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630 lines
19 KiB
HTML
630 lines
19 KiB
HTML
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original version by: Nikos Drakos, CBLU, University of Leeds
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<TITLE>Basic binning</TITLE>
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<H2><A NAME="SECTION00622000000000000000"></A><A NAME="sec:11"></A>
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<BR>
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Basic binning
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</H2>
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<P>
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<DL COMPACT>
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<DT>1. </DT>
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<DD>
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We have several patterns, say <IMG
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WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img72.png"
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ALT="$ P$">
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. Each <IMG
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WIDTH="13" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img73.png"
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ALT="$ k$">
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-th pattern, for <!-- MATH
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$k=1,\ldots,P$
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-->
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<IMG
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WIDTH="90" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img74.png"
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ALT="$ k=1,\ldots,P$">
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, is
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constituted by <IMG
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WIDTH="25" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img75.png"
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ALT="$ N_k$">
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angular intervals in the diffraction angle <!-- MATH
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$2\theta\equiv{\ensuremath{{2\theta}}}$
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-->
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<IMG
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WIDTH="57" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img76.png"
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ALT="$ 2\theta\equiv{\ensuremath{{2\theta}}}$">
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:
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<P><!-- MATH
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\begin{displaymath}
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b_{k,j}={\ensuremath{\left[{{\ensuremath{{2\theta}}}_{k,j}^{-},{\ensuremath{{2\theta}}}_{k,j}^{+}}\right]}},\qquad j=1,\ldots,N_k
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="270" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
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SRC="img77.png"
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ALT="$\displaystyle b_{k,j}={\ensuremath{\left[{{\ensuremath{{2\theta}}}_{k,j}^{-},{\ensuremath{{2\theta}}}_{k,j}^{+}}\right]}},\qquad j=1,\ldots,N_k
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$">
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</DIV><P>
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</P>
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of center
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<P><!-- MATH
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\begin{displaymath}
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\hat{b}_{k,j}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{{2\theta}}}_{k,j}^{+}+{\ensuremath{{2\theta}}}_{k,j}^{-}}}}}{{\ensuremath{\displaystyle{2}}}}}}}
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="140" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
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SRC="img78.png"
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ALT="$\displaystyle \hat{b}_{k,j}={\ensuremath{\displaystyle{\frac{{\ensuremath{\disp...
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...{+}+{\ensuremath{{2\theta}}}_{k,j}^{-}}}}}{{\ensuremath{\displaystyle{2}}}}}}}
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$">
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</DIV><P>
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</P>
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and width
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<P><!-- MATH
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\begin{displaymath}
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{\ensuremath{\left|{b_{k,j}}\right|}}={\ensuremath{{2\theta}}}_{k,j}^{+}-{\ensuremath{{2\theta}}}_{k,j}^{-}
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="145" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
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SRC="img79.png"
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ALT="$\displaystyle {\ensuremath{\left\vert{b_{k,j}}\right\vert}}={\ensuremath{{2\theta}}}_{k,j}^{+}-{\ensuremath{{2\theta}}}_{k,j}^{-}
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$">
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</DIV><P>
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</P>
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To each interval is associated a counting <IMG
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WIDTH="33" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img80.png"
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ALT="$ C_{k,j}$">
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, an efficiency correction factor <IMG
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WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img81.png"
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ALT="$ e_{k,j}$">
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, a
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monitor <IMG
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WIDTH="36" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img82.png"
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ALT="$ m_{k,j}$">
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(ionization chamber times acquisition time). All 'bad' intervals have been already flagged down and discarded.
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Efficiency corrections and monitors are supposed to be normalized to a suitable value.
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Note that intervals <IMG
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WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img83.png"
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ALT="$ b_{k,j}$">
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might have multiple overlaps and might not cover an compact angular
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range.
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</DD>
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<DT>2. </DT>
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<DD>Following Mighell's statistics[6] and normal scaling procedures, we first
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transform those numbers into associated intensities, intensity rates and relevant s.d.:
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<P><!-- MATH
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\begin{displaymath}
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I_{k,j}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{e_{k,j}}}}}{{\ensuremath{\displaystyle{m_{k,j}}}}}}}}{\ensuremath{\left({C_{k,j}+\min{\ensuremath{\left({1,C_{k,j}}\right)}}}\right)}}
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="232" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
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SRC="img84.png"
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ALT="$\displaystyle I_{k,j}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaysty...
