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Messages - Shu Wang

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Home Assignment 4 / Re: Problem 3
« on: October 23, 2012, 05:20:03 AM »
uh, say we assume some form of solution for X(x) and T(t), so there will be 3 coefficients. When we setup the matrix for A,B,C it becomes 2x3 matrix since we're given 2 B.C. I was wondering if we were to solve for eigenvalue for the matrix, do we need to take into account for all combinations of 2x2 matrices? (In other words, break the matrix into A&B, B&C, A&C). Or am I just completely off the question? o.O

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Misc Math / Re: Orthogonal systems: approximation
« on: October 21, 2012, 06:40:55 PM »
Theorem 1
b) You are right I think; we need a) and c) to say w is linearly independent, if we let it to be an element in K. However, if we consider that every 'alpha*u_n' product spans a dimension for K then it implies every element(u_n) in K is independent of one another. Hence if we allow w to be one of them, then it's independent as well. (say w = a_1*u_1--> a_n*u_n = 0 for n != 1)
c) I'm still a little confused. Shouldn't we change one of the alpha_n notation? maybe to w = summation(alpha_m*u_n)? Otherwise the expression for w and K are the same...

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Home Assignment 4 / Re: Problem 1
« on: October 20, 2012, 11:53:24 PM »
For X_n expression in a), if why don't we have w_n in front of the cosine, instead we have w?

Also for e) can we assume there are no degeneracy in the eigenfunctions/states? Otherwise they would be orthogonal with equal eigenvalue.

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Misc Math / Orthogonal systems: approximation
« on: October 20, 2012, 06:29:47 PM »
Hi all,
First, I have a trivial question on the definition of the {u_n}(finite orthonormal system) that's mentioned at the introduction of this section in lecture note #14.
Is this set/system defined in H, in which you did not mention whether it's infinite or finite? or does it span its own space(any) based on perhaps infinite/finite number of linear hulls including K, which in particular belongs to H?

Also  In the following Theorem 1
b) I think this implies w is linear independent of other elements in K, regardless of "unique" in a) or c)
c) do you mean "w in K"(summation of alpha times u_n) rather than equal to it?

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Misc Math / Re: Lecture Notes 7 -- Ex 3-4
« on: October 14, 2012, 01:31:28 PM »
rofl, thanks alot.

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Misc Math / Lecture Notes 7 -- Ex 3-4
« on: October 14, 2012, 01:25:44 AM »
sorry for stealing the post; I've got a question regarding to Example 3 and 4 on Lecture 7.
How do you turn the Robin boundary conditions to a function of psi and phi?

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Home Assignment 3 / Re: Problem5
« on: October 09, 2012, 05:09:42 AM »
Haha, I see what you mean now, and the the maxima is definitely messed up due to the du/dx.
By the way, when is the problem set due? since we didn't have the lecture on Monday.

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Home Assignment 3 / Re: Problem5
« on: October 08, 2012, 10:56:01 PM »
This may be a stupid question, but could you clarify how the "proof of maximum" breaks down while we're asked to find the maximum? Suppose if we can prove a way to find some maxima.

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Home Assignment 3 / Re: problem 3
« on: October 08, 2012, 07:07:43 PM »
Try completing the square and using the error function. It's a really messy integral.
$\newcommand{\erf}{\operatorname{erf}}$
Yes, you need to consider  $\erf(z)$ as an elementary function (and there is no compelling arguments why trigonometric functions are considered as such but not many others. In fact there are plenty of important special functions coming often from PDE, more precisely, from separation of variables--not $\erf$ but many others).

When you integrate erf(z) it always gives you zero because it's an odd function. When multiplied to any integrated function (and as alpha -> 0), the resulting functions are always 0. Does that make any sense?

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Home Assignment 2 / Re: Problem4
« on: October 01, 2012, 12:14:16 AM »
hi y'all, a quick question,
would it be appropriate to assume x & t are independent variables in this question? as like, they are presumably not correlated in any function of each other.

Also, what does it mean by rho = T = 1 ? (is T the the kinetic energy or something?)

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Home Assignment 2 / Re: Problem 2
« on: September 29, 2012, 02:21:41 PM »
for the last part, can we just assume the same solution as c) but state a few assumptions instead? it's because I dont think the general solution of the equation would vary since no other IC were stated.

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