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The Art of Problem Solving
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| LaTeX question |
[Jul. 13th, 2009|12:16 pm] |
So I have this diagram that I'm trying to include in my thesis. I have a couple different copies of it in different file types, seeing what works best. Right now, I'm using a jpg because I can't get the file to compile with bmp or eps.
Now, I'd really like to use the eps, in particular. Can anyone think of a reason that LaTeX would simply not recognize the eps extension (this is the error I get when trying to compile) ?
Thanks!
EDIT: The issue was PDF output. hurr |
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| Facts about Rotations |
[Jul. 10th, 2009|04:58 pm] |
I have two elementary questions on rotations which I couldn't find answers to on Wikipedia.
Working in R^n, if I fix an arbitrary order of the n choose 2 coordinate planes, can I always decompose any unitary matrix as a composition of rotations in those coordinate planes, in the chosen order? Also, if R and U are rotations in two different coordinate planes, can I write RU = U'R' where R' is a rotation in the same coordinate plane as R and U' is a rotation in the coordinate plane as U? References greatly appreciated :) |
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| A proof of the Monotone Convergence Theorem |
[Jul. 8th, 2009|03:43 pm] |
MCT: Let f_n be an increasing sequence of nonnegative measurable functions, bounded above by and converging up to f. Then \int f_n \uparrow \int f.
Cheney (GTM 208) has this proof of the MCT, due to Rudin:
First, f_n < f_{n+1} < f implies \int f_n < \int f_{n+1} < \int f, so lim \int f_n exists and is less than \int f.
Now, let g be a simple function with 0 <= g <= f, and let 0 < \theta < 1. Let A_n = {x | f_n(x) > \theta g(x)}. Then A_n is an increasing sequence of sets whose union is all of X, so \mu(A_n \cap E) \uparrow \mu(E). Writing g = \sum \lambda_i \Chi_E_i (where \Chi_E is the characteristic function of E), this gives \int g_n \Chi_A_n = \int \sum \lambda_i \Chi_{A_n \cap E_i} = \sum \lambda_i \mu{A_n \cap E_i} \uparrow \sum \lambda_i \mu{E_i} = \int g.
So now we have \theta \int g = lim_n \int \theta g_n \Chi_{A_n} < lim_n \int f_n. Letting \theta approach 1, we have \int g < lim_n \int f_n, and then taking the supremum over all simple functions g < f, we have \int f < lim_n \int f_n, which is the other inequality we wanted.
This is a cute little proof which is substantially shorter and more elegant than most other proofs I've seen of the MCT, but my question is, why do we need this \theta? Can't we just throw it out altogether and work directly with g instead of \theta g? |
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| (no subject) |
[Jul. 7th, 2009|04:46 pm] |
Have you ever done any analysis on the functionK = -4/φ4 (φΔφ - |∇φ|2), where ∇φ and Δφ are the gradient and Laplacian of φ: R2 → (0,∞), respectively?
Motivation: if φ is a conformal change of the flat Euclidean metric on the plane, then K is the new scalar curvature.
Edit 1: Let U be a compact, connected region in R2. Let κ be a large positive number. How can I determine if the PDEK = &kappa has a solution in U? I need that the solution φ is positive, but I don't care if it's unique.
Edit 2: My geometric intuition tells me that if κ and U are both too large, there should be no solution, since then there would be a geometric obstruction, but I need some way of understanding this better. Scratch that: consider stereographic projection of the sphere of radius 1/κ. In coordinates, we can get a conformal change φ of the Euclidean metric. This φ will then satisfy the equation K = κ on the whole plane. Hmmmmmm I don't understand this stuff well at all.
Edit 3: Talking to my friend, I realized I need to formulate this PDE problem better. I'm going to think on this for a few days and probably bug all of you again. Thanks! |
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| analysis and measure |
[Jul. 1st, 2009|05:04 pm] |
a few questions:
1. given a measurable set E (in R) with finite (lebesgue) measure, and any 0 < a < 1, there exists a subset A of E such that m(A) = a*m(E). i know this is true for a = 1/2, but how about all a in (0, 1)?
