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mes_3.tex

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