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...nsuremath{\left({C_{k,j}+\min{\ensuremath{\left({1,C_{k,j}}\right)}}}\right)}}
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$">
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</DIV><P>
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</P>
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<P><!-- MATH
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\begin{displaymath}
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\sigma_{I_{k,j}}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{e_{k,j}}}}}{{\ensuremath{\displaystyle{m_{k,j}}}}}}}}\sqrt{{\ensuremath{\left({C_{k,j}+1}\right)}}}
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="177" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
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SRC="img85.png"
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ALT="$\displaystyle \sigma_{I_{k,j}}={\ensuremath{\displaystyle{\frac{{\ensuremath{\d...
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...ath{\displaystyle{m_{k,j}}}}}}}}\sqrt{{\ensuremath{\left({C_{k,j}+1}\right)}}}
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$">
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</DIV><P>
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</P>
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<P><!-- MATH
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\begin{displaymath}
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r_{k,j}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{I_{k,j}}}}}{{\ensuremath{\displaystyle{{\ensuremath{\left|{b_{k,j}}\right|}}}}}}}}}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{e_{k,j}}}}}{{\ensuremath{\displaystyle{m_{k,j}{\ensuremath{\left|{b_{k,j}}\right|}}}}}}}}}{\ensuremath{\left({C_{k,j}+\min{\ensuremath{\left({1,C_{k,j}}\right)}}}\right)}}
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="323" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
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SRC="img86.png"
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ALT="$\displaystyle r_{k,j}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaysty...
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...nsuremath{\left({C_{k,j}+\min{\ensuremath{\left({1,C_{k,j}}\right)}}}\right)}}
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$">
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</DIV><P>
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</P>
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<P><!-- MATH
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\begin{displaymath}
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\sigma_{r_{k,j}}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{\sigma_{I_{k,j}}}}}}{{\ensuremath{\displaystyle{{\ensuremath{\left|{b_{k,j}}\right|}}}}}}}}}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{e_{k,j}}}}}{{\ensuremath{\displaystyle{{\ensuremath{\left|{b_{k,j}}\right|}}m_{k,j}}}}}}}}\sqrt{{\ensuremath{\left({C_{k,j}+1}\right)}}}
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="269" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
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SRC="img87.png"
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ALT="$\displaystyle \sigma_{r_{k,j}}={\ensuremath{\displaystyle{\frac{{\ensuremath{\d...
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...k,j}}\right\vert}}m_{k,j}}}}}}}}\sqrt{{\ensuremath{\left({C_{k,j}+1}\right)}}}
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$">
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</DIV><P>
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</P>
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</DD>
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<DT>3. </DT>
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<DD>
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We set up the final binned grid,
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composed of <IMG
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WIDTH="22" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img88.png"
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ALT="$ M$">
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binning intervals
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<P><!-- MATH
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\begin{displaymath}
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B_\ell=[{\ensuremath{{2\theta}}}_0+(\ell-1)B, {\ensuremath{{2\theta}}}_0+\ell B],\qquad \ell=1,\ldots,M
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="351" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img89.png"
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ALT="$\displaystyle B_\ell=[{\ensuremath{{2\theta}}}_0+(\ell-1)B, {\ensuremath{{2\theta}}}_0+\ell B],\qquad \ell=1,\ldots,M
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$">
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</DIV><P>
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</P>
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all contiguous and each having the same width <P><!-- MATH
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\begin{displaymath}
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{\ensuremath{\left|{B_\ell}\right|}}=B
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="66" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img90.png"
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ALT="$\displaystyle {\ensuremath{\left\vert{B_\ell}\right\vert}}=B$">
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</DIV><P>
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</P>
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and each centered in
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<P><!-- MATH
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\begin{displaymath}
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\hat{B}_\ell={\ensuremath{{2\theta}}}_0+(\ell-1/2)B,
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="166" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
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SRC="img91.png"
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ALT="$\displaystyle \hat{B}_\ell={\ensuremath{{2\theta}}}_0+(\ell-1/2)B,$">
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</DIV><P>
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</P>
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covering completely the angular range between <!-- MATH
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${\ensuremath{{2\theta}}}_0$
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-->
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<IMG
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WIDTH="28" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img92.png"
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ALT="$ {\ensuremath{{2\theta}}}_0$">
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and <!-- MATH
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${\ensuremath{{2\theta}}}_{max}={\ensuremath{{2\theta}}}_0+MB$
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-->
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<IMG
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WIDTH="140" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img93.png"
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ALT="$ {\ensuremath{{2\theta}}}_{max}={\ensuremath{{2\theta}}}_0+MB$">
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.
