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Need help with PDE

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Need help with PDE

Hello everybody,

I am a junior math student and I am taking partial differential equations course this semester, but I was not very good at ordinary differential equations. If you know good books or tutorials on PDE (which preferably covers basic facts about PDEs), would you please share them here?

Thanks in advance
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U Y


OOp's now i see the v=cos(theta)u+sin(theta)|N-1) may be ok as it is if we assume that 'u' is a normalized eigenvector in N-1 dimensions
though seems we should actually have an exp(i phi) for example(where 'i' is sqrt(-1) ) fronting the sin(theta)|N-1) expression to express the arbitrary phase relation and then take 0<(or =)theta<(or =)pi/2 with 0<(or =)phi<(or =)2pi
But his expression for omega_N cannot possibly be correct especially with the upside down V whatever that is supposed to mean and d theta which has NO meaning whatsoever. For one thing it would have to be delta theta and he has to give it a number for it's size. Anyway it is clear that start quote " the set of all v in the sphere... has volume at most (his expression). Anyway he says it is extremely small for N large and in fact this is ridiculous as it is independent of N and can be anywhere from 0 to 1 - there is absolutely NO reason why it should be limited. Another way of looking at it is that it MUST be 1 or unity as the eigenfunction must be normalized!!

2nd thought after reconsideration most of all above is not all errors. Briefly,first he considers the surface of a unit sphere of N dim's is because of the one constraint sum of all N vector rows(entries) each squared is 1 so this is automatically satisfied by considering the surface of the unit sphere. Not to go into too much exact detail because don't know exactly what set of angular variables he used - in the end it should not matter but if we knew the exact set he used it would streamline the derivations and his expressions. Anyway the hyper-surface area(or may call it simply a hyper- volume for N>4 and we are assuming N is very large) is

gamma(N/2)/gamma((N-1)/2)/sqrt(%pi)

and limit as N becomes very large of gamma(N/2)/gamma((N-1)/2) is very close to 0.7 times sqrt(N) which may account for his statement c_N ~ C sqrt(N) and his prior expression p7 of

omega_N=c_N (cos(theta))^(n-2) omega_(N-1) '^' d theta where note N=2^n but still am not sure of the meaning '^' d theta which may perhaps be misleading. Anyway it basically comes from integrating over all angles except one and one angular choice(note there will be total N-1 angles so we integrate over N-2 angles). Anyway the angular volume element for one choice may be taken as (sin(theta_1))^(N-2)*(sin(theta_2))^(N-3)*(sin(theta_3))^(N-4)*...*sin(theta_(N-2)) d theta_1 d theta_2
d theta_3...d theta_(N-2) d theta_(N-1) where have used '*' to denote multiplication and note the last angle, theta_(N-1) ,does not appear as a sin or cos etc. of anything in the d volume element and it and only it, that is theta_(N-1), goes -%pi to +%pi in the volume integration while all the other angles go 0 to %pi - the theta_(N-1) is akin the 'phi' angle in ordinary 3 dimensional conventions. NOw as mentioned this is only one choice of many angular choices but with this choice one (hyper)rectangular coordinate may be taken as cos(theta_1) - remember we are on surface of the hypersphere so the hyperradius=R say is unity. Anyway it is feasible with a rotation of axis it could be expressed as sin(theta_1)in which case the sin(theta_1) could be replaced by cos(theta_1) which after dropping the ' _1 ' suffix may go to cos(theta)^(N-2) in the integration volume element and would agree with the author's(Wallach) expression for omega_N on page 7 and in fact now it may be that he is expressing it as a differential volume element in which case his '^' d theta may make some sense.

one thing for almost sure though is his expression on p7

omega_N=c_N (cos(theta))^(n-2) omega_(N-1) '^' d theta

It should be ...(cos(theta))^(N-2)... that is with the capitalized N and not n because he is using the relation as per p6 that N=2^n

The above reply is not correct for a couple more reasons. First it is in N not N-1 dim's and my statement "...omega_N cannot possibly be correct especially with the upside down V whatever that is supposed to mean and d theta which has NO meaning..." is incorrect(that is the author's expression does have some merit) see another reply post by zmth below (or above of same date 12 May)

Also the reply "Anyway he says it is extremely small for N large and in fact this is ridiculous as it is independent of N and can be anywhere from 0 to 1 - there is absolutely NO reason why it should be limited. Another way of looking at it is that it MUST be 1 or unity as the eigenfunction must be normalized!!" is incorrect also ,again see the other reply of this date. It IS a fct of N and though have not verified his exact expression it could be of order magnitude correct by consideration of inequalities on the angles etc. per the other zmth reply 12 may 2011.

However will say beginning p8 2.3 "Grover's alsorithm" he has not done justice with the very brief explanation he gives. But consulting
another couple of articles

1) Quantum Mechanics helps in searching for a needle in a haystack
Lov K. Grover, 3C-404A Bell Labs, 600 Mountain Avenue, Murray Hill NJ 07974, lkgrover@bell-labs.com

2)10.1.1.137.5008.pdf 8 October 2002 'Quantum Learning Seminar Lecture 2: Grover's Algorithm " by David A. Meyer

makes the subject clear.

Nobody studies PDE anymore? =)

Has anyone read the supposedly elementary articlehttp://math.ucsd.edu/~nwallach/venice.pdf. If so then do you not agree that his p7 beginning 'Let omega_N be the O(N)invariant volume - should that not be O(1/N)since N must equal 2^(n-1) then also he says we write a state in the form v=cos(theta)u+sin(theta) |N-1) Also this cannot possibly be correct as the the u must be replaced by 1 and the
|N-1) by sqrt(-1) because only then will we get a magnitude equal 1 as it must. Then all the rest of that section must be wrong also as for example omega_(N-1)cannot possibly exist in the same space as omega_N
Also he says the set of all v in the sphere ... which is contradictory as before he spoke of only being on the surface of a sphere.

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