\batchmode \documentclass{report} \RequirePackage{ifthen} \usepackage{amssymb} \usepackage[dvips]{graphicx} \usepackage{verbatim} \usepackage{html} \usepackage{amsmath} \usepackage{latexsym,amssymb} \usepackage[mathscr]{eucal} \usepackage{amsthm,amsxtra,amscd,upref} \usepackage{layout,bm,dcolumn} \usepackage{graphicx,color} \usepackage{calc} \usepackage{framed} % \providecommand{\DST}[1]{{\ensuremath{\displaystyle{#1}}}}% \providecommand{\DSF}[2]{{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{#1}}}}{{\ensuremath{\displaystyle{#2}}}}}}}} % \providecommand{\vk}[1]{{\ensuremath{\underline{\bm{#1}}}}}% \providecommand{\mx}[1]{{\ensuremath{\bm{\mathsf{#1}}}}}% \providecommand{\lrb}[1]{{\ensuremath{\left({#1}\right)}}}% \providecommand{\lrs}[1]{{\ensuremath{\left[{#1}\right]}}}% \providecommand{\lrc}[1]{{\ensuremath{\left\{{#1}\right\}}}}% \providecommand{\lrv}[1]{{\ensuremath{\left|{#1}\right|}}} % \providecommand{\hell}{{\ensuremath{\hat{\jmath}}}}% 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\providecommand{\vhi}[1]{{\ensuremath{{\ensuremath{\underline{\bm{h}}}}_{#1}}}}% \providecommand{\vhr}{{\ensuremath{{\ensuremath{\underline{\bm{h}}}}}}}% \providecommand{\IMA}{{\ensuremath{\mathrm{i}}}}% \providecommand{\EE}{{\ensuremath{\mathrm{e}}}}% \providecommand{\half}{{\ensuremath{\frac{1}{2}}}} % \providecommand{\deltax}{{\ensuremath{\rho\cos(\beta)}}}% \providecommand{\deltay}{{\ensuremath{\rho\sin(\beta)}}}% \providecommand{\deltaz}{{\ensuremath{\delta_{z}}}}% \providecommand{\deltazs}{{\ensuremath{\widehat{{\ensuremath{\delta_{z}}}}}}}% \providecommand{\DD}[1]{{\ensuremath{\mathrm{d}{#1}\, }}}% \providecommand{\DDD}[2]{{\ensuremath{\mathrm{d}^{#1}{#2}\, }}} % \providecommand{\haf}{{\ensuremath{\scriptstyle{\frac{1}{2}}}}}% \providecommand{\unt}{{\ensuremath{\scriptstyle{\frac{1}{3}}}}}% \providecommand{\dut}{{\ensuremath{\scriptstyle{\frac{2}{3}}}}}% \providecommand{\xref}[1]{(\ref{#1})}% \providecommand{\eref}[1]{Eq.~(\ref{#1})}% \providecommand{\Beref}[1]{{\textcolor[rgb]{0,0,1}{Eq.~(\ref{#1})}}}% \providecommand{\eeref}[2]{Eqs.~(\ref{#1},\ref{#2})}% \providecommand{\eeeref}[3]{Eqs.~(\ref{#1},\ref{#2},\ref{#3})}% \providecommand{\aref}[1]{Appendix~\ref{#1}}% \providecommand{\sref}[1]{Sec.~\ref{#1}}% \providecommand{\cref}[1]{Chap.~\ref{#1}}% \providecommand{\tref}[1]{Tab.~\ref{#1}}% \providecommand{\fref}[1]{Fig.~\ref{#1}}% \providecommand{\Rref}[1]{Ref.~\cite{#1}}% \providecommand{\degC}{{\ensuremath{{}^{\mathrm{o}}}}} % \providecommand{\TT}{{\ensuremath{{2\theta}}}}% \providecommand{\TTz}{{\ensuremath{{{{\ensuremath{{2\theta}}}}_{0}}}}}% \providecommand{\TTe}{{\ensuremath{{{{\ensuremath{{2\theta}}}}_{e}}}}}% \providecommand{\TTB}{{\ensuremath{{{{\ensuremath{{2\theta}}}}_{B}}}}} \pagecolor[gray]{.7} \usepackage[]{inputenc} \makeatletter \makeatletter \count@=\the\catcode`\_ \catcode`\_=8 \newenvironment{tex2html_wrap}{}{}% \catcode`\<=12\catcode`\_=\count@ 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oddsidemargin=\the\oddsidemargin}\lthtmltypeout{}% \makeatletter \if@twoside\lthtmltypeout{latex2htmlLength evensidemargin=\the\evensidemargin}% \else\lthtmltypeout{latex2htmlLength evensidemargin=\the\oddsidemargin}\fi% \lthtmltypeout{}% \makeatother \setcounter{page}{1} \onecolumn % !!! IMAGES START HERE !!! {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline7641}% $ \chi ^2$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} \stepcounter{chapter} \stepcounter{section} \stepcounter{section} {\newpage\clearpage \lthtmlpictureA{tex2html_wrap7721}% \includegraphics[width=\textwidth]{multi_detector}% \lthtmlpictureZ \lthtmlcheckvsize\clearpage} \stepcounter{subsection} \stepcounter{section} {\newpage\clearpage \lthtmlpictureA{tex2html_wrap7727}% \includegraphics[width=\textwidth]{data_receiver}% \lthtmlpictureZ \lthtmlcheckvsize\clearpage} \stepcounter{section} \stepcounter{subsection} \stepcounter{subsection} \stepcounter{section} \stepcounter{section} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline7737}% $ d$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} \stepcounter{section} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline7740}% $ \Updownarrow$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage 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\Rightarrow \, \textrm{Header after script}\\ \\ \end{array} \right. \\ \\ \\ \\ \end{array} \par \\ % \\ \Rightarrow \, \textrm{Script after} \\ \end{array} % \right. \\ \\ \\ \end{array} \right. \\ \\ \\ \\ \end{array} \right. \\ \\ \Rightarrow \, \textrm{Stop script} \\ \\ \end{array} % \right. \\ \\ \end{array} \right. \end{displaymath}% \lthtmldisplayZ \lthtmlcheckvsize\clearpage} \stepcounter{section} {\newpage\clearpage \lthtmlpictureA{tex2html_wrap7767}% \includegraphics[width=\textwidth]{images/normal_acquisition.eps}% \lthtmlpictureZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlpictureA{tex2html_wrap7771}% \includegraphics[width=\textwidth]{images/gated_acquisition.eps}% \lthtmlpictureZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlpictureA{tex2html_wrap7775}% \includegraphics[width=\textwidth]{images/trigger_acquisition.eps}% \lthtmlpictureZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlpictureA{tex2html_wrap7779}% \includegraphics[width=\textwidth]{images/ro_trigger_acquisition.eps}% \lthtmlpictureZ \lthtmlcheckvsize\clearpage} \stepcounter{section} \stepcounter{section} \stepcounter{section} \stepcounter{subsection} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline7790}% $ (108602\&0xFFFFFFFE)>>1 = 54301$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline7792}% $ (108602\&0x1) =0$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} \stepcounter{subsection} \stepcounter{subsection} \stepcounter{chapter} \stepcounter{section} \stepcounter{subsection} {\newpage\clearpage \lthtmlpictureA{tex2html_wrap7801}% \includegraphics[width=\textwidth]{images/effiSiHardXRays2}% \lthtmlpictureZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline7803}% $ \mu$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlpictureA{tex2html_wrap7807}% \includegraphics[width=\textwidth]{images/effiThinkBackplanes}% \lthtmlpictureZ \lthtmlcheckvsize\clearpage} \stepcounter{subsection} {\newpage\clearpage \lthtmlpictureA{tex2html_wrap7812}% \includegraphics[width=\textwidth]{images/settings}% \lthtmlpictureZ \lthtmlcheckvsize\clearpage} \stepcounter{section} \stepcounter{chapter} \stepcounter{section} \stepcounter{subsection} \stepcounter{section} {\newpage\clearpage \lthtmlpictureA{tex2html_wrap7822}% \includegraphics[width=\textwidth]{images/thr_scan_expl}% \lthtmlpictureZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlpictureA{tex2html_wrap7826}% \includegraphics[width=\textwidth]{images/thr_scan_fluo}% \lthtmlpictureZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline7831}% $ E_0$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline7835}% $ E_t=E_0/2$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline7839}% $ E_f$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline7843}% $ E_t$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline7857}% $ E_fE_f+3$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline7863}% $ E_t4$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlpictureA{tex2html_wrap7886}% \includegraphics[width=\textwidth]{images/sample_with_fluorescence}% \lthtmlpictureZ \lthtmlcheckvsize\clearpage} \stepcounter{section} \stepcounter{subsection} {\newpage\clearpage \lthtmlpictureA{tex2html_wrap7892}% \includegraphics[width=\textwidth]{images/bad_ff_col}% \lthtmlpictureZ \lthtmlcheckvsize\clearpage} \stepcounter{subsection} {\newpage\clearpage \lthtmlpictureA{tex2html_wrap7897}% \includegraphics[width=\textwidth]{images/FFSetup}% \lthtmlpictureZ \lthtmlcheckvsize\clearpage} \stepcounter{section} \stepcounter{subsection} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline7904}% $ Vthreshold=7$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline7906}% $ Counts=500$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline7908}% $ Resolution=4$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline7910}% $ \pm$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlpictureA{tex2html_wrap7911}% \includegraphics[width=\textwidth]{images/noise_thresholdscanuntrimmed}% \lthtmlpictureZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlpictureA{tex2html_wrap7915}% \includegraphics[width=\textwidth]{images/trimbitdistribution}% \lthtmlpictureZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlpictureA{tex2html_wrap7919}% \includegraphics[width=\textwidth]{images/trimbitplot}% \lthtmlpictureZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlpictureA{tex2html_wrap7923}% \includegraphics[width=\textwidth]{images/noise_thresholdscantrimmed}% \lthtmlpictureZ \lthtmlcheckvsize\clearpage} \stepcounter{section} \stepcounter{subsection} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline7932}% $ \cdot$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} \stepcounter{section} \stepcounter{section} \stepcounter{subsection} \stepcounter{chapter} \stepcounter{section} \stepcounter{section} \stepcounter{section} \stepcounter{section} \stepcounter{chapter} \stepcounter{section} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline7946}% $ \alpha_{jm}$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline7948}% $ R_m$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline7950}% $ \Phi_m$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline7952}% $ D_m$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline7954}% $ c_m$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline7956}% $ o_m$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline7958}% $ k_m$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay7960}% $\displaystyle \alpha_{jm}=\Phi_m-{\ensuremath{\left({{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{180}}}}{{\ensuremath{\displaystyle{\pi}}}}}}}}\right)}}\arctan{\ensuremath{\left({{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{D_m-pj}}}}{{\ensuremath{\displaystyle{R_m}}}}}}}}\right)}}$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay7962}% $\displaystyle \alpha_{jm}=o_m+{\ensuremath{\left({{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{180}}}}{{\ensuremath{\displaystyle{\pi}}}}}}}}\right)}}c_mk_m+{\ensuremath{\left({{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{180}}}}{{\ensuremath{\displaystyle{\pi}}}}}}}}\right)}}\arctan{\ensuremath{\left[{{\ensuremath{\left({j-c_m}\right)}}k_m}\right]}}$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay7965}% $\displaystyle c_m$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay7967}% $\displaystyle =$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay7969}% $\displaystyle {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{D_m}}}}{{\ensuremath{\displaystyle{p}}}}}}};$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay7971}% $\displaystyle k_m$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay7975}% $\displaystyle {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{p}}}}{{\ensuremath{\displaystyle{R_m}}}}}}};$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay7977}% $\displaystyle o_m$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay7981}% $\displaystyle \Phi_m-{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{180}}}}{{\ensuremath{\displaystyle{\pi}}}}}}}{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{D_m}}}}{{\ensuremath{\displaystyle{R_m}}}}}}}.