679 lines
16 KiB
TeX
679 lines
16 KiB
TeX
\documentclass{article}
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\usepackage{iftex}
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\usepackage{ifthen}
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\usepackage{epsfig}
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\usepackage[utf8]{inputenc}
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\usepackage{xcolor}
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\ifPDFTeX
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\usepackage{newunicodechar}
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\makeatletter
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\def\doxynewunicodechar#1#2{%
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\@tempswafalse
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\edef\nuc@tempa{\detokenize{#1}}%
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\if\relax\nuc@tempa\relax
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\nuc@emptyargerr
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\else
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\edef\@tempb{\expandafter\@car\nuc@tempa\@nil}%
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\nuc@check
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\if@tempswa
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\@namedef{u8:\nuc@tempa}{#2}%
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\fi
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\fi
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}
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\makeatother
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\doxynewunicodechar{⁻}{${}^{-}$}% Superscript minus
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\doxynewunicodechar{²}{${}^{2}$}% Superscript two
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\doxynewunicodechar{³}{${}^{3}$}% Superscript three
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\fi
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\pagestyle{empty}
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\begin{document}
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$f(x_1,\ldots,x_n)$
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\pagebreak
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$g(x_1,\ldots,x_n)$
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\pagebreak
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$x_i$
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\pagebreak
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\[ A(t) = \frac{F(t) - \alpha B(t)}{F(t) + \alpha B(t)} \]
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\pagebreak
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\[ \Delta f_i^{\rm c} = \pm\sqrt{(\Delta f_i)^2 + (\Delta \mathrm{bkg})^2} = \pm\sqrt{f_i + \mathrm{bkg}} \]
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\pagebreak
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$ f_i^{\rm c} $
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\pagebreak
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$ f_i $
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\pagebreak
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$ \mathrm{bkg} $
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\pagebreak
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\[ \mathrm{bkg} = \frac{1}{N}\sum_{i=0}^N f_i \]
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\pagebreak
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\[ \Delta f_i^{\rm c} = \pm\sqrt{(\Delta f_i)^2 + (\Delta \mathrm{bkg})^2} = \pm\sqrt{f_i + (\Delta \mathrm{bkg})^2} \]
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\pagebreak
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$ N $
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\pagebreak
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\[ \Delta \mathrm{bkg} = \pm\frac{1}{N}\sqrt{\sum_{i=0}^N f_i} \]
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\pagebreak
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\[ A_{\rm RRF}(t) = A(t) \cdot 2\cos(\omega_{\rm RRF} t + \phi_{\rm RRF}) \]
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\pagebreak
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$ A(t) $
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\pagebreak
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$ A(t) = \frac{F(t) - \alpha B(t)}{F(t) + \alpha B(t)} $
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\pagebreak
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$ \omega_{\rm RRF} $
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\pagebreak
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$ \phi_{\rm RRF} $
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\pagebreak
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\[ \chi^2 = \sum_{i=1}^{N} \frac{(y_i^{\rm data} - y_i^{\rm theory})^2}{\sigma_i^2} \]
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\pagebreak
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\[ -2\ln L = 2\sum_{i=1}^{N} \left[y_i^{\rm theory} - y_i^{\rm data} \ln(y_i^{\rm theory})\right] \]
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\pagebreak
