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\mathrm{p}_{ij} = \left\{ \begin{array}{ll}
i^2/N^2 & j=i-1 \\
2i(N-i)/N^2 & j=i \\
(N-i)^2/N^2 & j=i 1 \\
0 & \mathrm{otherwise}
\end{array} \right.
&& {\rm scan}\;\;(\oplus)\;\;[a_0, \cdots , a_n] \\
&& = [a_0, a_0 \oplus a_1, \cdots , a_0 \oplus \cdots \oplus a_n] \\
&& \equiv [b_0, \cdots, b_n]
{\rm d}v &=& ae^{at}u {\rm d}t   e^{at}{\rm d}u\\
&=& ae^{at}u {\rm d}t   e^{at}(-au{\rm d}t - \sigma{\rm d}W_t)\\
&=& -\sigma e^{at}{\rm d}W_t
\left( {\rm E}\left[ \sum_{i=1}^N {\cal R}_i \right] \right)^2 &=& (\mu^N   \mu^{N-1}   \cdots   \mu)^2 \\
 &=& \left( \sum_{i=1}^N \mu^i \right)^2
{\rm d}y(t) &=& \frac{\partial g}{\partial t}{\rm d}t   \frac{\partial g}{\partial x}{\rm d}x   \frac{1}{2}\frac{\partial^2 g}{\partial x^2}({\rm d}x)^2