$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay7984}% $\displaystyle \Phi_m$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay7988}% $\displaystyle o_m+{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{180}}}}{{\ensuremath{\displaystyle{\pi}}}}}}}c_mk_m;$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay7990}% $\displaystyle R_m$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay7994}% $\displaystyle {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{p}}}}{{\ensuremath{\displaystyle{k_m}}}}}}};$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay7996}% $\displaystyle D_m$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8000}% $\displaystyle c_m p.$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} \stepcounter{section} \stepcounter{subsection} \stepcounter{subsubsection} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8005}% $ {\ensuremath{{2\theta}}}$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8007}% $ 2\theta$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} \stepcounter{subsubsection} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8010}% $\displaystyle {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\mathrm{d}{}\, }}\bf {\sigma}}}}}{{\ensuremath{\displaystyle{{\ensuremath{\mathrm{d}{}\, }}\Omega}}}}}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8012}% $ \Omega$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8014}% $\displaystyle {I_0}\Delta t \Delta\Omega{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\mathrm{d}{}\, }}\bf {\sigma}}}}}{{\ensuremath{\displaystyle{{\ensuremath{\mathrm{d}{}\, }}\Omega}}}}}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8016}% $ \Delta t$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8018}% $ \Delta\Omega$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8020}% $ I_0$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8024}% $ \Delta\Omega\propto \Delta {\ensuremath{{2\theta}}}$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} \stepcounter{subsection} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8035}% $ P$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8037}% $ k$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8039}% $ k=1,\ldots,P$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8041}% $ N_k$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8043}% $ 2\theta\equiv{\ensuremath{{2\theta}}}$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8045}% $\displaystyle b_{k,j}={\ensuremath{\left[{{\ensuremath{{2\theta}}}_{k,j}^{-},{\ensuremath{{2\theta}}}_{k,j}^{+}}\right]}},\qquad j=1,\ldots,N_k $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8047}% $\displaystyle \hat{b}_{k,j}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{{2\theta}}}_{k,j}^{+}+{\ensuremath{{2\theta}}}_{k,j}^{-}}}}}{{\ensuremath{\displaystyle{2}}}}}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8049}% $\displaystyle {\ensuremath{\left|{b_{k,j}}\right|}}={\ensuremath{{2\theta}}}_{k,j}^{+}-{\ensuremath{{2\theta}}}_{k,j}^{-} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8051}% $ C_{k,j}$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8053}% $ e_{k,j}$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8055}% $ m_{k,j}$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8057}% $ b_{k,j}$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8059}% $\displaystyle I_{k,j}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{e_{k,j}}}}}{{\ensuremath{\displaystyle{m_{k,j}}}}}}}}{\ensuremath{\left({C_{k,j}+\min{\ensuremath{\left({1,C_{k,j}}\right)}}}\right)}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8061}% $\displaystyle \sigma_{I_{k,j}}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{e_{k,j}}}}}{{\ensuremath{\displaystyle{m_{k,j}}}}}}}}\sqrt{{\ensuremath{\left({C_{k,j}+1}\right)}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8063}% $\displaystyle r_{k,j}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{I_{k,j}}}}}{{\ensuremath{\displaystyle{{\ensuremath{\left|{b_{k,j}}\right|}}}}}}}}}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{e_{k,j}}}}}{{\ensuremath{\displaystyle{m_{k,j}{\ensuremath{\left|{b_{k,j}}\right|}}}}}}}}}{\ensuremath{\left({C_{k,j}+\min{\ensuremath{\left({1,C_{k,j}}\right)}}}\right)}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8065}% $\displaystyle \sigma_{r_{k,j}}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\sigma_{I_{k,j}}}}}}{{\ensuremath{\displaystyle{{\ensuremath{\left|{b_{k,j}}\right|}}}}}}}}}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{e_{k,j}}}}}{{\ensuremath{\displaystyle{{\ensuremath{\left|{b_{k,j}}\right|}}m_{k,j}}}}}}}}\sqrt{{\ensuremath{\left({C_{k,j}+1}\right)}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8067}% $ M$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8069}% $\displaystyle B_\ell=[{\ensuremath{{2\theta}}}_0+(\ell-1)B, {\ensuremath{{2\theta}}}_0+\ell B],\qquad \ell=1,\ldots,M $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8071}% $\displaystyle {\ensuremath{\left|{B_\ell}\right|}}=B$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8073}% $\displaystyle \hat{B}_\ell={\ensuremath{{2\theta}}}_0+(\ell-1/2)B,$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8075}% $ {\ensuremath{{2\theta}}}_0$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8077}% $ {\ensuremath{{2\theta}}}_{max}={\ensuremath{{2\theta}}}_0+MB$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8079}% $ \ell$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8081}% $\displaystyle b_{k,j}$% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8082}% $\displaystyle \qquad {\ensuremath{\left|{ b_{k,j}\cap B_\ell }\right|}} > 0. $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8085}% $\displaystyle \qquad \hat{b}_{k,j}\in B_\ell . $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8089}% $ B_\ell$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8091}% $ N_E$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8093}% $ O_n$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8095}% $ O$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8097}% $ \sigma_{O_n}$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8099}% $ \nu_n$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8101}% $\displaystyle \langle O\rangle ={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{ \mathop{\sum}_{n=1}^{N_E}\nu_n O_n\sigma_{O_n}^{-2} }}}}{{\ensuremath{\displaystyle{ \mathop{\sum}_{n=1}^{N_E}\nu_n \sigma_{O_n}^{-2} }}}}}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8103}% $\displaystyle {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\left|{ b_{k,j}\cap B_\ell }\right|}}}}}}{{\ensuremath{\displaystyle{B}}}}}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8105}% $ k,j$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8109}% $\displaystyle X_\ell=\mathop{\sum_{k,j}}_{ {\ensuremath{\left|{ b_{k,j}\cap B_\ell }\right|}} > 0} {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\left|{ b_{k,j}\cap B_\ell }\right|}}}}}}{{\ensuremath{\displaystyle{B}}}}}}}\ r_{k,j}\ {\ensuremath{\left({\sigma_{r_{k,j}}}\right)}}^{-2} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8111}% $\displaystyle Y_\ell=\mathop{\sum_{k,j}}_{ {\ensuremath{\left|{ b_{k,j}\cap B_\ell }\right|}} > 0} {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\left|{ b_{k,j}\cap B_\ell }\right|}}}}}}{{\ensuremath{\displaystyle{B}}}}}}}\ {\ensuremath{\left({\sigma_{r_{k,j}}}\right)}}^{-2} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8113}% $\displaystyle R_\ell={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{X_\ell}}}}{{\ensuremath{\displaystyle{Y_\ell}}}}}}}; $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8115}% $\displaystyle \sigma_{R_\ell}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{\sqrt{Y_\ell}}}}}}}}. $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8117}% $ R_\ell$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8119}% $ \sigma_{R_\ell}$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8121}% $ B$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8123}% $ K$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8127}% $\displaystyle \mathop{\sum}_{\ell=1}^M{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{KR_\ell}}}}{{\ensuremath{\displaystyle{K^2\sigma_{R_\ell}^2}}}}}}}= {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{K}}}}}}} \mathop{\sum}_{\ell=1}^M{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{R_\ell}}}}{{\ensuremath{\displaystyle{\sigma_{R_\ell}^2}}}}}}}=M $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8131}% $\displaystyle K={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{ 1 }}}}{{\ensuremath{\displaystyle{ M }}}}}}}\mathop{\sum}_{\ell=1}^M{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{R_\ell}}}}{{\ensuremath{\displaystyle{\sigma_{R_\ell}^2}}}}}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8133}% $\displaystyle \hat{B}_\ell, \quad KR_\ell, \quad K\sigma_{R_\ell} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} \stepcounter{subsubsection} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8144}% $ X_\ell=0$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8146}% $ Y_\ell=0$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8148}% $ b$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8152}% $\displaystyle X_\ell={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\left|{ b\cap B_\ell }\right|}}}}}}{{\ensuremath{\displaystyle{B}}}}}}}\ {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{e(C+1)}}}}{{\ensuremath{\displaystyle{m|b|}}}}}}}\ {\ensuremath{\left({ {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{|b|m}}}}{{\ensuremath{\displaystyle{e\sqrt{C+1}}}}}}}} }\right)}}^{2} ={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\left|{ b\cap B_\ell }\right|}}}}}}{{\ensuremath{\displaystyle{B}}}}}}}{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{|b|m}}}}{{\ensuremath{\displaystyle{e}}}}}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8154}% $\displaystyle Y_\ell={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\left|{ b\cap B_\ell }\right|}}}}}}{{\ensuremath{\displaystyle{B}}}}}}}\ {\ensuremath{\left({ {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{|b|m}}}}{{\ensuremath{\displaystyle{e\sqrt{C+1}}}}}}}} }\right)}}^{2} ={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\left|{ b\cap B_\ell }\right|}}}}}}{{\ensuremath{\displaystyle{B}}}}}}}{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{|b|^2m^2}}}}{{\ensuremath{\displaystyle{e^2(C+1)}}}}}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8156}% $\displaystyle R_\ell={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{X_\ell}}}}{{\ensuremath{\displaystyle{Y_\ell}}}}}}}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{e(C+1)}}}}{{\ensuremath{\displaystyle{m|b|}}}}}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8160}% $\displaystyle \sigma_{R_\ell}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{\sqrt{Y_\ell}}}}}}}}= \sqrt{{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{B}}}}{{\ensuremath{\displaystyle{{\ensuremath{\left|{ b\cap B_\ell }\right|}}}}}}}}}} {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{e\sqrt{(C+1)}}}}}{{\ensuremath{\displaystyle{|b|m}}}}}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8164}% $\displaystyle \sqrt{{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{B}}}}{{\ensuremath{\displaystyle{{\ensuremath{\left|{ b\cap B_\ell }\right|}}}}}}}}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} \stepcounter{subsection} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8167}% $ \hat{b}_{j,k}$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8169}% $ \hat{B}_\ell$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} \stepcounter{subsection} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8172}% $ C_0$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8176}% $ \sqrt{C_0}$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8178}% $ n$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8180}% $\displaystyle P(n)={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{C_0^n{\ensuremath{\mathrm{e}}}^{-C_0} }}}}{{\ensuremath{\displaystyle{ n!