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$\begin{figure} \par\includegraphics[width=8.8cm,clip]{4260f1.eps} \end{figure}$ Figure 1: Two examples of EOSs employed in our calculations: constant-pressure phase transition (dashed line, EOS with $\Gamma _{\rm A}=\Gamma _{\rm B}=2.25$ , the density jump $\lambda =1.3$ , can be found in Table 2) and transition through a mixed-phase state (solid line, thin dotted lines mark n₁ and n₂; EOS MM of Table 1). Top panel: ${p}(\rho )$ diagrams. Bottom panel: density profiles in stellar core, for a $M=1.33~{M}_\odot$ neutron star models.
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In the text

$\begin{figure} \par\includegraphics[width=8.5cm,clip]{4260f2.eps} \end{figure}$ Figure 2: Examples of the three EOSs with phase transition through the mixed-phase state, considered in the present paper. The parameters of the EOSs are given in Table 1.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{4260f3.eps} \end{figure}$ Figure 3: Total angular momentum versus rotation frequency f ( left panels), and moment of inertia $I\equiv J/\Omega$ versus f ( right panels), for EOSs MSt and MUn. The stability criterion is easily applied to left panels. It is clear that for the MSt EOS back bending feature is not associated with an instability, with all configurations being stable. On the contrary, the MUn EOS produces back bending with a large segment of unstable configurations. Simultaneously, the I(f) curves for both EOSs are very similar and apparently show very similar back-bending shapes.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{4260f4.eps} \end{figure}$ Figure 4: Upper panel: stellar baryon mass vs radius for fixed frequency (dashed lines) and fixed total angular momentum (solid lines), for the model MM of mixed-phase EOS. Bottom panel: stellar angular momentum as a function of the rotational frequency for fixed baryon mass (indicated as a label, in solar masses) for the same MM EOS. This EOS corresponds to the marginal case from the point of view of stability - the curves $M(R_{\rm eq})_{J}$ and $J(f)_{M_{\rm b}}$ have flat horizontal regions. The regions of back-bending are drawn by thick lines.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{4260f5.eps} \end{figure}$ Figure 5: The same as in Fig. 4 but for the MSt EOS. Phase transition does not result in the stability loss - all configurations are stable.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{4260f6.eps} \end{figure}$ Figure 6: The same as in Fig. 4 but for MUn EOS. Upper panel: Phase transition results in the stability loss, i.e., there exists a region where $M_{\rm b}$ decreases with increasing $\rho _{\rm c}$ at fixed J (marked by dotted lines). Lower panel: plots in the J-f plane, where on the dotted segments J increases as $\rho _{\rm c}$ increases (unstable configurations). Both features indicate instability with respect to the axi-symmetric perturbations.
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In the text

$\begin{figure} \par\includegraphics[width=8.4cm,clip]{4260f7.eps} \end{figure}$ Figure 7: The definition of the frequency of the onset of back-bendingphenomenon, $f_{\rm on}$ , and corresponding mass, $M_{\rm b}^{\rm on}$ . In the presented example $f_{\rm on}=449$ Hz and $M_{\rm b}^{\rm on}=1.325 ~M_\odot$ .
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In the text

$\begin{figure} \par\includegraphics[width=8.4cm,clip]{4260f8.eps} \end{figure}$ Figure 8: The onset frequency of the back-bending phenomenon as a function of the departure of $\lambda$ from the maximum density jump $\lambda _{\rm max}$ ( $\lambda _{\rm max}-\lambda$ , left panel), and density jump $\lambda$ as a function of $f_{\rm on}$ ( right panel) for models presented in Table 2 (solid lines for $\Gamma _{\rm A}=2$ , dotted for $\Gamma _{\rm A}=2.25$ and dashed for $\Gamma _{\rm A}=2.5$ ).
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{4260f9.eps} \end{figure}$ Figure 9: The mass M of the star as a function of central baryon number density $n_{\rm c}$ for the MM model. Dashed lines - fixed rotational frequency. Solid lines - fixed total angular momentum J. The vertical lines correspond to the densities of transitions from the nuclear to mixed phase and from the mixed phase to the pure denser phase (here - quark matter) - see Table 1.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{4260f10.eps} \end{figure}$ Figure 10: Evolution of an isolated pulsar loosing angular momentum, after it reaches the instability region in J-f plane and then collapses. Arrows lead from unstable configuration to a collapsed stable one, with the same baryon mass and angular momentum. Dotted lines - unstable configurations.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{4260f11.eps} \end{figure}$ Figure 11: Changes of stellar parameters of a rotating solitary neutron star, due to a collapse which occurs after a pulsar loosing angularmomentum reaches an unstable configuration.
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In the text