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\[ w[n] = \frac{I_0\left(\beta\sqrt{1-\left(\frac{n-\alpha}{\alpha}\right)^2}\right)}{I_0(\beta)} \]
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\pagebreak
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$ I_0 $
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\pagebreak
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$ \alpha = (M-1)/2 $
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\pagebreak
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$ \beta $
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\pagebreak
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\[ y_{\rm filtered}[n] = \sum_{k=0}^{M-1} h[k] \cdot y_{\rm theory}[n-k] \]
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\pagebreak
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\[ -2\ln L_{\rm total} = \sum_{i=1}^{N_{\rm runs}} (-2\ln L_i) \]
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\pagebreak
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\[ -2\ln L = 2\sum_{j=1}^{N_{\rm bins}} \left[y_j^{\rm theory} - y_j^{\rm data} \ln(y_j^{\rm theory})\right] \]
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\pagebreak
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\[ N_{\rm bins,total} = \sum_{i=1}^{N_{\rm runs}} N_{\rm bins,i} \]
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\pagebreak
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\[ \chi^2_{\rm red} = \frac{\chi^2}{N_{\rm bins,total} - N_{\rm params}} \]
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\pagebreak
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\[ \frac{1}{\tau_{\rm eff}} = \frac{1}{\tau_{\mu}} + \lambda_{\rm capture} \]
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\pagebreak
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\[ \chi^2 = \sum_{i={\rm start}}^{\rm end} \frac{(N_i^{\rm data} - N_i^{\rm theory})^2}{\sigma_i^2} \]
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\pagebreak
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\[ \chi^2_{\rm expected} = \sum_{i} \frac{(N_i^{\rm data} - N_i^{\rm theory})^2}{N_i^{\rm theory}} \]
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\pagebreak
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\[ -2\ln L = 2\sum_{i} \left[N_i^{\rm theory} - N_i^{\rm data} \ln(N_i^{\rm theory})\right] \]
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\pagebreak
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\[ \chi^2 = \sum_{i={\rm start}}^{\rm end} \frac{(y_i^{\rm data} - y_i^{\rm theory})^2}{\sigma_i^2} \]
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\pagebreak
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\[ N(t) = N_0 e^{-t/\tau_\mu} P(t) + B \]
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\pagebreak
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\[ \chi^2 = \sum_{i} \frac{(N_i^{\rm data} - N_i^{\rm theory})^2}{\sigma_i^2} \]
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\pagebreak
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\[P_{\rm RRF}(t) = 2 \cdot P(t) \cdot \cos(\omega_{\rm RRF} t + \phi_{\rm RRF})
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\]
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\pagebreak
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\[P_{\rm RRF}(t) = A \cdot [\cos((\omega - \omega_{\rm RRF})t + \phi - \phi_{\rm RRF}) + \text{high freq.}] \cdot e^{-\lambda t}
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\]
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\pagebreak
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\[\sigma_{A}(t) = \frac{e^{t/\tau_\mu}}{N_0} \sqrt{N(t) + \left(\frac{N(t)-B}{N_0}\right)^2 \sigma_{N_0}^2}
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\]
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\pagebreak
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\[\sigma_{A_{\rm RRF}}^{\rm packed} = \frac{\sqrt{2}}{n} \sqrt{\sum_{i=1}^{n} \sigma_{A}^2(t_i)}
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\]
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\pagebreak
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\[\chi^2 = \sum_{i=t_{\rm start}}^{t_{\rm end}} \frac{[A_{\rm RRF}^{\rm data}(t_i) - A_{\rm RRF}^{\rm theory}(t_i)]^2}{\sigma_i^2}
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\]
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\pagebreak
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\[\chi^2_{\rm exp} = \sum_{i} \frac{[A_{\rm RRF}^{\rm data}(t_i) - A_{\rm RRF}^{\rm theory}(t_i)]^2}{A_{\rm RRF}^{\rm theory}(t_i)}
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\]
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\pagebreak
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\[N_0 = \frac{\sum_{i=0}^{n} M(t_i)}{n}
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\]
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\pagebreak
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\[\sigma_{N_0} = \frac{\sqrt{\sum w_i^2 \sigma_{M_i}^2}}{\sum w_i}
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\]
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\pagebreak
