deletions | additions       diff --git a/Introduction.tex b/Introduction.tex  index cf82889..bb48a52 100644  --- a/Introduction.tex  +++ b/Introduction.tex 
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\begin{equation}  \label{e.frequency_ba}  q^t_i=q_{min}+(q_{min}-q_{max})\beta^t, q^t_i=q_{min}+(q_{min}-q_{max})\beta,  \end{equation}  \begin{equation}  \label{e.velocity_ba}  \textbf{v}^{t+1}_i=\textbf{v}^t_i+(\textbf{x}^t_i-\textbf{g})q_i,  \end{equation}  where $\beta^t\sim{\cal $\beta\sim{\cal  U}(0,1)$, and $\textbf{g}$ stands for the best solution (bat) found so far (similar rationale is also employed by Equation~\ref{velocity_pso}). The Bat Algorithm works with the definition of ``temporary position", i.e., at each iteration we maintain two structures to store the current position of each bat: its usual position $\textbf{x}$ and the temporary position $\tilde{\textbf{x}}$, which is used to check whether the new generated solution is better than the previous one. 
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\begin{equation}  \label{e.position_tmp_bat}  \tilde{\textbf{x}}^{t+1}=\textbf{x}^t_i+\textbf{v}^{t+1}_i. \tilde{\textbf{x}}_i^{t+1}=\textbf{x}^t_i+\textbf{v}^{t+1}_i.  \end{equation}  Further, we apply a random walk with probability $p_r$ (also known as ``pulse rate"):  \begin{eqnarray}  \label{e.random_walk_bat}  \tilde{\textbf{x}}^{t+1} \tilde{\textbf{x}}_i^{t+1}  & = & \left\{ \begin{array}{ll} \tilde{\textbf{x}}^{t+1}+\epsilon\bar{A}^t \tilde{\textbf{x}}_i^{t+1}+\epsilon\bar{A}^t  & \mbox{{ with probability $p_r$}} \\ \tilde{\textbf{x}}^{t+1} \tilde{\textbf{x}}_i^{t+1}  & \mbox{{ with probability (1-$p_r$)},} \end{array}\right.  \end{eqnarray}  where $\bar{A}^t$ stands for the average of the loudness considering all agents at iteration $t$, and $\epsilon\in[-1,1]$. Finally, the new position of each agent is then computed as follows:  \begin{eqnarray}  \label{e.position_bat}  \textbf{x}^{t+1} \textbf{x}_i^{t+1}  & = & \left\{ \begin{array}{ll} \tilde{\textbf{x}}^{t+1} \tilde{\textbf{x}}_i^{t+1}  & \mbox{{ if $f(\tilde{\textbf{x}}^{t+1}) $f(\tilde{\textbf{x}}_i^{t+1})  and $\xi $\x^i  \\ \textbf{x}^{t+1} & \mbox{{ otherwise}}  \end{array}\right.  \end{eqnarray}