diplom/mes_3.tex

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\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}