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mun
 Post subject: function
PostPosted: Mon, 16 Apr 2012 19:05:01 UTC 
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If P(x)+P(1-x)=k where k is constant .P(0)=1 and P(1)=41 .how can we find P(x),where P(x) is algebraic function.

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Shadow
 Post subject: Re: function
PostPosted: Mon, 16 Apr 2012 22:40:44 UTC 
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mun wrote:
If P(x)+P(1-x)=k where k is constant .P(0)=1 and P(1)=41 .how can we find P(x),where P(x) is algebraic function.


Well, start by finding k...

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outermeasure
 Post subject: Re: function
PostPosted: Tue, 17 Apr 2012 10:52:56 UTC 
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Shadow wrote:
mun wrote:
If P(x)+P(1-x)=k where k is constant .P(0)=1 and P(1)=41 .how can we find P(x),where P(x) is algebraic function.


Well, start by finding k...


Let P(x)=k/2+f(x-1/2). Then f satisfies ...

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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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