GLYPH<s45>GLYPH<s54> GLYPH<s45>GLYPH<s51> GLYPH<s48> GLYPH<s51> GLYPH<s54> GLYPH<s48> GLYPH<s49> GLYPH<s50> GLYPH<s51> GLYPH<s52> GLYPH<s88> GLYPH<s77> GLYPH<s40>GLYPH<s67> GLYPH<s41> GLYPH<s78>GLYPH<s40>GLYPH<s83> GLYPH<s41> GLYPH<s78>GLYPH<s40>GLYPH<s67> GLYPH<s41> GLYPH<s78>GLYPH<s40>GLYPH<s83> GLYPH<s41> GLYPH<s80> GLYPH<s40>GLYPH<s67> GLYPH<s41> GLYPH<s84> GLYPH<s32> GLYPH<s47> GLYPH<s32> GLYPH<s84> GLYPH<s97> GLYPH<s49>GLYPH<s32> GLYPH<s47>GLYPH<s32> GLYPH<s110> GLYPH<s49>GLYPH<s47>GLYPH<s51> GLYPH<s97> GLYPH<s115> GLYPH<s80> GLYPH<s40>GLYPH<s83> GLYPH<s41> FIG. 3: Mean-field phase diagram of a pairing Bose gas with a wide Feshbach resonance. P(S) denotes the stable molecular-BEC phase; P(C) denotes the collapsing molecular-BEC phase; N(S) denotes the stable normal phase; N(C) denotes the collapsing normal phase; M(C) denotes the collapsing coherent mixture of atoms and molecules; X denotes no solution according to the mean- field theory. The Dashed line is the mean-field collapse temperature of the normal phase with attractive interactions. Recently similar phase diagrams were obtained in Ref. [20] within the approach pioneered by Nozi` eres and Schmitt-Rink (NSR) [21]. In the NSR approach, pairing fluctuations are treated by the t -matrix approximation, and the total particle density includes not only the mean-field density, but also the density of thermal molecules. Although there are some discrepancies, most features of our mean-field phase diagram Fig. 3 qualitatively agree with those in Ref. [20]. The condensation temperature of the MBEC phase is significantly reduced in the NSR approach. In the limit of weakly-repulsive interaction, it can recover the condensation temperature of an ideal molecular Bose gas, T$_{p}$ = T$_{a}$/ 2 5 / $^{3}$. Both NSR and mean-field phase diagrams show that there is a transition from the molecular BEC to the mixture state. As shown in the next section, the mixture state is mechanically unstable in the mean-field approximation. In Ref. [20], the normal state was found to be stable in the simple NSR approach, but unstable close to the resonance in the extended NSR approach by taking into account interaction between molecules . In the mean-field approach, the normal state is unstable near the resonance only on the attractive-interaction side. 9