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</DD>
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<DT>4. </DT>
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<DD>
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For bin <IMG
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WIDTH="11" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img94.png"
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ALT="$ \ell$">
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, we consider only and all the experimental intervals
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<P><!-- MATH
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\begin{displaymath}
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b_{k,j}\qquad\text{such\ that}\qquad {\ensuremath{\left|{ b_{k,j}\cap B_\ell }\right|}} > 0.
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img95.png"
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ALT="$\displaystyle b_{k,j}$"> such that<IMG
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WIDTH="139" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img96.png"
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ALT="$\displaystyle \qquad {\ensuremath{\left\vert{ b_{k,j}\cap B_\ell }\right\vert}} > 0.
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$">
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</DIV><P>
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</P>
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More restrictively, one may require to consider only and all the experimental intervals
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<P><!-- MATH
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\begin{displaymath}
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b_{k,j}\qquad\text{such\ that}\qquad \hat{b}_{k,j}\in B_\ell .
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="29" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img95.png"
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ALT="$\displaystyle b_{k,j}$"> such that<IMG
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WIDTH="103" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
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SRC="img97.png"
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ALT="$\displaystyle \qquad \hat{b}_{k,j}\in B_\ell .
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$">
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</DIV><P>
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</P>
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</DD>
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<DT>5. </DT>
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<DD>
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In order to estimate the rate in each <IMG
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WIDTH="11" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img94.png"
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ALT="$ \ell$">
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-th bin,
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we use all above selected rate estimates concerning bin <IMG
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WIDTH="23" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img98.png"
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ALT="$ B_\ell$">
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and we get
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a better one with the weighted average method.
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<BR>
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In the weighted average method, we suppose to have a number <IMG
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WIDTH="28" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img99.png"
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ALT="$ N_E$">
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of estimates <IMG
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WIDTH="25" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img100.png"
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ALT="$ O_n$">
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of the same observable <IMG
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WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
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SRC="img101.png"
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ALT="$ O$">
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,
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each one with a known s.d. <!-- MATH
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$\sigma_{O_n}$
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-->
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<IMG
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WIDTH="32" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img102.png"
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ALT="$ \sigma_{O_n}$">
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and each (optionally) repeated with a frequency
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<IMG
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WIDTH="21" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img103.png"
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ALT="$ \nu_n$">
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.
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Then
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<P><!-- MATH
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\begin{displaymath}
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\langle O\rangle ={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{
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\mathop{\sum}_{n=1}^{N_E}\nu_n
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O_n\sigma_{O_n}^{-2}
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}}}}{{\ensuremath{\displaystyle{
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\mathop{\sum}_{n=1}^{N_E}\nu_n
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\sigma_{O_n}^{-2}
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}}}}}}}
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="146" HEIGHT="121" ALIGN="MIDDLE" BORDER="0"
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SRC="img104.png"
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ALT="$\displaystyle \langle O\rangle ={\ensuremath{\displaystyle{\frac{{\ensuremath{\...
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...remath{\displaystyle{
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\mathop{\sum}_{n=1}^{N_E}\nu_n
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\sigma_{O_n}^{-2}
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}}}}}}}
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$">
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</DIV><P>
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</P>
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Clearly the place of the frequencies in our case can be taken by coefficients
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<P><!-- MATH
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\begin{displaymath}
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{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\left|{ b_{k,j}\cap B_\ell }\right|}}}}}}{{\ensuremath{\displaystyle{B}}}}}}}
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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<IMG
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WIDTH="77" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
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SRC="img105.png"
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ALT="$\displaystyle {\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ens...
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...ert{ b_{k,j}\cap B_\ell }\right\vert}}}}}}{{\ensuremath{\displaystyle{B}}}}}}}
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$">
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</DIV><P>
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</P>
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that weigh the <IMG
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WIDTH="28" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img106.png"
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ALT="$ k,j$">
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-th estimate by its relative extension within bin <IMG
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WIDTH="23" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img98.png"
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ALT="$ B_\ell$">
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.
