In this article, we study the small sphere limit of the WangYau quasi-local energy defined in Wang and Yau (Phys Rev Lett 102(2):021101, 2009, Commun Math Phys 288(3):919942, 2009). Given a point <i>p</i> in a spacetime <i>N</i>, we consider a canonical family of surfaces approaching <i>p</i> along its future null cone and evaluate the limit of the WangYau quasi-local energy. The evaluation relies on solving an optimal embedding equation whose solutions represent critical points of the quasi-local energy. For a spacetime with matter fields, the scenario is similar to that of the large sphere limit found in Chen etal. (Commun Math Phys 308(3):845863, 2011). Namely, there is a natural solution which is a local minimum, and the limit of its quasi-local energy recovers the stress-energy tensor at <i>p</i>. For a vacuum spacetime, the quasi-local energy vanishes to higher order and the solution of the optimal embedding

We find a concise relation between the moduli , of a rational Narain lattice (,) and the corresponding momentum lattices of left and right chiral algebras via the Gauss product. As a byproduct, we find an identity which counts the cardinality of a certain double coset space defined for isometries between the discriminant forms of rank two lattices.

We give close formulas for the counting functions of rational curves on complete intesection Calabi-Yau manifolds in terms of special solutions of generalized hypergeometric differential systems. For the one modulus cases we derive a differential equation for the Mirror map, which can be viewed as a generalization of the Schwarzian equation. We also derive a nonlinear seventh order differential equation which directly governs the Prepotential.

We find two conditions related to the {\it news functions} of the Bondi's radiating vacuum spacetimes. We provide a complete proof of the positivity of the Bondi mass by using Schoen-Yau's method under one condition and by using Witten's method under another condition.

We derive an upper bound on the free energy of a Bose gas at density $ \varrho $ and temperature T. In combination with the lower bound derived previously by Seiringer (Commun. Math. Phys. 279(3): 595–636, 2008), our result proves that in the low density limit, i.e., when $ a^3 \varrho \ll 1$, where $a$ denotes the scattering length of the pair-interaction potential, the leading term of $\Delta f$, the free energy difference per volume between interacting and ideal Bose gases, is equal to $ 4\pi a(2\varrho^{2}-[\varrho-\varrho_{c}]^{2}_{+}) $. Here, $\varrho_c (T)$ denotes the critical density for Bose–Einstein condensation (for the ideal Bose gas), and $ [⋅]_{+}= max \{⋅ , 0\}$ denotes the positive part.