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\[ P(t) = a_0 + a_1 t + a_2 t^2 + a_3 t^3 \]
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\pagebreak
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$a_0, a_1, a_2, a_3$
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\pagebreak
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\[ P(t) = \texttt{param[0]} + \texttt{param[1]} \cdot t
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+ \texttt{param[2]} \cdot t^2 + \texttt{param[3]} \cdot t^3 \]
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\pagebreak
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$a_0$
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\pagebreak
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$a_1$
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\pagebreak
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$a_2$
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\pagebreak
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$a_3$
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\pagebreak
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\[ \Delta A = \pm\frac{2}{(F+B)^2}\sqrt{B^2(\Delta F)^2 + F^2(\Delta B)^2} \]
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\pagebreak
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$\Delta\mathrm{bkg} = \sqrt{\mathrm{bkg}}$
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\pagebreak
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\[ \Delta f_i^{\rm c} = \pm\left[ (\Delta f_i)^2 + (\Delta \mathrm{bkg})^2 \right]^{1/2} =
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\pm\left[ f_i + \mathrm{bkg} \right]^{1/2}, \]
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\pagebreak
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$ f_i^{\rm c} = f_i - \mathrm{bkg} $
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\pagebreak
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\[ \Delta f_i^{\rm c} = \pm \sqrt{ (\Delta f_i)^2 + (\Delta \mathrm{bkg})^2 } =
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\pm \sqrt{f_i + (\Delta \mathrm{bkg})^2} \]
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\pagebreak
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$ \Delta \mathrm{bkg} $
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\pagebreak
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\[ \Delta \mathrm{bkg} = \pm\frac{1}{N}\left[\sum_{i=0}^N (\Delta f_i)^2\right]^{1/2} =
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\pm\frac{1}{N}\left[\sum_{i=0}^N f_i \right]^{1/2},\]
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\pagebreak
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$N$
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\pagebreak
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$ A_i = (f_i^c-b_i^c)/(f_i^c+b_i^c) $
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\pagebreak
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$ \delta A_i = 2 \sqrt{(b_i^c)^2 (\delta f_i^c)^2 + (\delta b_i^c)^2 (f_i^c)^2}/(f_i^c+b_i^c)^2$
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\pagebreak
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$ A_i = (\alpha f_i^c-b_i^c)/(\alpha \beta f_i^c+b_i^c) $
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\pagebreak
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\[ A_i = \frac{f_i^{\rm c} - b_i^{\rm c}}{f_i^{\rm c} + b_i^{\rm c}} \]
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\pagebreak
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\[ \Delta A_i = \pm\frac{2}{(f_i^{\rm c}+b_i^{\rm c})^2}\sqrt{
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(b_i^{\rm c})^2 (\Delta f_i^{\rm c})^2 +
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(f_i^{\rm c})^2 (\Delta b_i^{\rm c})^2} \]
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\pagebreak
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$ b_i^{\rm c} $
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\pagebreak
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$ A_i = (f_i^{\rm c}-b_i^{\rm c})/(f_i^{\rm c}+b_i^{\rm c}) $
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\pagebreak
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$ \delta A_i = \frac{2}{(f_i^{\rm c}+b_i^{\rm c})^2}\sqrt{(b_i^{\rm c})^2 (\delta f_i^{\rm c})^2 + (f_i^{\rm c})^2 (\delta b_i^{\rm c})^2} $
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\pagebreak
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$ \sigma_{\rm packed} = \sqrt{\sum \sigma_i^2}/N_{\rm pack} $
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\pagebreak
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$ A_i = (\alpha f_i^{\rm c} - b_i^{\rm c})/(\alpha \beta f_i^{\rm c} + b_i^{\rm c}) $
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\pagebreak
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$ A_R(t) = A(t) \cdot 2\cos(\omega_R t + \phi_R) $
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\pagebreak
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$ T(t) $
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\pagebreak
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$ T_R(t) = T(t) \cdot 2\cos(\omega_R t + \phi_R) $