2. this one is more involved. suppose f is differentiable on (a, b) and continuous on [a, b]. then there exists a sequence of polynomials that converges to f uniformly on [a, b]. since these polynomials are (infinitely) differentiable, can we conclude that the derivatives of the polynomials converge pointwise to df on (0, 1)? obviously we can't hope for uniform convergence since derivatives are not necessarily continuous, but derivatives are baire class one functions. this would be consistent with being the pointwise limit of continuous polynomials.
we can take this further, too. since the polynomials are differentiable not just once, but say, r times, what does the pointwise limit of the rth derivative say about f? f is merely continuous, possibly differentiable nowhere. does this give us a sense of an artificial derivative for arbitrary continuous functions? |
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| tensors |
[Jun. 29th, 2009|03:26 am] |
I need to prepare a talk on tensor algebra, so I am trying to refresh my memory about that... I have a question about the conditions (as explicit as possible) when a tensor A can be decomposed into a product of two tensors of lesser degree, as A=B x C. (The talk is related to defining pure states in Quantum Mechanics). Are there any simple methods that allow to produce a decomposition given the high degree tensor A? Is it unique? Is there a simple book that contains all that?
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| differential geometry |
[Jun. 27th, 2009|09:41 pm] |
Hey guys
I'm looking for a proof of the following theorem:
The total Gaussian curvature of an oriented surface M in R^3 equals the algebraic area of the image of its Gauss map.
I've checked in O'Neill's Diff Geom, and while he states how to prove it, I don't understand his notation. Trying to go back through the book to figure out his notation is quite tedious, and so far yielding no fruit.
Thanks! |
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| simple question about dynamical systems |
[Jun. 27th, 2009|07:24 pm] |
I have a discrete-time dynamical system
x(t+1) = f(x(t)),
where f: Rn &rarr Rn is a smooth C&infin function. The zero vector is
an equilibrium of this system: f(0)=0. Moreover, 0 is an attractive equilibrium, meaning that limt x(t)=0 no matter what x(0) you start with.
My question: I am interested in how big a solution can get in the course of its trajectory. Let M(x(0))= supt ||x(t)||2 where x(t) is the trajectory that begins at x(0). Let M = sup||x(0)||2 &le 1 M(x(0)). Can M be infinite?
My first gut reaction is no, that one should be able to do something like pick a convergent subsequence, and use continuity of f to get a trajectory that does not go to 0. However, I can't seem to make this argument work. |
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| applying Freedman's theorem |
[Jun. 26th, 2009|02:13 am] |
I've got a smooth manifold M (which actually depends on a lot of parameters) which is either diffeomorphic to S2 x S2, CP2 # CP2 or CP2 # - CP2, and I'm trying to find out which one.
By Freedman's theorem, the homemorphism classifcation boils down to their intersection forms, which I can crunch out (in terms of the parameters).
However, how do I practically tell when two intersection forms are equivalent? In terms of matrices, if A is the matrix associated to the intersection form, then BAB^t is equivalent to A for any invertible B. Notice that in particular, the determinant and trace are NOT invariants of the intersection form.
More specifically, what are some invariants (other than signature) which distinguish two intersection forms?
Or even more specifically, how do I know algebraically that the intersection forms of S2 x S2 and CP2 # - CP2 are different?
(The intersection form for S2 x S2 is given by the matrix
[0, 1
1, 0]
While the intersection for CP2 #-CP2 is given by the matrix
[1, 0
0, -1]
They both have +-1 as eigenvalues, so their signatures are the same.) |
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| (no subject) |
[Jun. 25th, 2009|05:51 pm] |
I'd like to use the fact that the spectral norm of a matrix X is about equal to the 1/(2p)-th power of the frobenius norm of X^p to estimate the tails of the distribution of ||X|| for a random matrix (in my application the entries of X *are not i.i.d.*) Something along this line is used in proving the semicircle law for GOE/GUE, but I haven't found any good introductory references. Any references to useful material?
Or really any ideas on how else to approach the question of bounding the tails of the norm of X, when X does not have i.i.d. entries. |
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| Math Tutor Rates in Dallas? |
[Jun. 25th, 2009|02:03 pm] |
| [ | mood |
| |  curious | ] |
I just moved to Dallas about 2 weeks ago, and I have a job offer for a Mathematics tutoring position at a private high school in North Dallas. Trouble is, I don't know what to charge.
My Qualifications: BA in Math, and 9 years experience tutoring high school and college students. I don't have a teaching certificate.
Previous to this, I was working for a govt-subsidized program for $10/hr. I'm pretty sure this is below market rate, especially for a private school, but I don't know by how much.
I'll also likely be a teacher's aide for a Physics class (grading papers, administering tests, helping out during labs). I'm not sure how this factors into the equation, either.
My guess is that if I were going on my own, I could probably charge at least $25/hr, but I don't know if the school would be willing to pay that much, particularly since I'd also be doing things like grading papers and administering tests, which are more simple types of work. I saw in their school paper that they're letting a History teacher go because of the economy. Does $15/hr seem reasonable? $17/hr?
