First-principles calculations of the crystal structure and the elastic properties of tetragonal WN 2 have been performed with the plane-wave pseudopotential density functional theory method. The calculated structural parameters and elastic constants at zero pressure and temperature are in excellent agreement with the available theoretical results. The dependence of the elastic constants C ij , the aggregate elastic moduli B, G and the anisotropies on pressure have been investigated. WN 2 is a brittle system below about 66 GPa, whereas it becomes ductile under high pressure. By the elastic stability criteria, it is predicted that tetragonal WN 2 are not stable above 232.1 GPa.
Finding theta function (or q-)analogues for well-known trigonometric identities is an interesting topic. In this paper, we first introduce the definition of q-analogues for tanz and cotz and then apply the theory of elliptic functions to establish a theta function identity. From this identity we deduce two q-trigonometric identities involving tanqz and cotq z, which are theta function analogues for two well-known trigonometric identities concerning tanz and cot z. Some other q-trigonometric identities are also given.2000 Mathematics Subject Classification. 33E05, 11F11, 11F12.
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