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\pagebreak
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$ \omega_R $
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\pagebreak
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$ \phi_R $
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\pagebreak
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$ A_{\rm RRF}(t) = A(t) \cdot 2\cos(\omega_{\rm RRF} t + \phi_{\rm RRF}) $
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\pagebreak
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\[ \delta A_i = \frac{2}{(f_i^{\rm c}+b_i^{\rm c})^2}\sqrt{(b_i^{\rm c})^2 (\delta f_i^{\rm c})^2 + (f_i^{\rm c})^2 (\delta b_i^{\rm c})^2} \]
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\pagebreak
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$ w[n] = \frac{I_0(\beta\sqrt{1-(n/\alpha)^2})}{I_0(\beta)} $
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\pagebreak
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$ h[n] = \frac{\sin(\omega_c n)}{\pi n} \cdot w[n] $
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\pagebreak
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\[ y_{\rm filtered}[i] = \sum_{j=0}^{M-1} h[j] \cdot y_{\rm theory}[i-j] \]
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\pagebreak
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\[ \chi^2_{\rm total} = \sum_{i=1}^{N_{\rm runs}} \chi^2_i \]
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\pagebreak
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\[ \chi^2_i = \sum_{j=1}^{N_{\rm bins,i}} \frac{(y_j^{\rm data} - y_j^{\rm theory})^2}{\sigma_j^2} \]
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\pagebreak
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\[ A_i = \frac{F_i - \alpha B_i}{F_i + \alpha B_i} \]
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\pagebreak
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\[ -2\ln L = 2\sum_{i} \left[N_i^{\rm theory} - N_i^{\rm data}\ln(N_i^{\rm theory})\right] \]
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\pagebreak
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\[ P(n|\\lambda) = \frac{\\lambda^n e^{-\\lambda}}{n!} \]
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\pagebreak
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\[ \chi^2 = \sum_{i={\rm start}}^{\rm end} \frac{(y_i - f(x_i))^2}{\sigma_i^2} \]
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\pagebreak
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\[\chi^2 = \sum_{i=t_{\rm start}}^{t_{\rm end}} \frac{[N_i - N_{\rm theo}(t_i)]^2}{\sigma_i^2}
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\]
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\pagebreak
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\[N_{\rm theo}(t) = N_0 e^{-t/\tau_\mu} [1 + P(t)] + B
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\]
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\pagebreak
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\[\text{correction} = \text{packing} \times (t_{\rm res} \times 1000)
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\]
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\pagebreak
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\[\chi^2_{\rm exp} = \sum_{i=t_{\rm start}}^{t_{\rm end}} \frac{[N_i - N_{\rm theo}(t_i)]^2}{N_{\rm theo}(t_i)}
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\]
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\pagebreak
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\[-2\ln\mathcal{L} = 2 \sum_{i} \left[ N_{\rm theo}(t_i) - N_i + N_i \ln\frac{N_i}{N_{\rm theo}(t_i)} \right]
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\]
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\pagebreak
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\[P(N_i | N_{\rm theo}) = \frac{N_{\rm theo}^{N_i} e^{-N_{\rm theo}}}{N_i!}
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\]
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\pagebreak
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\[-2\ln\mathcal{L}_{\rm exp} = 2 \sum_{i} N_i \ln\frac{N_i}{N_{\rm theo}(t_i)}
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\]
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\pagebreak
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\[N(t) = N_0 e^{-t/\tau_\mu} [1 + P(t)] + B
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\]
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\pagebreak
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$ \lceil \frac{t_{\rm start} - t_{\rm data,0}}{\Delta t} \rceil $
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\pagebreak
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$ \lfloor \frac{t_{\rm end} - t_{\rm data,0}}{\Delta t} \rfloor + 1 $
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\pagebreak
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\[ N(t) = N_0 e^{-t/\tau} [ 1 + A(t) ] + \mathrm{Bkg} \]
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\pagebreak
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\[ A(t) = (-1) + e^{+t/\tau}\, \frac{N(t)-\mathrm{Bkg}}{N_0}. \]
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\pagebreak