}}}}}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8182}% $\displaystyle \mathop{\sum}_{n=0}^{+\infty} P(n)=1\ ; $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8184}% $\displaystyle \langle n\rangle=\mathop{\sum}_{n=0}^{+\infty} nP(n)=C_0\ ; $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8186}% $\displaystyle \langle n^2\rangle=\mathop{\sum}_{n=0}^{+\infty} n^2 P(n)=C_0^2+C_0\ ; $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8188}% $\displaystyle \sigma_{C_0}=\sqrt{\langle n^2\rangle-\langle n\rangle^2}=\sqrt{C_0} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8192}% $\displaystyle \chi^2 = \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}} {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\left({F_j-O_j}\right)}}^2 }}}}{{\ensuremath{\displaystyle{ \sigma_j^2 }}}}}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8194}% $ O_j$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8196}% $ F_j$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8198}% $ \sigma_j$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8200}% $\displaystyle \chi_{(0)}^2 = \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}} {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\left({F_j-C_j}\right)}}^2 }}}}{{\ensuremath{\displaystyle{ C_j }}}}}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8202}% $\displaystyle \chi_{(1)}^2 = \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}} {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\left({F_j-{\ensuremath{\left({C_j+\min{\ensuremath{\left({1,C_j}\right)}}}\right)}}}\right)}}^2 }}}}{{\ensuremath{\displaystyle{ C_j+1 }}}}}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} \stepcounter{subsection} \stepcounter{subsubsection} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8208}% $ N_{\mathrm{obs}}$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8210}% $ C_j,\quad j=1\ldots N_{\mathrm{obs}}$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8212}% $ x$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8216}% $\displaystyle x=\langle x\rangle={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{ N_{\mathrm{obs}}}}}}}}} \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}C_j\ . $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8218}% $\displaystyle \sigma_x=\sqrt{\langle x^2\rangle-\langle x\rangle^2}=\sqrt{ {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{ N_{\mathrm{obs}}}}}}}}} \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}C_j^2-{\ensuremath{\left({ {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{ N_{\mathrm{obs}}}}}}}}} \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}C_j }\right)}} } $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8220}% $\displaystyle \sigma_x={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{ N_{\mathrm{obs}}}}}}}}}\sqrt{ \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}C_j } =\sqrt{{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\langle x\rangle}}}}{{\ensuremath{\displaystyle{N_{\mathrm{obs}}}}}}}}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} \stepcounter{subsubsection} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8223}% $ C_j=0$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8225}% $ N_{\mathrm{obs}}^*$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8227}% $\displaystyle x=\langle x\rangle^*={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{ N_{\mathrm{obs}}^*}}}}}}} \mathop{\sum}_ {\stackrel{1\leqslant j\leqslant N_{\mathrm{obs}}}{{C_j>0}}} C_j={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{ N_{\mathrm{obs}}^*}}}}}}} \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}C_j = {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{N_{\mathrm{obs}}}}}}{{\ensuremath{\displaystyle{N_{\mathrm{obs}}^*}}}}}}}\langle x\rangle $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8229}% $\displaystyle \sigma_{x^*}= {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{N_{\mathrm{obs}}}}}}{{\ensuremath{\displaystyle{N_{\mathrm{obs}}^*}}}}}}}\sigma_x = \sqrt{{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{N_{\mathrm{obs}}}}}}{{\ensuremath{\displaystyle{N_{\mathrm{obs}}^*}}}}}}}} \sqrt{{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\langle x\rangle}}}}{{\ensuremath{\displaystyle{N_{\mathrm{obs}}^*}}}}}}}}=\sqrt{{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\langle x\rangle^*}}}}{{\ensuremath{\displaystyle{N_{\mathrm{obs}}^*}}}}}}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8231}% $ C_j$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} \stepcounter{subsubsection} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8238}% $\displaystyle \chi^2 = \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}} {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\left({x-O_j}\right)}}^2 }}}}{{\ensuremath{\displaystyle{ \sigma_j^2 }}}}}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8240}% $\displaystyle x= \langle x \rangle_{\!\mathrm{w}}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{ \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}} {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{O_j }}}}{{\ensuremath{\displaystyle{ \sigma_j^2 }}}}}}} }}}}{{\ensuremath{\displaystyle{ \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}} {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1 }}}}{{\ensuremath{\displaystyle{ \sigma_j^2 }}}}}}} }}}}}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8244}% $\displaystyle \sigma_{\langle x \rangle_{\!