THE OPTIMAL CONVERGENCE ORDER OF THE DISCONTINUOUS FINITE ELEMENT METHODS FOR FIRST ORDER HYPERBOLIC SYSTEMS

In this paper, a discontinuous finite element method for the positive and symmetric, first-order hyperbolic systems （steady and nonsteady state） is constructed and analyzed by using linear triangle elements, and the O（h^2）-order optimal error estimates are derived under the assumption of strongly regular triangulation and the Ha-regularity for the exact solutions. The convergence analysis is based on some superclose estimates of the interpolation approximation. Finally, we discuss the Maxwell equations in a two-dimensional domain, and numerical experiments are given to validate the theoretical results.     

Interlayer vibration of twisted bilayer graphene: A first-principles study

Multi-layered structures of graphene and other two dimensional materials are novel organizations with fascinating properties. In this work, we investigate the interlayer lattice dynamics of the Moiré superlattice of twisted bilayer graphene (tBLG) using first-principles calculations. Due to the structural complexity, we calculate the frequencies of the interlayer breathing and shear vibrational modes of the tBLG without exploring all the degree of freedoms of the tBLG. Our calculation shows that the frequencies of breathing and shear modes display pronounced dependence on twisting angle, and compare well with the state of arts experimental measurements. We hint that the strategy employed here paves a route to the study of the interlayer vibrational properties of double layered two dimensional systems.     

Global versus local superintegrability of nonlinear oscillators

Liouville (super)integrability of a Hamiltonian system of differential equations is based on the existence of globally well-defined constants of the motion, while Lie point symmetries provide a local approach to conserved integrals. Therefore, it seems natural to investigate in which sense Lie point symmetries can be used to provide information concerning the superintegrability of a given Hamiltonian system. The two-dimensional oscillator and the central force problem are used as benchmark examples to show that the relationship between standard Lie point symmetries and superintegrability is neither straightforward nor universal. In general, it turns out that superintegrability is not related to either the size or the structure of the algebra of variational dynamical symmetries. Nevertheless, all of the first integrals for a given Hamiltonian system can be obtained through an extension of the standard point symmetry method, which is applied to a superintegrable nonlinear oscillator describing the motion of a particle on a space with non-constant curvature and spherical symmetry.     

二维g-AlN材料中n型掺杂探索：基于第一性原理的带电缺陷计算

电荷转移能级和缺陷形成能的计算对探索半导体材料的n型和p型掺杂效率具有重要的指导意义。基于第一性原理方法，结合二维带电缺陷计算理论，系统计算了二维类石墨烯氮化铝(graphene-like AlN, g-AlN)中四种（C_Al，Si_Al，Ge_Al，Sn_Al）可能的n型掺杂体系的结构、磁学、电学以及缺陷性质。结果表明，四种体系的最稳定价态均为+1价和0价，Sn_Al具有较深的施主能级，不具备为二维g-AlN提供n型载流子的条件，而C_Al，Si_Al，Ge_Al表现为浅能级施主特性，均能在一定条件下成为理想的施主杂质，其中Si_Al具有最浅的施主特性以及最低的缺陷形成能，因而是二维g-AlN中实现n型掺杂的首选掺杂剂。另外，在p型二维g-AlN中，四种掺杂原子都会成为有效的空穴捕获中心，严重降低p型载流子导电率。研究数据将会为实验上实现二维g-AlN n型掺杂提供理论解释和指导。     

Limit cycles of discontinuous piecewise differential systems formed by linear centers in R2 and separated by two circles

We show that discontinuous planar piecewise differential systems formed by linear centers and separated by two concentric circles can have at most three limit cycles. Usually is a difficult problem to provide the exact upper bound that a class of differential systems can exhibit. Here we also provide examples of such systems with zero, one, two, or three limit cycles.     

The eight first vibrational states of the water molecule: measurements and analysis

In this work the spectrum of the water molecule has been the subject of extensive experimental and theoretical investigations. More than 1500 transitions have been assigned in a far infrared emission spectrum recorded in the 50-500 cm⁻¹ region and experimental rotational energies have been obtained for the second triad of interacting states, (0 3 0), (1 1 0), and (0 1 1), up to J=10 using flame spectroscopy. These new data along with a large data set of already published experimental rotational energies, high-resolution infrared transitions, and microwave lines, involving the eight first vibrational states, have been analyzed using the theoretical formalism developed for water in a previous paper (J. Mol. Spectrosc. 206 (2001) 83), which accounts for the anomalous centrifugal distortion as well as for the Fermi- and Coriolis-type couplings.     

Time sensitive functionals of marked Cox processes

This paper continues studies on a multivariate marked Cox process C_t observed upon some random epochs τ={τ₁,τ₂,…} initiated in [J. Math. Anal. Appl. 293 (2004) 1-13]. The goal is to connect the continuous time parameter process C_t with C_τ for which closed-form transforms were presented in that paper. This work does not only restore some or much of the information on C_t lost due to its limited observation, but it also manages to scrutinize the behavior of C_τ around the first passage time of C_τ (that takes place upon one of the observation epochs τ) within some random time intervals. Again, analytically tractable formulas for functionals of C_τ are derived.     

On Several Two-Boundary Problems for a Particular Class of Lévy Processes

Several two-boundary problems are solved for a special Lévy process: the Poisson process with an exponential component. The jumps of this process are controlled by a homogeneous Poisson process, the positive jump size distribution is arbitrary, while the distribution of the negative jumps is exponential. Closed form expressions are obtained for the integral transforms of the joint distribution of the first exit time from an interval and the value of the overshoot through boundaries at the first exit time. Also the joint distribution of the first entry time into the interval and the value of the process at this time instant are determined in terms of integral transforms.     

Harnack's inequality for a nonlinear eigenvalue problem on metric spaces

We prove Harnack's inequality for first eigenfunctions of the p-Laplacian in metric measure spaces. The proof is based on the famous Moser iteration method, which has the advantage that it only requires a weak (1,p)-Poincaré inequality. As a by-product we obtain the continuity and the fact that first eigenfunctions do not change signs in bounded domains.