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$ N(t) $
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\pagebreak
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\[ \Delta A(t) = \frac{e^{t/\tau}}{N_0}\,\sqrt{\frac{N(t)}{p}} \]
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\pagebreak
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$ p $
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\pagebreak
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\[ N(t_i) = \frac{1}{p}\, \sum_{j=i}^{i+p} n_j \]
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\pagebreak
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$ n_j $
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\pagebreak
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\[0 \leq t_0 \leq N_{\rm bins}
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\]
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\pagebreak
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\[t_{\rm start} = (\text{fgb} + n_0 - t_0) \times \Delta t
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\]
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\pagebreak
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\[t_{\rm end} = (\text{lgb} - n_1 - t_0) \times \Delta t
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\]
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\pagebreak
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\[t_{\rm start} = (\text{fgb} - t_0) \times \Delta t
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\]
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\pagebreak
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\[t_{\rm end} = (\text{lgb} - t_0) \times \Delta t
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\]
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\pagebreak
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\[\text{scale factor} = \text{packing} \times (t_{\rm res} \times 1000)
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\]
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\pagebreak
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\[\chi^2_{\rm exp} = \sum_{i=t_{\rm start}}^{t_{\rm end}} \frac{[A_{\rm RRF}^{\rm data}(t_i) - A_{\rm RRF}^{\rm theory}(t_i)]^2}{A_{\rm RRF}^{\rm theory}(t_i)}
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\]
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\pagebreak
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\[\mathcal{L} = \prod_i \frac{\mu_i^{n_i} e^{-\mu_i}}{n_i!}
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\]
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\pagebreak
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$\mu_i$
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\pagebreak
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$n_i$
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\pagebreak
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\[t_{\rm start} = ({\rm fgb} + n_0 - t_0) \times \Delta t
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\]
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\pagebreak
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\[t_{\rm end} = ({\rm lgb} - n_1 - t_0) \times \Delta t
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\]
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\pagebreak
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\[i_{\rm start} = \lceil \frac{t_{\rm start} - t_{\rm data,0}}{\Delta t_{\rm data}} \rceil
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\]
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\pagebreak
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\[i_{\rm end} = \lfloor \frac{t_{\rm end} - t_{\rm data,0}}{\Delta t_{\rm data}} \rfloor + 1
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\]
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\pagebreak
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$t_{\rm data,0}$
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\pagebreak
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$\Delta t_{\rm data}$
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\pagebreak
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\[M(t) = [N(t) - B] \cdot e^{+t/\tau_\mu}
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\]
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\pagebreak
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$\sigma_M = e^{+t/\tau_\mu} \sqrt{N(t) + \sigma_B^2}$
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\pagebreak
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$w = 1/\sigma_M^2$
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\pagebreak
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\[A(t) = \frac{M(t)}{N_0} - 1
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\]
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\pagebreak
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\[\sigma_A(t) = \frac{e^{+t/\tau_\mu}}{N_0} \sqrt{N(t) + \left(\frac{N(t)-B}{N_0}\right)^2 \sigma_{N_0}^2}
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\]
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\pagebreak