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</DD>
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<DT>6. </DT>
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<DD>
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Now
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we can simply accumulate registers
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<P><!-- MATH
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\begin{displaymath}
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X_\ell=\mathop{\sum_{k,j}}_{ {\ensuremath{\left|{ b_{k,j}\cap B_\ell }\right|}} > 0}
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{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\left|{ b_{k,j}\cap B_\ell }\right|}}}}}}{{\ensuremath{\displaystyle{B}}}}}}}\ r_{k,j}\ {\ensuremath{\left({\sigma_{r_{k,j}}}\right)}}^{-2}
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\end{displaymath}
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-->
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</P>
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<DIV ALIGN="CENTER">
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|
<IMG
|
|
WIDTH="288" HEIGHT="87" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img107.png"
|
|
ALT="$\displaystyle X_\ell=\mathop{\sum_{k,j}}_{ {\ensuremath{\left\vert{ b_{k,j}\cap...
|
|
...aystyle{B}}}}}}}\ r_{k,j}\ {\ensuremath{\left({\sigma_{r_{k,j}}}\right)}}^{-2}
|
|
$">
|
|
</DIV><P>
|
|
</P>
|
|
and
|
|
<P><!-- MATH
|
|
\begin{displaymath}
|
|
Y_\ell=\mathop{\sum_{k,j}}_{ {\ensuremath{\left|{ b_{k,j}\cap B_\ell }\right|}} > 0}
|
|
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{{\ensuremath{\left|{ b_{k,j}\cap B_\ell }\right|}}}}}}{{\ensuremath{\displaystyle{B}}}}}}}\ {\ensuremath{\left({\sigma_{r_{k,j}}}\right)}}^{-2}
|
|
\end{displaymath}
|
|
-->
|
|
</P>
|
|
<DIV ALIGN="CENTER">
|
|
<IMG
|
|
WIDTH="254" HEIGHT="87" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img108.png"
|
|
ALT="$\displaystyle Y_\ell=\mathop{\sum_{k,j}}_{ {\ensuremath{\left\vert{ b_{k,j}\cap...
|
|
...th{\displaystyle{B}}}}}}}\ {\ensuremath{\left({\sigma_{r_{k,j}}}\right)}}^{-2}
|
|
$">
|
|
</DIV><P>
|
|
</P>
|
|
so that we can extract an intensity rate estimate (counts per unit diffraction angle and per unit time at constant incident intensity) as
|
|
<P><!-- MATH
|
|
\begin{displaymath}
|
|
R_\ell={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{X_\ell}}}}{{\ensuremath{\displaystyle{Y_\ell}}}}}}};
|
|
\end{displaymath}
|
|
-->
|
|
</P>
|
|
<DIV ALIGN="CENTER">
|
|
<IMG
|
|
WIDTH="72" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img109.png"
|
|
ALT="$\displaystyle R_\ell={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{X_\ell}}}}{{\ensuremath{\displaystyle{Y_\ell}}}}}}};
|
|
$">
|
|
</DIV><P>
|
|
</P>
|
|
<P><!-- MATH
|
|
\begin{displaymath}
|
|
\sigma_{R_\ell}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{\sqrt{Y_\ell}}}}}}}}.
|
|
\end{displaymath}
|
|
-->
|
|
</P>
|
|
<DIV ALIGN="CENTER">
|
|
<IMG
|
|
WIDTH="90" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img110.png"
|
|
ALT="$\displaystyle \sigma_{R_\ell}={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{\sqrt{Y_\ell}}}}}}}}.
|
|
$">
|
|
</DIV><P>
|
|
</P>
|
|
Now optionally we can transforms rates in intensities (multiplying
|
|
both <IMG
|
|
WIDTH="23" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img111.png"
|
|
ALT="$ R_\ell$">
|
|
and <!-- MATH
|
|
$\sigma_{R_\ell}$
|
|
-->
|
|
<IMG
|
|
WIDTH="30" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img112.png"
|
|
ALT="$ \sigma_{R_\ell}$">
|
|
by <IMG
|
|
WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img113.png"
|
|
ALT="$ B$">
|
|
).
|
|
We can use any other scaling factor <IMG
|
|
WIDTH="19" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img114.png"
|
|
ALT="$ K$">
|
|
as we wish instead of <IMG
|
|
WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img113.png"
|
|
ALT="$ B$">
|
|
.