---------
Speaking of teaching certificates, does anybody here have any experience with acquiring them in Texas/Dallas? I've looked at a few websites of a few certification companies... unfortunately, I had some difficulty in my senior-level courses (in addition to planning a wedding), and so my GPA is a little lower than what they usually require (though my 'Major' GPA is higher). How flexible are they on the requirements?
Thanks.
x-posted to ![[info]](http://l-stat.livejournal.com/img/community.gif) dfwtx |
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| Measure theory question |
[Jun. 24th, 2009|10:36 pm] |
I'm going back through some of my old homework problems on measure theory in preparation for the analysis qual, and ran across this one. I remember being distinctly unsatisfied by my argument for it, and wonder if you can provide any insight here.
Cheney 8.2.2 - Is there an example of a set X and a measure \mu on 2^X such that \mu(X) = 1 and \mu({x}) = 0 for every x \in X?
My argument was that X has got to be uncountable (otherwise, because of countable additivity, \mu(X) = 0), and that every uncountable set has to contain a non-measurable subset. However, this chapter is before you learn about non-measurable subsets, so I am opposed to this argument on philosophical grounds (plus it is hand-wavy as sin). Any hints or suggestions? Kthx |
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| (no subject) |
[Jun. 22nd, 2009|03:44 pm] |
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I'd like an upper bound for the norm of a random projection composed with a fixed projection (these projections may be onto subspaces with different dimensions), that holds with high probability. Any ideas where I can find such a bound in the literature? |
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| classifying functors |
[Jun. 21st, 2009|10:26 pm] |
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In an expository paper on category theory and quantization, John Baez says there is no natural functor from SYMP (the category of symplectic manifolds, with symplectomorphisms as morphisms) to HILB (the category of Hilbert spaces, with unitary operators as morphisms) that would correspond to the quantization of a classical system. This got me thinking about the existence of functors in general. I have only a basic familiarity with category theory -- so the answer to this question may in fact be quite elementary -- but is there a systematic way of characterizing functors between categories? Is there, say, some category-theoretic analogue of Hom(W,V) like Func(C,D)? If such an object even makes sense to study, could you point me to some references where they are discussed? |
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| Smooth operator |
[Jun. 17th, 2009|06:08 pm] |
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How much smoothness do you need on a Riemannian metric g do you need to be able to define a smooth connection and geodesics? If g ∈ C1 you're in the clear, right? Does C2 suffice for smooth curvature tensors? Geometers sometimes talk about a C2+α assumption. What's that for? |
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| converse of the Poincare Lemma |
[Jun. 12th, 2009|01:57 am] |
Poincare's Lemma in its most general form says that if a smooth manifold M is contractible, then all closed forms on M are exact, i.e., the de Rham cohomology groups H^{k}(M) for k>0 are all trivial. Is the converse of this statement true? In other words, is contractibility not only a sufficient but also a necessary condition for triviality of higher cohomology groups?
By the homotopy invariance of de Rham cohomology (from which the Poincare Lemma follows immediately as a corollary), a homotopy equivalence between manifolds gives rise to isomorphisms between all their cohomology groups. One way to go about proving the converse of the Poincare Lemma, then, would be to work backwards from cohomology isomorphisms to homotopy equivalences. Is this possible? For then the triviality of the higher cohomology groups of a non-trivial smooth manifold M would give cohomology isomorphisms with the corresponding groups for the smooth manifold consisting of a single point (since the cohomology groups for a single point are obviously trivial), and one could then deduce a homotopy equivalence between M and a point, i.e., the contractibility of M. |
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| exactness of symplectic forms |
[Jun. 11th, 2009|04:14 pm] |
Consider a 2n-dimensional symplectic manifold (M,w) with closed non-degenerate 2-form w. We know that w^n is a volume form for M. If M is closed (compact and without boundary), then w^n cannot be exact; otherwise, by Stokes's Theorem, we would have Vol(M)=\int_M w^n = \int_M d\alpha = \int_{boundary M} \alpha = \int_{\emptyset} \alpha = 0. But this contradicts the fact that Vol(M) is nonzero, so w^n is not exact. Then the cohomology class [w^n] in H^{2n}(M) is not 0. But [w^n]=[w]^n (how exactly does this equivalence work?), so [w] in H^{2}(M) cannot be 0 either. All this amounts to saying that on closed symplectic manifolds, w is NOT exact.
So far so good. But in his her notes on symplectic geometry, da Silva runs through this argument at the top of page 7 and concludes "Exact symplectic forms can only exist on noncompact manifolds."
This doesn't seem right to me. The logical negation of "closed" is not "noncompact." What about compact symplectic manifolds with boundary, like the symplectic disc in the plane? A Stokes argument won't work here, since the disc does have boundary and therefore the integrals above don't come out to be zero. |
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