$\begin{figure} \par\includegraphics[angle=-90,width=8.8cm,clip]{4260f12.eps} \end{figure}$ Figure 12: Total gravitational mass of the star as a function of its angular momentum, for fixed baryon number of the star for our EOS model MUn. The central density is increasing along this curve as marked by the arrows. The upper segment (dotted) corresponds to the unstable configurations. Two cusps reflect strict property that the mass and angular momentum have simultaneous extrema along the path with fixed baryon number.
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In the text

$\begin{figure} \par\includegraphics[angle=-90,width=8.8cm,clip]{4260f13.eps} \end{figure}$ Figure 13: The relative mass-energy increase due to rotation of the star at fixed baryon mass, $\varepsilon (f)\equiv [(M(f)-M(0))/M(0)]_{M_{\rm b}}$ , for the EOS with a phase transition (MM model), is shown using solid line. Three solid lines are labeled by the gravitational mass of the non-rotating configuration (in solar masses). Dotted lines correspond to the $\varepsilon (f)$ curves calculated for the standard model, Eq. (3), with $\alpha =4$ .
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{4260f14.eps} \end{figure}$ Figure 14: The evolution of the pulsar period when the energy loss is described by the magnetic dipole braking with $\alpha =4$ . Solid curves - results for our model MM with different values of baryon mass. Dotted lines correspond to the standard model, Eq. (3). The unit of time (horizontal axis) is $1000~{\rm yr}/\kappa_{28}$ where $\kappa _{28}= \kappa /10^{28}$ [cgs] is parameter entering Eq. (2).
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{4260f15.eps} \end{figure}$ Figure A.1: Examples of phase transitions considered in the text; constant pressure phase transition ( left), and the phase transition through the mixed-phase state ( right).

In the text

$\begin{figure} \par\includegraphics[width=8.5cm,clip]{4260f2.eps} \end{figure}$	Figure 2: Examples of the three EOSs with phase transition through the mixed-phase state, considered in the present paper. The parameters of the EOSs are given in Table 1.
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$\begin{figure} \par\includegraphics[width=8.8cm,clip]{4260f5.eps} \end{figure}$	Figure 5: The same as in Fig. 4 but for the MSt EOS. Phase transition does not result in the stability loss - all configurations are stable.
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$\begin{figure} \par\includegraphics[width=8.4cm,clip]{4260f7.eps} \end{figure}$	Figure 7: The definition of the frequency of the onset of back-bendingphenomenon, $f_{\rm on}$ , and corresponding mass, $M_{\rm b}^{\rm on}$ . In the presented example $f_{\rm on}=449$ Hz and $M_{\rm b}^{\rm on}=1.325 ~M_\odot$ .
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$\begin{figure} \par\includegraphics[width=8.8cm,clip]{4260f9.eps} \end{figure}$	Figure 9: The mass M of the star as a function of central baryon number density $n_{\rm c}$ for the MM model. Dashed lines - fixed rotational frequency. Solid lines - fixed total angular momentum J. The vertical lines correspond to the densities of transitions from the nuclear to mixed phase and from the mixed phase to the pure denser phase (here - quark matter) - see Table 1.
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$\begin{figure} \par\includegraphics[width=8.8cm,clip]{4260f10.eps} \end{figure}$	Figure 10: Evolution of an isolated pulsar loosing angular momentum, after it reaches the instability region in J-f plane and then collapses. Arrows lead from unstable configuration to a collapsed stable one, with the same baryon mass and angular momentum. Dotted lines - unstable configurations.
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$\begin{figure} \par\includegraphics[width=8.8cm,clip]{4260f11.eps} \end{figure}$	Figure 11: Changes of stellar parameters of a rotating solitary neutron star, due to a collapse which occurs after a pulsar loosing angularmomentum reaches an unstable configuration.
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