\mathrm{w}}} = {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{ 1 }}}}{{\ensuremath{\displaystyle{\sqrt{ \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}} {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1 }}}}{{\ensuremath{\displaystyle{ \sigma_j^2 }}}}}}} }}}}}}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8246}% $\displaystyle \mathsf{GoF}= \sqrt{ {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{ \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}} {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{O_j^2 }}}}{{\ensuremath{\displaystyle{ \sigma_j^2 }}}}}}} -{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{ {\ensuremath{\left[{ \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}} {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{O_j }}}}{{\ensuremath{\displaystyle{ \sigma_j^2 }}}}}}} }\right]}}^2 }}}}{{\ensuremath{\displaystyle{ \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}} {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1 }}}}{{\ensuremath{\displaystyle{ \sigma_j^2 }}}}}}} }}}}}}} }}}}{{\ensuremath{\displaystyle{ N_{\mathrm{obs}}-1 }}}}}}} } $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8248}% $\displaystyle {\sigma}_{\langle x \rangle_{\!\mathrm{w}}}^{\mathrm{corrected}} = \mathsf{GoF}\ \sigma_{\langle x \rangle_{\!\mathrm{w}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} \stepcounter{subsubsection} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8251}% $ O_j=C_j$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8253}% $ \sigma_j^2=C_j$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8255}% $\displaystyle \langle x \rangle_{\!\mathrm{w(1)}}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{ {N_{\mathrm{obs}}} }}}}{{\ensuremath{\displaystyle{ \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}} {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1 }}}}{{\ensuremath{\displaystyle{ C_j }}}}}}} }}}}}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8263}% $\displaystyle \sigma_{\langle x \rangle_{\!\mathrm{w(1)}}} = {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{ 1 }}}}{{\ensuremath{\displaystyle{\sqrt{ \mathop{\sum}_{\stackrel{1\leqslant j\leqslant N_{\mathrm{obs}}}{{C_j>0}}} {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1 }}}}{{\ensuremath{\displaystyle{ C_j }}}}}}} }}}}}}}}=\sqrt{{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\langle x \rangle_{\!\mathrm{w(1)}}}}}}{{\ensuremath{\displaystyle{ N_{\mathrm{obs}}^* }}}}}}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8265}% $\displaystyle \mathsf{GoF}_{(1)}= \sqrt{ {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{ \mathop{\sum}_{\stackrel{1\leqslant j\leqslant N_{\mathrm{obs}}}{{C_j>0}}} \!\!\!\!C_j -{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{ {\ensuremath{\left[{ N_{\mathrm{obs}}^* }\right]}}^2 }}}}{{\ensuremath{\displaystyle{ \mathop{\sum}_{\stackrel{1\leqslant j\leqslant N_{\mathrm{obs}}}{{C_j>0}}} {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1 }}}}{{\ensuremath{\displaystyle{ C_j }}}}}}} }}}}}}} }}}}{{\ensuremath{\displaystyle{ N_{\mathrm{obs}}^*-1 }}}}}}} } =\sqrt{ {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{N_{\mathrm{obs}}^*}}}}{{\ensuremath{\displaystyle{N_{\mathrm{obs}}^*-1}}}}}}} {\ensuremath{\left({ \langle x\rangle^*-\langle x \rangle_{\!\mathrm{w(1)}} }\right)}} } $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8267}% $ \langle x\rangle^*$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8269}% $\displaystyle {\sigma}_{\langle x \rangle_{\!\mathrm{w(1)}}}^{\mathrm{corrected}} = \mathsf{GoF}_{(1)}\ \sigma_{\langle x \rangle_{\!\mathrm{w(1)}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} \stepcounter{subsubsection} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8272}% $ O_j=C_j+\min{\ensuremath{\left({1,C_j}\right)}}$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8274}% $ \sigma_j^2=C_j+1$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8276}% $\displaystyle \langle x \rangle_{\!\mathrm{w(2)}}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{ {N_{\mathrm{obs}}^*} }}}}{{\ensuremath{\displaystyle{ \mathop{\sum}_{\stackrel{1\leqslant j\leqslant N_{\mathrm{obs}}}{{C_j>0}}} {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1 }}}}{{\ensuremath{\displaystyle{ C_j+1 }}}}}}} }}}}}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8278}% $\displaystyle \sigma_{\langle x \rangle_{\!\mathrm{w(2)}}} = {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{ 1 }}}}{{\ensuremath{\displaystyle{\sqrt{ \mathop{\sum}_{\stackrel{1\leqslant j\leqslant N_{\mathrm{obs}}}{{C_j>0}}} {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1 }}}}{{\ensuremath{\displaystyle{ C_j+1 }}}}}}} }}}}}}}}=\sqrt{{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\langle x \rangle_{\!\mathrm{w(2)}}}}}}{{\ensuremath{\displaystyle{ N_{\mathrm{obs}}^* }}}}}}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8280}% $\displaystyle \mathsf{GoF}_{(2)}= \sqrt{ {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{ \mathop{\sum}_{\stackrel{1\leqslant j\leqslant N_{\mathrm{obs}}}{{C_j>0}}} \!\!\!\!C_j+N_{\mathrm{obs}}^* -{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{ {\ensuremath{\left[{ N_{\mathrm{obs}}^* }\right]}}^2 }}}}{{\ensuremath{\displaystyle{ \mathop{\sum}_{\stackrel{1\leqslant j\leqslant N_{\mathrm{obs}}}{{C_j>0}}} {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1 }}}}{{\ensuremath{\displaystyle{ C_j+1 }}}}}}} }}}}}}} }}}}{{\ensuremath{\displaystyle{ N_{\mathrm{obs}}^*-1 }}}}}}} } =\sqrt{ {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{N_{\mathrm{obs}}^*}}}}{{\ensuremath{\displaystyle{N_{\mathrm{obs}}^*-1}}}}}}} {\ensuremath{\left({ \langle x\rangle^*-\langle x \rangle_{\!\mathrm{w(2)}}+1 }\right)}} } $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8284}% $\displaystyle {\sigma}_{\langle x \rangle_{\!\mathrm{w(2)}}}^{\mathrm{corrected}} = \mathsf{GoF}_{(2)}\ \sigma_{\langle x \rangle_{\!