The mechanical stability,elastic,and thermodynamic properties of the anti-perovskite superconductors MNNi 3(M=Zn,Mg,Al) are investigated by means of the first-principles calculations.The calculated structural parameters and elastic properties of MNNi 3 are in good agreement with the experimental and the other theoretical results.From the elastic constants under high pressure,we predict that ZnNNi 3,MgNNi 3,and AlNNi 3 are not stable at the pressures above 61.2 GPa,113.3 GPa,and 122.4 GPa,respectively.By employing the Debye model,the thermodynamic properties,such as the heat capacity and the thermal expansion coefficient,under pressures and at finite temperatures are also obtained successfully.
In this paper, we establish a general theta function identity. It is a common origin of many theta function identities. From which many classical and new modular equations are derived. All the proofs are elementary.
