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$\begin{figure} \par\includegraphics[clip,width=8.8cm]{0416fig1.eps}\end{figure}$ Figure 1: The internal chemical profile corresponding to a $1.06 ~ M_\odot$ CO model in which chemical rehomogenization induced by crystallization has been neglected. The gray area marks the domain of crystallization. $M_{\rm c}$ means the crystallized mass of the model.
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In the text

$\begin{figure} \par\includegraphics[clip,width=8.8cm,clip]{0416fig2.eps}\end{figure}$ Figure 2: Same as Fig. 1, but in the case in which chemical rehomogenization induced by crystallization has been taken into account.
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In the text

$\begin{figure} \par\includegraphics[clip,width=8.7cm]{0416fig3.eps}\end{figure}$ Figure 3: The internal chemical profile corresponding to our initial ONe white dwarf model in terms of the outer mass fraction.
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In the text

$\begin{figure} \par\includegraphics[clip,width=8.7cm]{0416fig4.eps}\end{figure}$ Figure 4: Same as Fig. 3, but for an ONe white dwarf model in the ZZ Ceti instability strip. The crystallized region is displayed as a gray area.
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In the text

$\begin{figure} \par\includegraphics[clip,width=8.8cm]{0416fig5.eps}\end{figure}$ Figure 5: The squared Brunt-Väisälä frequency (N²) - solid line - in terms of the outer mass fraction, corresponding to the same CO white dwarf model shown in Fig. 1. Inset: the Ledoux term B. For the sake of completeness, we also show using a dotted line the acoustic (Lamb) frequency corresponding to $\ell = 2$ . The gray area marks the domain of the crystallized core. Here, because of the hard-sphere approximation, the detailed shape of the Brunt-Väisälä frequency is irrelevant to pulsations and hence it is not shown.
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In the text

$\begin{figure} \par\includegraphics[clip,width=8.8cm,clip]{0416fig6.eps}\end{figure}$ Figure 6: Same as in Fig. 5, but for the CO white dwarf model analyzed in Fig. 2.
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In the text

$\begin{figure} \par\includegraphics[clip,width=250pt]{0416fig7.eps}\end{figure}$ Figure 7: Same as Fig. 6, but for the ONe white dwarf model analyzed in Fig. 4. Dashed lines correspond to the case in which the Ledoux term B is neglected in the region $-2 \protect\la\log~ (1-M_{\rm r}/M_*) \protect\la -1$ (see Sect. 3.2 for additional details).
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In the text

$\begin{figure} \par\includegraphics[clip,width=8.8cm]{0416fig8.eps}\end{figure}$ Figure 8: Asymptotic period spacing, $\Delta P_{(\ell =2)}$ , as a function of the effective temperature, for the CO and ONe evolutionary cooling sequences (thick and thin curves, respectively). See text for details.
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In the text

$\begin{figure} \par\includegraphics[clip,width=16cm]{0416fig9.eps}\end{figure}$ Figure 9: The forward period spacing ( upper panels) and the kinetic energy ( lower panels) in terms of the periods of $\ell = 2$ pulsation modes. The left panels correspond to a CO white dwarf model in which crystallization has not been taken into account. The central panels show the same quantities for a model at the same $T_{\rm eff}$ in which crystallization has been considered but phase separation has not. Finally, the right panels show the case in which both crystallization and phase separation have been taken into account.
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In the text

$\begin{figure} \par\includegraphics[clip,width=11.5cm]{0416fi10.eps}\end{figure}$ Figure 10: Same as Fig. 9, but for the case of an ONe white dwarf.
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In the text

$\begin{figure} \par\includegraphics[clip,width=250pt]{0416fi11.eps}\end{figure}$ Figure 11: Same as the upper-right panel of Fig. 10, but for the case in which the effect of the strong step in the chemical profile at $-2 \protect\la\log~ (1-M_{\rm r}/M_*) \protect\la -1$ (see Fig. 4) on the shape of the Brunt-Väisälä frequency has been neglected by setting B= 0 in that region (see Fig. 7).
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In the text

$\begin{figure} \par\includegraphics[clip,width=8.8cm]{0416fig1.eps}\end{figure}$	Figure 1: The internal chemical profile corresponding to a $1.06 ~ M_\odot$ CO model in which chemical rehomogenization induced by crystallization has been neglected. The gray area marks the domain of crystallization. $M_{\rm c}$ means the crystallized mass of the model.
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$\begin{figure} \par\includegraphics[clip,width=8.8cm,clip]{0416fig2.eps}\end{figure}$	Figure 2: Same as Fig. 1, but in the case in which chemical rehomogenization induced by crystallization has been taken into account.
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$\begin{figure} \par\includegraphics[clip,width=8.7cm]{0416fig3.eps}\end{figure}$	Figure 3: The internal chemical profile corresponding to our initial ONe white dwarf model in terms of the outer mass fraction.
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$\begin{figure} \par\includegraphics[clip,width=8.7cm]{0416fig4.eps}\end{figure}$	Figure 4: Same as Fig. 3, but for an ONe white dwarf model in the ZZ Ceti instability strip. The crystallized region is displayed as a gray area.
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$\begin{figure} \par\includegraphics[clip,width=8.8cm,clip]{0416fig6.eps}\end{figure}$	Figure 6: Same as in Fig. 5, but for the CO white dwarf model analyzed in Fig. 2.
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$\begin{figure} \par\includegraphics[clip,width=250pt]{0416fig7.eps}\end{figure}$	Figure 7: Same as Fig. 6, but for the ONe white dwarf model analyzed in Fig. 4. Dashed lines correspond to the case in which the Ledoux term B is neglected in the region $-2 \protect\la\log~ (1-M_{\rm r}/M_*) \protect\la -1$ (see Sect. 3.2 for additional details).
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$\begin{figure} \par\includegraphics[clip,width=8.8cm]{0416fig8.eps}\end{figure}$	Figure 8: Asymptotic period spacing, $\Delta P_{(\ell =2)}$ , as a function of the effective temperature, for the CO and ONe evolutionary cooling sequences (thick and thin curves, respectively). See text for details.
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$\begin{figure} \par\includegraphics[clip,width=11.5cm]{0416fi10.eps}\end{figure}$	Figure 10: Same as Fig. 9, but for the case of an ONe white dwarf.
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$\begin{figure} \par\includegraphics[clip,width=250pt]{0416fi11.eps}\end{figure}$	Figure 11: Same as the upper-right panel of Fig. 10, but for the case in which the effect of the strong step in the chemical profile at $-2 \protect\la\log~ (1-M_{\rm r}/M_*) \protect\la -1$ (see Fig. 4) on the shape of the Brunt-Väisälä frequency has been neglected by setting B= 0 in that region (see Fig. 7).
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