Consider $N$ bosons in a finite box  $ \Lambda = [0,L]^3 \subset R^3$ interacting via a two-body smooth repulsive short range potential. We construct a variational state which gives the following upper bound on the ground state energy per particle $$ \bar{lim|_{\rho \to 0} \bar{lim|_{L \to \infty, N/L^3 \to \rho} ( \ frac{e_0 (\rho) - 4\pi a \rho }{ (4\pi a)^{5/2} (\rho)^{3/2} } ) \le \frac{16}{15 \pi ^2} , $$ where $a$ is the scattering length of the potential. Previously, an upper bound of the form $C16/15\pi^2$ for some constant $C >1$ was obtained in (Erdös et al. in Phys. Rev. A 78:053627, 2008). Our result proves the upper bound of the prediction by Lee and Yang (Phys. Rev. 105(3):1119–1120, 1957) and Lee et al. (Phys. Rev. 106(6):1135–1145, 1957).

Consider an N-Boson system interacting via a two-body repulsive short-range potential $V$ in a three dimensional box  of side length $L$. We take the $limit N,L \to \infty $ while keeping the density $\rho = N/L^3 fixed and small. We prove a new lower bound for its ground state energy per particle $$ \frac{E(N,\Lambda)}{N} \ge 4 \pi a \rho [1- O(\rho^{1/3} | log \rho |^3) ] $$ as $\rho \to 0$, where $a$ is the scattering length of $V$.

The leading term of the ground state energy/particle of a dilute gas of bosons with mass m in the thermodynamic limit is $ 2\pi \hbar^2 a \varrho/m$ when the density of the gas is $ \varrho$, the interaction potential is non-negative and the scattering length $a$ is positive. In this paper, we generalize the upper bound part of this result to any interaction potential with positive scattering length, i.e, $a > 0$ and the lower bound part to some interaction potentials with shallow and/or narrow negative parts.

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.

Sharp edges, ridges, valleys and prongs are critical for the appearance and an accurate representation of a 3D model. In this paper, we propose a novel approach that deals with the global shape of features in a robust way. Based on a remeshing algorithm which delivers an isotropic mesh in a feature sensitive metric, features are recognized on multiple scales via integral invariants of local neighborhoods. Morphological and smoothing operations are then used for feature region extraction and classification into basic types such as ridges, valleys and prongs. The resulting representation of feature regions is further used for feature-specific editing operations.