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Sunday, December 20th, 2009

mathematics [ derralf ]

4:24p LaTeX continued fractions In LaTeX, is there way to define a custom continued fraction formula \mycfrac{an}{bn} with a single continuous zickzack fraction line? I am not quite satisfied with this code \textnormal{\LARGE K}_{n=1}^\infty \frac{\left.a_n\right|}{\left|b_n\right.} which gives me

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Monday, December 14th, 2009

mathematics [ czarandy ]

10:40p sqrt Does anyone know a good proof that:

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Friday, December 11th, 2009

mathematics [ oonh ]

1:33p factoring the dottie number ( factoring the dottie number )

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mathematics [ sans_galois ]

1:11a analysis -- one last question! Ok, just one more question before my final tomorrow: Do there exist non-zero compact multiplier operators on L^2[0,1]? I want to say yes, and I want to try the operator Mf(x) = xf(x), because we've done a little bit with it in the past. However, we've done very few examples of actually showing an operator is compact using the definition. Any thoughts are welcome!

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Thursday, December 10th, 2009

mathematics [ sans_galois ]

2:41p expectation and variance Consider the two-dimensional (non-negative) random variable (X,Y), where Y = a + bX + cX^2 + εX, and where εX is a random variable having E(εX) = 0 and Var(εX)=σ2x. 1. What is E(Y|X) and Var(Y|X)? 2. Suppose X is normally distributed with E(X) = μ' and Var(X) = (σ')2. Express Var(Y) in terms of a, b, c, μ', σ')2, and σ2, where you can include integrals in your answer. The way this question is posed is confusing to me, for whatever reason, but unless I'm misunderstanding something, isn't E(Y|X) just E(Y)? And for 2, I don't see how it helps to know that X is normally distributed....I don't know, this question just has me confused Thanks for youe help

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Wednesday, December 9th, 2009

mathematics [ jakemcbatuga ]

7:47p Sets & Shit. I need to know if this conditional statement is necessarily true: If A is a non-empty set bounded below, then ∀δ>0 ∃x∈A in [inf A, inf A + δ)?

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mathematics [ llyrfish ]

12:20a Orientation on Lie groups Let's say you have a top-dimensional left-invariant form ω on a Lie group G, and that this ω determines the orientation on G. Denote by Lg:G\rightarrow G the left-multiplication map, which is a diffeomorphism. I'm pretty sure that Lg is then orientation-preserving, and I'm pretty sure this is a proof: since ω is left-invariant, (Lg)*ω = ω, thus Lg preserved the guy we picked to give our orientation, so it's orientation-preserving. Is this the right way to prove this? (In case you're wondering, I'm on my way to proving something about Haar measure.)

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Tuesday, December 8th, 2009

mathematics [ sans_galois ]

12:55a functional analysis So my analysis final is coming up on Friday, and I've been going over some homework problems. (1) A compact operator is self-adjoint if and only if all of its eigenvalues are real. This, I think, is false. One direction is certainly true (self-adjoint --> real e-vals), but the other, I suspect is not. However, I've been trying to come up with a compact operator with real eigenvalues that is not self-adjoint... to no avail. Thoughts? (2) If H is a Hilbert space, A is in B(H), and AT = TA for every compact operator T, then A is a multiple of the identity. Not sure where to begin. I know that if T is compact and normal, then Aφ(T) = φ(T)A, but that doesn't seem helpful.... Any help would be great. I imagine I might have another question or two tomorrow or Wednesday. Thanks a lot!

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