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A-R-Q
 Post subject: Evaluate the integral (6)
PostPosted: Sun, 26 Feb 2012 17:25:42 UTC 
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Please see link:
http://i1084.photobucket.com/albums/j40 ... ttempt.png
Image
I did the RHS of the compound inequality, but I don't know how to do LHS
I would like a hint.

I tried with 0, b.c 0 =< sin^2(x) <= 1, but then the 0 messed it up.
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outermeasure
 Post subject: Re: Evaluate the integral (6)
PostPosted: Sun, 26 Feb 2012 17:48:34 UTC 
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A-R-Q wrote:
Please see link:
http://i1084.photobucket.com/albums/j40 ... ttempt.png
Image
I did the RHS of the compound inequality, but I don't know how to do LHS
I would like a hint.

I tried with 0, b.c 0 =< sin^2(x) <= 1, but then the 0 messed it up.


Use a sharper bound: \sin x\geq\cdots when \frac{\pi}{4}\leq x\leq\frac{\pi}{2}.

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\begin{aligned} Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\ Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\ Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\ Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\ Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\ Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4 \end{aligned}
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Shadow
 Post subject: Re: Evaluate the integral (6)
PostPosted: Mon, 27 Feb 2012 00:14:26 UTC 
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outermeasure wrote:
A-R-Q wrote:
Please see link:
http://i1084.photobucket.com/albums/j40 ... ttempt.png
Image
I did the RHS of the compound inequality, but I don't know how to do LHS
I would like a hint.

I tried with 0, b.c 0 =< sin^2(x) <= 1, but then the 0 messed it up.


Use a sharper bound: \sin x\geq\cdots when \frac{\pi}{4}\leq x\leq\frac{\pi}{2}.


There is an easy bound from the proof that $\lim{x\to 0}{\sin x\over x}=1...

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