arXiv subjects
Publications and source records attributed to Pan Liu.
We prove several Liouville type results for the stationary Hall-MHD and MHD equations. In particular, we show that the velocity and magnetic field, belonging to some Lorentz spaces, must be zero.
In this paper, we empirically analyze the spatial distribution of Chinese cities using a method based on triangle transition. This method uses a regular triangle mapping from the observed cities and its three neighboring cities to analyze their distribution of mapping positions. We find that obvious center-gathering tendency for the relationship between cities and its nearest three cities, indicating the spatial competition between cities. Moreover, we observed the competitive trends between neighboring cities with similar economic volume, and the remarkable cooperative tendency between neighboring cities with large difference on economy. The threshold of the ratio of the two cities' economic volume on the transition from competition to cooperation is about 1.2. These findings are helpful in the understanding of the cities economic relationship, especially in the study of competition and cooperation between cities.
2H MoS2 has been intensively studied because of layer-dependent electronic structures and novel physical properties. Though the metastable 1T MoS2 with the [MoS6] octahedron was observed from the microscopic area, the true crystal structure of 1T phase has not been determined strictly. Moreover, the true physical properties have not been demonstrated from experiments due to the challenge for the preparation of pure 1T MoS2 crystals. Here, we successfully synthesized the 1T MoS2 single crystals and re-determined the crystal structure of 1T MoS2 from single-crystal X-ray diffraction. 1T MoS2 crystalizes in space group P-3m1 with a cell of a = b = 3.190(3) Å and c = 5.945(6) Å. The individual MoS2 layer consists of MoS6 octahedron sharing edge with each other. More surprisingly, the bulk 1T MoS2 crystals undergo a superconducting transition of Tc = 4 K, which is the first observation of superconductivity in pure 1T MoS2 phase.
New fractional $r$-order seminorms, $TGV^r$, $r\in \mathbb R$, $r\geq 1$, are proposed in the one-dimensional (1D) setting, as a generalization of the integer order $TGV^k$-seminorms, $k\in\mathbb{N}$. The fractional $r$-order $TGV^r$-seminorms are shown to be intermediate between the integer order $TGV^k$-seminorms. A bilevel training scheme is proposed, where under a box constraint a simultaneous optimization with respect to parameters and order of derivation is performed. Existence of solutions to the bilevel training scheme is proved by $Γ$-convergence. Finally, the numerical landscape of the cost function associated to the bilevel training scheme is discussed for two numerical examples.
The Ambrosio-Tortorelli approximation scheme with weighted underlying metric is investigated. It is shown that it Γ-converges to a Mumford-Shah image segmentation functional depending on the weight $ωdx$, where $ω\in SBV(Ω)$, and on its value $ω^-$.
We proved uniqueness and instability of the symmetric subsonic--sonic flow solution of the compressible potential flow equation in a surface with convergent areas of cross--sections. Such a surface may be regarded as an approximation of a two--dimensional convergent nozzle in aerodynamics. Mathematically these are uniqueness and nonexistence results of a nonlinear degenerate elliptic equation with Bernoulli type boundary conditions. The proof depends on maximum principles and a generalized Hopf boundary point lemma which was proved in the paper.
In this paper we study the solutions of the $Q$-curvature equation on a 4-dimensional Riemannian manifold $(M,g)$ with $\int_MQdV_g=8π^2$, proving some sufficient conditions for the existence.
Let $(M,g)$ be a 2-dimensional compact Riemannian manifold. In this paper, we use the method of blowing up analysis to prove several Moser-Trdinger type inequalities for vector bundle over $(M,g)$. We also derive an upper bound of such inequalities under the assumption that blowing up occur.