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\[A_{\rm RRF}(t) = 2 \cdot A(t) \cdot \cos(\omega_{\rm RRF} t + \phi_{\rm RRF})
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\]
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\pagebreak
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$A_{\rm packed} = \frac{1}{n}\sum_{i=1}^{n} A_{\rm RRF}(t_i)$
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\pagebreak
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$\sigma_{\rm packed} = \frac{\sqrt{2}}{n}\sqrt{\sum_{i=1}^{n} \sigma_A^2(t_i)}$
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\pagebreak
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\[N_0 = \frac{1}{n} \sum_{i=0}^{n-1} M(t_i)
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\]
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\pagebreak
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\[n_{\rm end} = {\rm round}\left( \left\lceil \frac{T \cdot f_{\rm max}}{2\pi} \right\rceil \cdot \frac{2\pi}{f_{\rm max} \cdot \Delta t} \right)
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\]
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\pagebreak
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\[\sigma_{N_0} = \frac{\sqrt{\sum_{i=0}^{n-1} w_i^2 \sigma_{M_i}^2}}{\sum_{i=0}^{n-1} w_i}
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\]
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\pagebreak
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$ B = \frac{1}{n}\sum_{i={\rm start}}^{{\rm end}} N(i) $
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\pagebreak
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$ \sigma_B = \sqrt{\frac{1}{n-1}\sum_{i={\rm start}}^{{\rm end}} (N(i) - B)^2} $
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\pagebreak
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\[ P(t) = c \]
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\pagebreak
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\[ P(t) = A \]
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\pagebreak
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\[ P(t) = \exp\left(-\lambda t\right) \]
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\pagebreak
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\[ P(t) = \exp\left(-\lambda [t-t_{\rm shift}] \right) \]
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\pagebreak
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\[ P(t) = \exp\left(-[\lambda t]^\beta\right) \]
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\pagebreak
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\[ P(t) = \exp\left(-[\lambda (t-t_{\rm shift})]^\beta\right) \]
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\pagebreak
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\[ P(t) = \exp\left(-\frac{1}{2} [\sigma t]^2\right) \]
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\pagebreak
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\[ P(t) = \exp\left(-\frac{1}{2} [\sigma (t-t_{\rm shift})]^2\right) \]
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\pagebreak
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\[ P(t) = \frac{1}{3} + \frac{2}{3} \left[1-(\sigma t)^2\right] \exp\left[-\frac{1}{2} (\sigma t)^2\right] \]
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\pagebreak
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\[ P(t) = \frac{1}{3} + \frac{2}{3} \left[1-(\sigma \{t-t_{\rm shift}\})^2\right] \exp\left[-\frac{1}{2} (\sigma \{t-t_{\rm shift}\})^2\right] \]
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\pagebreak
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\[ = 1-\frac{2\sigma^2}{(2\pi\nu)^2}\left[1-\exp\left(-1/2 \{\sigma t\}^2\right)\cos(2\pi\nu t)\right] +
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\frac{2\sigma^4}{(2\pi\nu)^3}\int^t_0 \exp\left(-1/2 \{\sigma \tau\}^2\right)\sin(2\pi\nu\tau)\mathrm{d}\tau \]
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\pagebreak
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$\sigma$
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\pagebreak
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$\nu$
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\pagebreak
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$t_{\rm shift}$
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\pagebreak
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$(\mu\mathrm{s})$
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\pagebreak
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\[ = \frac{1}{2\pi \imath}\int_{\gamma-\imath\infty}^{\gamma+\imath\infty}
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\frac{f_{\mathrm{G}}(s+\Gamma)}{1-\Gamma f_{\mathrm{G}}(s+\Gamma)} \exp(s t) \mathrm{d}s,
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\mathrm{~where~}\,f_{\mathrm{G}}(s)\equiv \int_0^{\infty}G_{\mathrm{G,LF}}(t)\exp(-s t) \mathrm{d}t\]
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\pagebreak
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$G_{\mathrm{G,LF}}(t)$
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\pagebreak
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$\Gamma$
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\pagebreak