|
|
The best cosmetic scaling is the one where
|
|
<P><!-- MATH
|
|
\begin{displaymath}
|
|
\mathop{\sum}_{\ell=1}^M{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{KR_\ell}}}}{{\ensuremath{\displaystyle{K^2\sigma_{R_\ell}^2}}}}}}}=
|
|
{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{1}}}}{{\ensuremath{\displaystyle{K}}}}}}}
|
|
\mathop{\sum}_{\ell=1}^M{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{R_\ell}}}}{{\ensuremath{\displaystyle{\sigma_{R_\ell}^2}}}}}}}=M
|
|
\end{displaymath}
|
|
-->
|
|
</P>
|
|
<DIV ALIGN="CENTER">
|
|
<IMG
|
|
WIDTH="216" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img115.png"
|
|
ALT="$\displaystyle \mathop{\sum}_{\ell=1}^M{\ensuremath{\displaystyle{\frac{{\ensure...
|
|
...\displaystyle{R_\ell}}}}{{\ensuremath{\displaystyle{\sigma_{R_\ell}^2}}}}}}}=M
|
|
$">
|
|
</DIV><P>
|
|
</P>
|
|
as if the intensities were simply counts.
|
|
Therefore <IMG
|
|
WIDTH="19" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img114.png"
|
|
ALT="$ K$">
|
|
is given by
|
|
<P><!-- MATH
|
|
\begin{displaymath}
|
|
K={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{
|
|
1
|
|
}}}}{{\ensuremath{\displaystyle{
|
|
M
|
|
}}}}}}}\mathop{\sum}_{\ell=1}^M{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{R_\ell}}}}{{\ensuremath{\displaystyle{\sigma_{R_\ell}^2}}}}}}}
|
|
\end{displaymath}
|
|
-->
|
|
</P>
|
|
<DIV ALIGN="CENTER">
|
|
<IMG
|
|
WIDTH="119" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img116.png"
|
|
ALT="$\displaystyle K={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{
|
|
1
|
|
...
|
|
...h{\displaystyle{R_\ell}}}}{{\ensuremath{\displaystyle{\sigma_{R_\ell}^2}}}}}}}
|
|
$">
|
|
</DIV><P>
|
|
</P>
|
|
|
|
<P>
|
|
In output then we give 3-column files
|
|
with columns
|
|
<P><!-- MATH
|
|
\begin{displaymath}
|
|
\hat{B}_\ell, \quad KR_\ell, \quad K\sigma_{R_\ell}
|
|
\end{displaymath}
|
|
-->
|
|
</P>
|
|
<DIV ALIGN="CENTER">
|
|
<IMG
|
|
WIDTH="141" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img117.png"
|
|
ALT="$\displaystyle \hat{B}_\ell, \quad KR_\ell, \quad K\sigma_{R_\ell}
|
|
$">
|
|
</DIV><P>
|
|
</P>
|
|
</DD>
|
|
</DL>
|
|
|
|
<P>
|
|
<BR><HR>
|
|
<!--Table of Child-Links-->
|
|
<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
|
|
|
|
<UL>
|
|
<LI><A NAME="tex2html808"
|
|
HREF="node51.html">Special nasty cases</A>
|
|
</UL>
|
|
<!--End of Table of Child-Links-->
|
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<HR>
|
|
<!--Navigation Panel-->
|
|
<A NAME="tex2html806"
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HREF="node51.html">
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<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
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SRC="file:/usr/share/latex2html/icons/next.png"></A>
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<A NAME="tex2html802"
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HREF="node46.html">
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<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
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SRC="file:/usr/share/latex2html/icons/up.png"></A>
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<A NAME="tex2html796"
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HREF="node49.html">
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<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
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SRC="file:/usr/share/latex2html/icons/prev.png"></A>
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<A NAME="tex2html804"
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HREF="node1.html">
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<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
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SRC="file:/usr/share/latex2html/icons/contents.png"></A>
|
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<BR>
|
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<B> Next:</B> <A NAME="tex2html807"
|
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HREF="node51.html">Special nasty cases</A>
|
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<B> Up:</B> <A NAME="tex2html803"
|
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HREF="node46.html">How are different positions</A>
|
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<B> Previous:</B> <A NAME="tex2html797"
|
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HREF="node49.html">Observables</A>
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<B> <A NAME="tex2html805"
|
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HREF="node1.html">Contents</A></B>
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<!--End of Navigation Panel-->
|
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<ADDRESS>
|
|
Thattil Dhanya
|
|
2018-09-28
|
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</ADDRESS>
|
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</BODY>
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</HTML>
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