Sharp results for spherical metric on flat tori with conical angle 6π\pi at two symmetric points

A conjecture about the existence or nonexistence of solutions to the curvature equation (1.1) defined on a rectangle torus $E_{\tau},$ $\tau\in i\mathbb{R}_{>0}$ with four conical singularties at its symmetric points is proposed in [3]. See Conjecture 1. For the purposes to understand this problem, in this paper, we study the following equation: \Delta u+e^{u}=8\pi(\delta_{0}+\delta_{\frac{\omega_{k}}{2}})\text{in}E_{\tau}\,\tau\in\mathbb{H}\, \label{a} where $\frac{\omega_{k}}{2}$ is one of the half periods of $E_{\tau}$, i.e., the case $(m_{0},m_{1},$ $m_{2},m_{3})$ $=(1,1,0,0)$, $(1,0,1,0)$, $(1,0,0,1)$ for $k=1,2,3,$ respectively. Among others, we prove that the existence of \textit{non-even family of solutions} (see the definition in Section 1 ) is related to the existence of solutions for the equation with single conical singularity: $$\Delta u+e^{u}=8\pi\delta_{0}\text{ in }E_{\tau}\text{, }\tau\in \mathbb{H}\text{.}$$ Consequently, equation (0.1) does not have any non-even family of solutions for all $k=1,2,3$. As an application, we completely understand the solution structure of the equation (0.1) for rectangle torus and give a confirmative answer for this conjecture in this three cases. See Theorem 1.3

Phase-transition radiation of water

The radiative relaxation mechanism of water between its different phases is studied to understand an uncommon radiation phenomenon observed in the first-order phase-transition process of water. This kind of radiation is often referred to as phase-transition radiation, whose nature is different from the Planckian radiation because its strength can be even stronger than blackbody radiation at the same temperature. In the theoretical approach of this study, analytical thermodynamic models for condensed-state water are presented to study its energetic behaviors at temperatures ranging from a few degrees K to near the critical point. Changes of energetic behaviors of water molecules during phase-transitions are of special interest and are linked to the direct emission of infrared radiation. A two-level energy transition model is proposed to investigate the characteristic radiation during vapor condensation, leading to a newly defined absorption coefficient for phase-transition radiation in the radiative transfer equation. The reported characteristic radiation for vapor condensation at wavelength 4-8 micron meter is attributed to the radiative relaxation with one hydrogen-bond formation in liquid-water during vapor condensation. In addition to the theoretical modeling, optical measurements are also included in this study to examine the energy transmission characteristics in vapor-liquid mixtures of water in the 3-5 micron meter spectral range. Results from the infrared transmission experiments and the associated theoretical predictions by the Monte Carlo radiative transfer analysis suggest that the probability for condensation radiation occurrence is one out of 20 million collisions between water-vapor molecules and liquid-water droplets