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\[ = 1/3 + 2/3 [1 - \lambda t] \exp(-\lambda t) \]
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\pagebreak
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$\lambda$
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\pagebreak
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\[ = 1-\frac{a}{2\pi\nu}j_1(2\pi\nu t)\exp\left(-at\right)-
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\left(\frac{a}{2\pi\nu}\right)^2 \left[j_0(2\pi\nu t)\exp\left(-at\right)-1\right]-
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a\left[1+\left(\frac{a}{2\pi\nu}\right)^2\right]\int^t_0 \exp\left(-a\tau\right)j_0(2\pi\nu\tau)\mathrm{d}\tau) \]
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\pagebreak
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$a$
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\pagebreak
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\[ = \frac{1}{2\pi \imath}\int_{\gamma-\imath\infty}^{\gamma+\imath\infty}
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\frac{f_{\mathrm{L}}(s+\Gamma)}{1-\Gamma f_{\mathrm{L}}(s+\Gamma)} \exp(s t) \mathrm{d}s,
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\mathrm{~where~}\,f_{\mathrm{L}}(s)\equiv \int_0^{\infty}G_{\mathrm{L,LF}}(t)\exp(-s t) \mathrm{d}t\]
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\pagebreak
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$G_{\mathrm{L,LF}}(t)$
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\pagebreak
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\[ \nu_c \gg \gamma_\mu \Delta_{\rm L} \]
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\pagebreak
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\[ = 1/3 + 2/3 \left(1-(\sigma t)^2-\lambda t\right) \exp\left(-1/2(\sigma t)^2-\lambda t\right)\]
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\pagebreak
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\[ = 1/3 + 2/3 \left(1-(\sigma t)^\beta \right) \exp\left(-(\sigma t)^\beta / \beta\right)\]
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\pagebreak
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$\beta$
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\pagebreak
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\[ = \frac{1}{3}\exp\left(-\sqrt{\frac{4\lambda^2(1-q)t}{\gamma}}\right)+\frac{2}{3}\left(1-\frac{q\lambda^2t^2}{\sqrt{\frac{4\lambda^2(1-q)t}{\gamma}+q\lambda^2t^2}}\right)\exp\left(-\sqrt{\frac{4\lambda^2(1-q)t}{\gamma}+q\lambda^2t^2}\right)\]
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\pagebreak
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$\gamma$
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\pagebreak
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$q$
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\pagebreak
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\[ = \frac{1}{6}\left(1-\frac{\nu t}{2}\right)\exp\left(-\frac{\nu t}{2}\right)+\frac{1}{3}\left(1-\frac{\nu t}{4}\right)\exp\left(-\frac{\nu t + 2.44949\lambda t}{4}\right)\]
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\pagebreak
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\[ = \exp\left[-\frac{\sigma^2}{\gamma^2}\left(e^{-\gamma t}-1+\gamma t\right)\right] \]
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\pagebreak
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\[ = \cos(2\pi\nu t + \varphi) \]
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\pagebreak
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$\varphi$
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\pagebreak
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\[ = \alpha\,\cos\left(2\pi\nu t+\frac{\pi\varphi}{180}\right)\exp\left(-\lambda_{\mathrm{T}}t\right)+(1-\alpha)\,\exp\left(-\lambda_{\mathrm{L}}t\right)\]
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\pagebreak
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$\alpha$
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\pagebreak
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$\lambda_{\rm T}$
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\pagebreak
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$\lambda_{\rm L}$
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\pagebreak
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\[ = \alpha\,\left[\cos(2\pi\nu t)-\frac{\sigma^2 t}{2\pi\nu}\sin(2\pi\nu t)\right]\exp(-[\sigma t]^2/2)+(1-\alpha)\,\exp(-(\lambda t)^\beta)\]
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\pagebreak
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$\alpha=2/3$
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\pagebreak
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\[ = \alpha\,\left[\cos(2\pi\nu t)-\frac{a}{2\pi\nu}\sin(2\pi\nu t)\right]\exp(-a t)+(1-\alpha)\,\exp(-(\lambda t)^\beta)\]
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\pagebreak
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\[ = j_0(2\pi\nu t + \varphi) \]
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\pagebreak
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\[ = \alpha\,j_0\left(2\pi\nu t+\frac{\pi\varphi}{180}\right)\exp\left(-\lambda_{\mathrm{T}}t\right)+(1-\alpha)\,\exp\left(-\lambda_{\mathrm{L}}t\right)\]
