interesting. My interest is building an "intuition" for concepts from physics and I feel like I can do that by focusing on the math. For example, my first introduction to ##L^p## spaces was simply that ##iff## p=2 can...
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    Interesting Subspaces of ##L^p## Spaces

    It was interesting, unfortunately most of it was to advanced for me. I struggled through it but to be honest I had to give up on the topological stuff. I mentioned to my professor that I was reading this and he guaranteed me a grad school recommendation if I wrote a report on the topic of...
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    Interesting Subspaces of ##L^p## Spaces

    That is extremely interesting indeed. So much to think about now....
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    Interesting Subspaces of ##L^p## Spaces

    Wow, good stuff. I guess the first thing I need to do work through why ##C_0## isn't the subspace with the least amount of requirements for the momentum operator to be self-adjoint like I thought it was. Quick question, I thought ##C_0## was first considered by Banach? Am I mistaken...
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    Interesting Subspaces of ##L^p## Spaces

    ##C_0=\{f\in L^p: f(x)\rightarrow 0 ## as ## x\rightarrow infinity\}## This is an interesting subspace because it is the subspace of ##L^p## in which the momentum operator from physics is self adjoint. It seems that there should be more to be said about the importance of ##C_0## though...
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    Levi-Civita Tensor

    I have been trying to think about the Levi-Civita tensor in the context of Group Theory. Is there a group that it is symmetric to? I'm sorry if this is a double post but I don't think my original identical post submitted correctly. Thanks, Nate
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