User:Shanth phy/sandbox


 * $$ \begin{align}

\mathbf{P} &= (M_H,0,0,0) \\ \mathbf{p_1} &= \left( \sqrt{m^2 + p_1^2}, \vec{p}_1 \right) \\ s_1 &= \left( \mathbf{P} - \mathbf{p_1}\right)^2 \\ &= \mathbf{P}^2 + \mathbf{p_1}^2 -2\mathbf{P}\cdot \mathbf{p_1} \\ &= -M_H^2 -m^2 -2\left(-M_H \sqrt{m^2 + p_1^2}\right) \\ \Rightarrow s_1 &= 2M_H \sqrt{m^2 + p_1^2} -M_H^2 -m^2 \\ \\ \hline \\ \textrm{So, } \quad p_1 &= \sqrt{ \left(\frac{s_1 + M_H^2 +m^2}{2M_H} \right)^2 -m^2} \\ \textrm{This} \, \, \textrm{means,} \quad \int d^3\vec{p}_1 &= \int d\Omega \int_0^{p_1^{cutoff}} dp_1 = 4\pi \int_0^{p_1^{cutoff}} dp_1 \\ &= 4\pi \int_{-(M_H-m)^2}^{s_1^{cutoff}}\, ds_1 \frac{\frac{s_1 + M_H^2 +m^2}{2M_H}}{\sqrt{ \left(\frac{s_1 + M_H^2 +m^2}{2M_H} \right)^2 -m^2}} \\ &= 4\pi \int_{-(M_H-m)^2}^{s_1^{cutoff}}\, ds_1 \frac{ {s_1 + M_H^2 +m^2} }{\sqrt{ 2\left( s_1 + M_H^2 +m^2 \right)^2 -4M_H^2 m^2}} \end{align}$$

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 * $$ \begin{align}

\textrm{Maximum} \,\, \textrm{points} \,\, \textrm{allowable} \,\, \textrm{with} \,\, k \,\, \textrm{correct} \,\, \textrm{answers}, f_{max}(k) &= (0,1,2,3,4,5,6) \rightarrow (0, 3, 5, 7, 8, 9, 9) \\ \textrm{Minimum} \,\, \textrm{points} \,\, \textrm{allowable} \, \,\textrm{with} \,\, k \,\, \textrm{correct} \,\, \textrm{answers}, f_{min}(k) &= (0,1,2,3,4,5,6) \rightarrow (0, 0, 1, 2, 4, 6, 9) \\ \textrm{Actual} \,\, \textrm{points} \,\, \textrm{allowed} \,\, \textrm{on} \,\, \textrm{day} \,\,i \,\, \textrm{with} \,\, k_i \,\, \textrm{correct} \,\, \textrm{answers}, f(k_i) &= x_i \\ \textrm{Unforced} \,\, \textrm{points} \,\, \textrm{allowed} \,\, \textrm{on} \,\, \textrm{day} \,\,i \,\, \textrm{with} \,\, k_i \,\, \textrm{correct} \,\, \textrm{answers}, UfPA_i &= x_i - f_{min}(k_i) \\ UfPA_{max}(k) &= f_{max}(k_i) - f_{min}(k_i) \\ &= (0, 1, 2, 3, 4, 5, 6) \rightarrow (0, 3, 4, 5, 4, 3, 0) DE_i &= \frac{f_{max}(k_i) - x_i}{f_{max}(k_i) - f_{min}(k_i)} \\ DE &= \langle DE_i \rangle_{i=1\rightarrow N} = \frac{1}{N}\sum_{i=1}^{N} \frac{UfPA_{max}(k_i) - UfPA_i}{UfPA_{max}(k_i)} = \frac{1}{N}\sum_{i=1}^{N} \frac{f_{max}(k_i) - x_i}{f_{max}(k_i) - f_{min}(k_i)}\\ &= 1 - \left\langle \frac{UfPA_i}{UfPA_{max}(k_i)} \right\rangle_{i=1\rightarrow N} \\ UfPA &= \sum_{i=1}^{N} UfPA_i = N \cdot \overline{UfPA} \end{align}$$