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\pagebreak
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\begin{eqnarray*} &=& \frac{\sigma_{-}}{\sigma_{+}+\sigma_{-}}\exp\left[-\frac{\sigma_{-}^2t^2}{2}\right]
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\left\lbrace\cos\left(2\pi\nu t+\frac{\pi\varphi}{180}\right)+
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\sin\left(2\pi\nu t+\frac{\pi\varphi}{180}\right)\mathrm{Erfi}\left(\frac{\sigma_{-}t}{\sqrt{2}}\right)\right\rbrace+\\
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& & \frac{\sigma_{+}}{\sigma_{+}+\sigma_{-}}
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\exp\left[-\frac{\sigma_{+}^2t^2}{2}\right]\left\lbrace\cos\left(2\pi\nu t+\frac{\pi\varphi}{180}\right)-
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\sin\left(2\pi\nu t+\frac{\pi\varphi}{180}\right)\mathrm{Erfi}\left(\frac{\sigma_{+}t}{\sqrt{2}}\right)\right\rbrace
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\end{eqnarray*}
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\pagebreak
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$\sigma_{-}$
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\pagebreak
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$\sigma_{+}$
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\pagebreak
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$\Delta_{\rm eff}^2 = (1+R_b^2) \Delta_0^2$
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\pagebreak
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$R_b$
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\pagebreak
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$\Delta_0$
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\pagebreak
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$1/\mu\mathrm{sec}$
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\pagebreak
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\[ = \frac{1}{3} + \frac{2}{3}\,\frac{1}{\left(1+(R_b \Delta_0 t)^2\right)^{3/2}}\,
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\left(1 - \frac{(\Delta_0 t)^2}{\left(1+(R_b \Delta_0 t)^2\right)}\right)\,
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\exp\left[\frac{(\Delta_0 t)^2}{2\left(1+(R_b \Delta_0 t)^2\right)}\right] \]
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\pagebreak
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$R_{\rm b} = \Delta_{\rm GbG}/\Delta_0$
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\pagebreak
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\[ = \frac{1}{\sqrt{1+(R_b \gamma\Delta_0 t)^2}}\,
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\exp\left[-\frac{(\gamma\Delta_0 t)^2}{2(1+(R_b \gamma\Delta_0 t)^2)}\right]\,
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\cos(\gamma B_{\rm ext} t + \varphi) \]
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\pagebreak
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$\nu = \gamma B_{\rm ext}$
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\pagebreak
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\begin{eqnarray*}\Theta(t) &=& \frac{\exp(-\nu_c t) - 1 - \nu_c t}{\nu_c^2} \\
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P_{Z}^{\rm dyn}(t) &=& \sqrt{\frac{1}{1+4 R_b^2 \Delta_0^2 \Theta(t)}}\,\exp\left[-\frac{2 \Delta_0^2 \Theta(t)}{1+4 R_b^2 \Delta_0^2 \Theta(t)}\right]
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\end{eqnarray*}
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\pagebreak
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$\nu_c$
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\pagebreak
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\begin{eqnarray*}\Theta(t) &=& \frac{\exp(-\nu_c t) - 1 - \nu_c t}{\nu_c^2} \\
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P_{X}^{\rm dyn}(t) &=& \sqrt{\frac{1}{1+2 R_b^2 \Delta_0^2 \Theta(t)}}\,\exp\left[-\frac{\Delta_0^2 \Theta(t)}{1+2 R_b^2 \Delta_0^2 \Theta(t)}\right]\,\cos(\gamma B_{\rm ext} t + \varphi)
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\end{eqnarray*}
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\pagebreak
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\[ P(t) = c_0 + c_1 t + c_2 t^2 + c_3 t^3 = \sum_{k=0}^{3} c_k t^k \]
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\pagebreak
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$c_k$
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\pagebreak
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$c_0$
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\pagebreak
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$c_1$
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\pagebreak
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$c_2$
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\pagebreak
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$c_3$
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\pagebreak
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$P(t)$
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\pagebreak
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\end{document}
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