\mathrm{w(2)}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} \stepcounter{subsubsection} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8291}% $\displaystyle \epsilon_x = {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\sigma_x}}}}{{\ensuremath{\displaystyle{x}}}}}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8295}% $ O(\epsilon_x^2)$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8299}% $ \propto\epsilon_x$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8303}% $ O(\epsilon_x^3)$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} \stepcounter{subsubsection} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8306}% $ N_{\mathrm{obs}}=2$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8308}% $ C_1$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8310}% $ C_2$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8312}% $\displaystyle \langle x \rangle={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{C_1+C_2}}}}{{\ensuremath{\displaystyle{2}}}}}}}; \qquad \sigma_x={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\sqrt{C_1+C_2}}}}}{{\ensuremath{\displaystyle{2}}}}}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8314}% $\displaystyle \langle x \rangle_{\mathrm{w(2)}}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{2(C_1+1)(C_2+1)}}}}{{\ensuremath{\displaystyle{C_1+C_2+2}}}}}}}; \qquad \sigma_{\langle x \rangle_{\mathrm{w(2)}}}=\sqrt{{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{(C_1+1)(C_2+1)}}}}{{\ensuremath{\displaystyle{C_1+C_2+2}}}}}}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8316}% $ C_1,C_2$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8318}% $ \lambda$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8320}% $\displaystyle E{\ensuremath{\left({\langle x \rangle}\right)}} = \mathop{\sum}_{m,n=0}^{+\infty} {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{n+m}}}}{{\ensuremath{\displaystyle{2}}}}}}}P(n)P(m)=\mathop{\sum}_{m=0}^{+\infty} {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{m}}}}{{\ensuremath{\displaystyle{2}}}}}}}{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\lambda^m{\ensuremath{\mathrm{e}}}^{-\lambda}}}}}{{\ensuremath{\displaystyle{m!}}}}}}} +\mathop{\sum}_{n=0}^{+\infty} {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{n}}}}{{\ensuremath{\displaystyle{2}}}}}}}{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\lambda^n{\ensuremath{\mathrm{e}}}^{-\lambda}}}}}{{\ensuremath{\displaystyle{n!}}}}}}} =\lambda $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8322}% $\displaystyle E{\ensuremath{\left({\sigma_x^2}\right)}} = \mathop{\sum}_{m,n=0}^{+\infty} {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{n+m}}}}{{\ensuremath{\displaystyle{4}}}}}}}P(n)P(m)={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{ \lambda}}}}{{\ensuremath{\displaystyle{2}}}}}}};\qquad E{\ensuremath{\left({\sigma_x}\right)}} =\sqrt{{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\lambda}}}}{{\ensuremath{\displaystyle{2}}}}}}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8324}% $\displaystyle \langle x \rangle_{\mathrm{w(2)}}=\langle x \rangle + 1 -{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{(C_1-C_2)^2}}}}{{\ensuremath{\displaystyle{4(\langle x \rangle+1)}}}}}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8326}% $\displaystyle E{\ensuremath{\left({{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{(C_1-C_2)^2}}}}{{\ensuremath{\displaystyle{4(\langle x \rangle+1)}}}}}}}}\right)}} = \mathop{\sum}_{m,n=0}^{+\infty} {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{(n-m)^2}}}}{{\ensuremath{\displaystyle{2(n+m+2) }}}}}}}P(n)P(m)={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\mathrm{e}}}^{-2\lambda}}}}}{{\ensuremath{\displaystyle{2}}}}}}} \mathop{\sum}_{m,n=0}^{+\infty} {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{(n-m)^2}}}}{{\ensuremath{\displaystyle{(n+m+2) }}}}}}}{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\lambda^{n+m}}}}}{{\ensuremath{\displaystyle{n!m!}}}}}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8328}% $ s=n+m$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8330}% $ s=0\ldots +\infty$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8332}% $ n-m=s-2k$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8334}% $ k=0\ldots s$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8336}% $\displaystyle E{\ensuremath{\left({{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{(C_1-C_2)^2}}}}{{\ensuremath{\displaystyle{4(\langle x \rangle+1)}}}}}}}}\right)}} = {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\mathrm{e}}}^{-2\lambda}}}}}{{\ensuremath{\displaystyle{2}}}}}}} \mathop{\sum}_{s=0}^{+\infty} \mathop{\sum}_{k=0}^{s} {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{(s-2k)^2(s+1)}}}}{{\ensuremath{\displaystyle{(s+2)! }}}}}}}{\lambda^{s}} \binom{s}{k}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{2}}}}}}}-{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{2\lambda}}}}}}}+{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1-{\ensuremath{\mathrm{e}}}^{-2\lambda}}}}}{{\ensuremath{\displaystyle{4\lambda^2}}}}}}} %{n!m!} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8338}% $\displaystyle {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{E{\ensuremath{\left({\langle x \rangle_{\mathrm{w(2)}}-\langle x \rangle}\right)}}}}}}{{\ensuremath{\displaystyle{E{\ensuremath{\left({\langle x \rangle}\right)}}}}}}}}}= {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{2\lambda}}}}}}}+{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{2\lambda^2}}}}}}}-{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1-{\ensuremath{\mathrm{e}}}^{-2\lambda}}}}}{{\ensuremath{\displaystyle{4\lambda^3}}}}}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8340}% $ \langle x \rangle$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8342}% $\displaystyle \epsilon = {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\sigma_x}}}}{{\ensuremath{\displaystyle{\langle x \rangle}}}}}}} = {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\lambda^{1/2}}}}}{{\ensuremath{\displaystyle{\sqrt{2} \lambda}}}}}}}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{\sqrt{2\lambda}}}}}}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8344}% $\displaystyle {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{E{\ensuremath{\left({\langle x \rangle_{\mathrm{w(2)}}-\langle x \rangle}\right)}}}}}}{{\ensuremath{\displaystyle{E{\ensuremath{\left({\langle x \rangle}\right)}}}}}}}}}= O(\epsilon^2) $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} \stepcounter{subsubsection} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8349}% $ \lambda=1,10,100,\ldots,1000000$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8353}% $ N=10^8$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8359}% $ \xi_\lambda=\sqrt{\lambda/N}$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8361}% $ \epsilon_\lambda=\sqrt{\lambda/N}/\lambda=1/\sqrt{N\lambda}$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8363}% $ E_\lambda$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8367}% $ e_\lambda=E_\lambda/\lambda$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8369}% $ e_\lambda/\epsilon_\lambda$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8371}% $ \lambda =$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8373}% $ \xi_\lambda = $% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8375}% $ \epsilon_\lambda$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8377}% $ {\langle x \rangle_{\!\mathrm{w(1)}}}$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8379}% $ {\langle x \rangle_{\!\mathrm{w(2)}}}$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8387}% $ e_\lambda$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8731}% $ {\langle x \rangle_{\!\mathrm{w(1)}}}\ :$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8735}% $ {\langle x \rangle^*}\ $% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8737}% $ \lambda<100$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8739}% $ {\langle x \rangle}\ $% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} \stepcounter{subsection} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8764}% $\displaystyle X_0=\eta_0 C_0 $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8766}% $ \eta_0$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8768}% $ X$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8770}% $\displaystyle P'(X)=P(X/\eta_0)=P(n)\qquad\Biggl|\Biggr.\qquad \frac{X}{\eta_0}\equiv n\in\mathbb{Z} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8774}% $\displaystyle \langle X\rangle=\mathop{\sum}_{n=0}^{+\infty} \eta_0 nP(n)=\eta_0 C_0=X_0\ ; $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8776}% $\displaystyle \langle X^2\rangle=\mathop{\sum}_{n=0}^{+\infty} \eta_0^2 n^2 P(n)=\eta_0^2(C_0^2+C_0)=X_0^2+\eta_0 X_0\ ; $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8778}% $\displaystyle \sigma_X=\sqrt{\langle X^2\rangle-\langle X\rangle^2}=\sqrt{\eta_0 X_0}=\eta_0\sqrt{C_0}=\sqrt{\eta_0}\sqrt{X_0} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8780}% $ \sigma_X=\sqrt{\langle X\rangle}=\sqrt{X_0}$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8782}% $\displaystyle \sigma_X=\sqrt{\eta_0}\sqrt{X_0}=\eta_0\sqrt{C_0}=\eta_0\sigma_{C_0} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_inline8786}% $ \sigma_{\eta_0}$% \lthtmlinlinemathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8788}% $\displaystyle \widehat{P}(\eta)={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{ {\ensuremath{\mathrm{e}}}^{ -\frac{1}{2} {\ensuremath{\left({ \frac{\eta-\eta_0}{\sigma_{\eta_0}} }\right)}}^2 } }}}}{{\ensuremath{\displaystyle{ \sigma_{\eta_0}\sqrt{2\pi} }}}}}}} $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8790}% $\displaystyle \int_{-\infty}^{+\infty}{\ensuremath{\mathrm{d}{\eta}\, }}\mathop{\sum}_{n=0}^{+\infty} P(n)\widehat{P}(\eta)=1\ ; $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8792}% $\displaystyle \langle X\rangle=\int_{-\infty}^{+\infty}{\ensuremath{\mathrm{d}{\eta}\, }}\mathop{\sum}_{n=0}^{+\infty} \widehat{P}(\eta)\eta nP(n)= \mathop{\sum}_{n=0}^{+\infty} nP(n) \int_{-\infty}^{+\infty}{\ensuremath{\mathrm{d}{\eta}\, }} \widehat{P}(\eta)\eta = \eta_0 C_0=X_0\ ; $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8794}% $\displaystyle \langle X^2\rangle=\int_{-\infty}^{+\infty}{\ensuremath{\mathrm{d}{\eta}\, }}\mathop{\sum}_{n=0}^{+\infty} \widehat{P}(\eta)\eta^2 n^2 P(n)= \int_{-\infty}^{+\infty}{\ensuremath{\mathrm{d}{\eta}\, }}\widehat{P}(\eta)\eta^2 \mathop{\sum}_{n=0}^{+\infty} n^2 P(n) = (\eta_0^2+\sigma_{\eta_0}^2)(C_0^2+C_0)\ ; $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} {\newpage\clearpage \lthtmlinlinemathA{tex2html_wrap_indisplay8796}% $\displaystyle {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\sigma_X}}}}{{\ensuremath{\displaystyle{\langle X\rangle}}}}}}}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\sqrt{\langle X^2\rangle-\langle X\rangle^2}}}}}{{\ensuremath{\displaystyle{\langle X\rangle}}}}}}} ={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\sqrt{ \eta_0^2 C_0+\sigma_{\eta_0}^2C_0^2+\sigma_{\eta_0}^2 C_0 }}}}}{{\ensuremath{\displaystyle{\eta_0C_0}}}}}}}= \sqrt{ {\ensuremath{\left({{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{ \sigma_{C_0}}}}}{{\ensuremath{\displaystyle{C_0}}}}}}}}\right)}}^2 +{\ensuremath{\left({{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\sigma_{\eta_0}}}}}{{\ensuremath{\displaystyle{\eta_0}}}}}}}}\right)}}^2+{\ensuremath{\left({{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\sigma_{\eta_0}}}}}{{\ensuremath{\displaystyle{\eta_0}}}}}}}{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\sigma_{C_0}}}}}{{\ensuremath{\displaystyle{C_0}}}}}}}}\right)}}^2 }\approx\sqrt{ {\ensuremath{\left({{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{ \sigma_{C_0}}}}}{{\ensuremath{\displaystyle{C_0}}}}}}}}\right)}}^2 +{\ensuremath{\left({{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\sigma_{\eta_0}}}}}{{\ensuremath{\displaystyle{\eta_0}}}}}}}}\right)}}^2 } $% \lthtmlindisplaymathZ \lthtmlcheckvsize\clearpage} \stepcounter{section} \end{document}