126 lines
3.8 KiB
TeX
Executable File
126 lines
3.8 KiB
TeX
Executable File
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\documentclass{report}
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%\batchmode
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\usepackage{latexsym}
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\usepackage[T2A]{fontenc}
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\usepackage[cp1251]{inputenc}
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\usepackage[russian]{babel}
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\usepackage{mathtext}
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\righthyphenmin=2
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\oddsidemargin=0pt
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\textwidth=18cm
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\topmargin=0cm
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\textheight=23cm
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\begin{document}
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$S_{12}$ -- ïëîùàäü ïåðåñå÷åíèÿ 1 è 2 îêðóæíîñòåé;
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$S_{23}$ -- ïëîùàäü ïåðåñå÷åíèÿ 2 è 3 îêðóæíîñòåé;
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$S_2$ -- ïëîùàäü 2é îêðóæíîñòè.
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Ïëîùàäü ôèãóðû íàéäåì ïî ôîðìóëå \ref{first}:
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\begin{equation}
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\label{first}
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{\bf mes}\:\Omega_{inters} = S_2 - (S_2-S_{12}) - (S_2-S_{23})
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\end{equation}
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Èëè, ïîñëå ðàñêðûòèÿ ñêîáîê:
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\begin{equation}
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\label{second}
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{\bf mes}\:\Omega_{inters} = S_{12}+S_{23}-S_2
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\end{equation}
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$S_{12}$ è $S_{23}$ áûëè íàéäåíû ðàíåå, äëÿ ìîìåíòîâ II ïîðÿäêà, $S_2=\pi r_2^2$.  ðåçóëüòàòå ïîäñòàíîâêè ïîëó÷àåì ôîðìóëó \ref{end}:
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\begin{equation}
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\label{end}
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\begin{array}{rcl}
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{\bf mes}\:\Omega_{inters} = r_2^2\cdot
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\left(
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\arccos\left[\frac{1}{2R_{12}r_2}\cdot\left(r_2^2-r_1^2+R_{12}^2\right)\right]+
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\arccos\left[\frac{1}{2R_{23}r_2}\cdot\left(r_2^2-r_1^2+R_{23}^2\right)\right]-\pi
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\right)+\\
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{}+
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r_1^2\arccos\left[\frac{1}{2R_{12}r_1}\cdot\left(r_1^2-r_2^2+R_{12}^2\right)\right]+
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r_3^2\arccos\left[\frac{1}{2R_{23}r_3}\cdot\left(r_3^2-r_2^2+R_{23}^2\right)\right]-\\
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{}-2\cdot\left(
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\sqrt{p_1(p_1-r_1)(p_1-r_2)(p_1-R_{12})}+
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\sqrt{p_2(p_2-r_3)(p_2-r_2)(p_2-R_{23})}
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\right)
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\end{array}
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\end{equation}
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\begin{equation}
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\label{p1}
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p_1=\frac{1}{2}\cdot\left(r_1+r_2+R_{12}\right)
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\end{equation}
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\begin{equation}
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\label{p2}
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p_2=\frac{1}{2}\cdot\left(r_3+r_2+R_{23}\right)
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\end{equation}
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Ðàññìîòðèì ÷àñòíûå ñëó÷àè:
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\begin{enumerate}
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\item $r_i=r_j \ne r_k, i \ne j \ne k$:
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\begin{equation}
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\label{rierj}
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\bf K_{\lambda}^{(3)}(r_i,r_i,r_k) = \left<\lambda(r_i)\lambda(r_i)\lambda(r_k)\right>-\nu_f \left[\left<\lambda(r_i)\lambda(r_i)\right>+2\left<\lambda(r_i)\lambda(r_k)\right>\right]+2\nu_f^3
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\end{equation}
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\begin{equation}
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\label{lililkf}
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\begin{array}{rcl}
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\bf\left<\lambda(r_i)\lambda(r_i)\lambda(r_k)\right> =
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Prob(r_i \in \Omega_f \land r_i \in \Omega_f \land r_k \in \Omega_f) = {}\\
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\bf{}=Prob \left[r_i \in \Omega_f \mid (r_i \in \Omega_f \land r_k \in \Omega_f)\right]\times{}\\
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\bf{}\times Prob\left[r_i \in \Omega_f \mid r_k \in \Omega_f\right]Prob\left[r_k \in \Omega_f\right]
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\end{array}
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\end{equation}
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\begin{equation}
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\label{lililks}
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\begin{array}{rcl}
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\bf\left<\lambda(r_i)\lambda(r_i)\lambda(r_k)\right> =
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Prob(r_i \in \Omega_f \land r_k \in \Omega_f)Prob(r_i \in \Omega_f \mid r_k \in \Omega_f)Prob(r_k \in \Omega_f)={}\\
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\bf{}=Prob(r_i \in \Omega_f \mid r_k \in \Omega_f) \cdot\nu_f\cdot Prob(r_i \in \Omega_f \mid r_k \in \Omega_f)Prob(r_k \in \Omega_f)
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\end{array}
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\end{equation}
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\begin{equation}
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\label{lililkt}
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\bf\left<\lambda(r_i)\lambda(r_i)\lambda(r_k)\right> =
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\frac{mes(\Omega_f^i \cap \Omega_f^k)}{mes(\Omega_f)}\cdot\nu_f\cdot\frac{mes(\Omega_f^i \cap \Omega_f^k)}{mes\Omega_f}\cdot\nu_f=\left(\frac{mes(\Omega_f^i \cap \Omega_f^k)}{mes\widehat{\Omega}}\right)^2
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\end{equation}
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\begin{equation}
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\label{lili}
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\bf\left<\lambda(r_i)\lambda(r_i)\right>=Prob(r_i \in \Omega_f \land r_i \in \Omega_f)=Prob(r_i \in \Omega_f)=\nu_f
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\end{equation}
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\begin{equation}
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\label{lilk}
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\bf\left<\lambda(r_i)\lambda(r_k)\right>=Prob(r_i \in \Omega_f \mid r_k \in \Omega_f)\cdot\nu_f=\frac{mes(\Omega_f^i \cap \Omega_f^k)}{mes\widehat{\Omega}}
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\end{equation}
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\begin{equation}
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\label{kiikend}
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\begin{array}{rcl}
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\bf K_{\lambda}^{(3)}(r_i,r_i,r_k)=
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\left[\frac{mes(\Omega_f^i \cap \Omega_f^k)}{mes\widehat{\Omega}}\right]^2-
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\nu_f\cdot\left[\nu_f+2\frac{mes(\Omega_f^i \cap \Omega_f^k)}{mes\widehat{\Omega}}\right]+2\nu_f^3={}\\
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\bf{}=\frac{mes(\Omega_f^i \cap \Omega_f^k)}{mes\widehat{\Omega}}\left[\frac{mes(\Omega_f^i \cap \Omega_f^k)}{mes\widehat{\Omega}}-2\nu_f\right]-\nu_f^2+2\nu_f^3
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\end{array}
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\end{equation}
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\end{enumerate}
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\end{document}
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