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    Black Holes - the two points of view.

    I think it's the case that time dilation prevents you from ever seeing something reach the singularity. After all, as the mass M increases, the field strength at the event horizon is ~1/M, so we can make it as small as we like.
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    Diff eq. and linear algebra concurrently and before calc III?

    This makes sense to me. Linear algebra and differential equations are closely related, and the intuition you gain from that connection will help with the perhaps more confusing multivariate calculus.
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    Physics Apprehensive about being a quantum physicist.

    It's somewhat silly to be asking this question at your age. I doubt you know any calculus (I didn't at your age), so how can you say you know any physics? Sure it might all sound cool, but why don't you give it a try before worrying yourself about these things? Also, your performance now might...
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    Should I take Number Theory

    If you go into quantum gravity, things like infinite Galois theory, cohomology, and algebraic geometry are very important tools from algebraic number theory that one wouldn't necessarily learn in an abstract algebra course. There's also that cool connection between calculating Etale motives and...
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    Set theory union question

    Well if A is contained in B and B is contained in A, then A=B.
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    Does anyone else think ETS is obscene?

    That's a naive assumption. Often a monopoly is the most efficient configuration of firms. This can occur when either the entry cost is high, the profits low, or especially in economies of scale, where returns increase much faster than costs of expanding. Regulation is hard to do correctly. The...
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    Definition of a homeomorphism between topological spaces

    The trivial function (R->{0}) maps (non-empty) open sets to closed sets, but the preimage of every open set is open, namely the empty set.
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    On the Reality of the Psi-Function

    What's the difference between being part of physical reality and being part of a calculation that describes physical reality? Seems like a silly discussion to me.
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    Math 55 at Harvard

    I covered much of the same material (but much more algebra, galois theory, representation, etc, and less analysis) in two freshman level classes this year. On top of that I took some set theory and some chaos theory. In that sense it's less than a year's work, but certainly an absurd amount for...
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    Quotient Groups and their Index

    Lattice isomorphism theorem. Assuming those are normal subgroups, (G/K)/(K/H) is isomorphic to G/H. Of course, it's not hard to just count the number of cosets. G = g_1K + ... + g_nK where n = [G:K]. K = k_1H + ... + k_mH, where m = [H:K]. Then G = g_1(k_1 + ... + k_m)H + ... +...
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    Other Should I Become a Mathematician?

    I know a guy who does statistical mechanics of the brain. Stat mech is a very mathematical branch of physics, you might find it interesting. Beyond that I don't know much.
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    Model theory, category theory and universal algebra

    Universal algebra is very much an aspect of model theory, so I'll just define the other two. Model theory studies the abstract notion of a mathematical theory. With model theory you can study things like the consistency of certain statements, how to construct models of more complicated...
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    Integral of products?

    It's similar to a convolution, which has some nice properties, but other than that they usually need to be dealt with on a case by case basis. On the other hand if you wish to perform a contour integral you can get the residues of f(z)g(z) with the Laurent series for f(z) and g(z) in certain cases.
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    Fundamental theorem of algebra

    I suppose I also used numbers so it's a number theory proof. And I had to found it in ZF so it's a set theory proof. It might as well also be called a combinatorics proof since that's how Sylow's theorem is proven. When asked for an analytic proof of the theorem, I would've supplied the complex...
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    Fundamental theorem of algebra

    It's a really trivial application of the IVT. Monic, odd polynomials are large and positive for large x and large and negative for small x, hence they cross the axis somewhere. I hardly consider this an analytic proof.
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    Black box!

    Fascinating problem. f(0),f(1),...,f(7) gives a system of 8 equations mod M. They seem to be linearly independent, but I haven't checked. If so, these can be diagonalized to give solutions to the coefficents of f mod M. Then we just need a way of finding the modulus.
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    Fundamental theorem of algebra

    Here's one: Let f(x) be a polynomial with real coefficients of degree n with splitting field K/R. K(i) is a Galois extension of R, so let G=Aut(K(i)/R). If P is any 2-sylow subgroup of G, then the fixed field of P is an odd extension of R, which must be real, and since complex roots come in...
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    How do you Derive equations

    The algorithm for deriving equations is quite similar to the algorithm for proving theorems. Write down true statements until you're satisfied with what you've done. Of course, there are many heuristic refinements, but you'll have to figure those out on your own with practice practice practice!
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    Tensor product exercise

    Well it does, it's just easier to show surjectivity using a sum of simple tensors. In fact you need to since arbitrary elements of the codomain do not have preimages that are simple tensors.
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    Galois Extension of Q isomorphic to Z/3Z

    Consider the splitting field of any cubic polynomial with a perfect square discriminant. The root of the discriminant is rational, so it is fixed by all the Galois automorphisms. This means that there are no 2-cycles in the Galois group. If the polynomial is irreducible, then its Galois group is...
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    Tensor product exercise

    Fair enough. I think this would lead to a nicer proof, but I think your method works anyway. You need to consider it acting on all the tensors in your product, not just the simple ones. Every tensor in the matrix group over K can be expressed as a sum of its values in each index. These can be...
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    Tensor product exercise

    K tensor over F of the n dimensional F-vector space is isomorphic to the n dimensional K-vector space by easily proven properties of the tensor product over the direct sum. Perhaps this fact is useful? I'm new to this, as well.
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    Boolean Algebra

    Interpreting the 0 or 1 as a value for each member is not the best way to visualize them, in my opinion. There is an elegant (and simple!) theorem called the Stone representation theorem that says any boolean algebra is isomorphic (as a ring) to an algebra of sets, specifically some subset of a...
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    Testing Long Exams

    I don't know about the rest of you, but these kill my ability to think straight.
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    Ideals of Rings

    Suppose you have an ideal of the form mR+nR. How can you express this as a principal ideal?
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    Post your Fall 2010 schedule!

    Yay sophomore year! No more gen ed requirements! real analysis general topology graduate algebra graduate quantum mechanics graduate stat mech When I say graduate I mean what 1st year grad students take to pass their quals. I love this place. :]
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    Characteristic of a field

    Consider the subfield generated by 1. Its order must divide the order of the field, by Lagrange's theorem, since a field is an additive group. What does this tell you?
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    F = MA Contest and physics olympiad prep

    Just do a lot of problems. There are past tests online. I found the F=ma test was very similar to physics GREs--at least in terms of the mechanics. The later tests are much different, being free response. Those are what you really should study for if you hope to get far in IPhO. Building...
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    Schools High school

    You definitely don't need to be a valedictorian to get into a good college. I got about 8 or 9 Bs in my high school career (out of 12 classes a year, so about a 3.8 gpa) and ended up at caltech. You have to be self-motivated and work really hard to stand out. Don't settle for just learning from...
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    Jackson's Text is not my Friend-Any E+M Text Recommendations?

    I loved Shankar as well and found many of the same things I loved about Shankar in Griffiths. Griffiths is not as rigorous as Jackson of course, but it will prepare you well. Have you read Goldstein yet as well?