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mun
 Post subject: Number of solution
PostPosted: Mon, 20 Feb 2012 09:39:37 UTC 
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If frac(x^2)=frac(1/x) where frac means fractional part .how many positive roots exist .here x^2 lies between 2 and 3.

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outermeasure
 Post subject: Re: Number of solution
PostPosted: Mon, 20 Feb 2012 10:11:57 UTC 
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mun wrote:
If frac(x^2)=frac(1/x) where frac means fractional part .how many positive roots exist .here x^2 lies between 2 and 3.


Monotonicity.

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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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mun
 Post subject: Re: Number of solution
PostPosted: Sat, 25 Feb 2012 15:47:07 UTC 
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outermeasure wrote:
mun wrote:
If frac(x^2)=frac(1/x) where frac means fractional part .how many positive roots exist .here x^2 lies between 2 and 3.


Monotonicity.

sir monotonocity is not helping .please check again

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outermeasure
 Post subject: Re: Number of solution
PostPosted: Sat, 25 Feb 2012 16:34:33 UTC 
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mun wrote:
outermeasure wrote:
mun wrote:
If frac(x^2)=frac(1/x) where frac means fractional part .how many positive roots exist .here x^2 lies between 2 and 3.


Monotonicity.

sir monotonocity is not helping .please check again


Work harder. It will help along the line if you are thinking about it in the right way.

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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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mun
 Post subject: Re: Number of solution
PostPosted: Sun, 26 Feb 2012 15:40:59 UTC 
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outermeasure wrote:
mun wrote:
outermeasure wrote:
mun wrote:
If frac(x^2)=frac(1/x) where frac means fractional part .how many positive roots exist .here x^2 lies between 2 and 3.


Monotonicity.

sir monotonocity is not helping .please check again


Work harder. It will help along the line if you are thinking about it in the right way.

Good evening Sir,
periodicity of {x} (fractional part function) is one.but how can we so sure that in interval \sqrt2<x<\sqrt3. {1/x}will follow one particular behaviour. I mean to say that it can decrease or increase so can cut {x^2} at more that one point.

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