An unconditionally energy stable finite difference scheme for a stochastic Cahn-Hilliard equation

In this work, the MMC-TDGL equation, a stochastic Cahn-Hilliard equation is solved numerically by using the finite difference method in combination with a convex splitting technique of the energy functional. For the non-stochastic case, we develop an unconditionally energy stable difference scheme which is proved to be uniquely solvable. For the stochastic case, by adopting the same splitting of the energy functional, we construct a similar and uniquely solvable difference scheme with the discretized stochastic term. The resulted schemes are nonlinear and solved by Newton iteration. For the long time simulation, an adaptive time stepping strategy is developed based on both first- and second-order derivatives of the energy. Numerical experiments are carried out to verify the energy stability, the efficiency of the adaptive time stepping and the effect of the stochastic term.Comment: This paper has been accepted for publication in SCIENCE CHINA Mathematic

Intertemporal decision-making and loss aversion

This thesis is devoted to a novel approach to intertemporal decision-making. Specifically, it applies the reference-dependent model to intertemporal choices and shows that one can model intertemporal choice even without the assumption of time discounting.In Chapter I, I present a brief review of the background literature, which could be of help for our understanding of intertemporal decision-making. Specifically, it focuses on three prominent economic theories, which are the most relevant for my work on intertemporal decision-making, namely, the exponential discounted utility theory, the hyperbolic discounting theory, the theory of reference-dependent preferences, and the similarity-based procedure.In Chapter II, I present the intertemporal reference-dependent model composed of an intrinsic consumption utility and a reference-dependent gain-loss utility which are additively separable over time. Three simple ways of reference point determination - choice-set dependent, choice-set independent, and hybrid reference points are also explored. In addition, I also argue that the intertemporal reference-dependent model may offer an alternative explanation of the experimental observations by Rubinstein (2003). Furthermore, in Chapter III, I extend the intertemporal reference-dependent model, where an individual is uncertain about her future circumstances, and her reference levels follow a random walk process. While the model does not explicitly include time discounting and return on saving, it nevertheless may offer an alternative explanation of present bias and negative time preference (future bias), and may also offer an alternative explanation of dynamically inconsistent preferences over time.Finally, in Chapter IV, using a survey-based within-subject choice experiment, I explore whether subjects' behaviour is consistent with a number of existing and emerging theories, namely hyperbolic discounting theory, Rubinstein's similarity procedure, and the novel intertemporal reference-dependent model. The main result of the survey-based within-subject choice experiment is that none of these three theories explain subjects' behaviour. However, I do find that, among these three theories, the novel intertemporal reference-dependent model performs no worse than Rubinstein's similarity procedure, and hyperbolic discounting theory performs the worst with subjects' behaviour on subset of questions. Moreover, I could reject the hypothesis that the subjects made their choices at random. Finally, I also found that their behaviour is not consistent with the Independence of Irrelevant Alternatives

A Review of the Chinese Real Estate Market

The Chinese real estate market has undergone rapid growth within the past decade. Although the growth of the real estate market has helped the Chinese economy, there are many underlying problems to such a hurried rate of growth. This paper briefly looks at the background of the real estate market leading up to the current situation, and then discusses some of the potential problems and government responses

Microscopic theory of quantum anomalous Hall effect in graphene

We present a microscopic theory to give a physical picture of the formation of quantum anomalous Hall (QAH) effect in graphene due to a joint effect of Rashba spin-orbit coupling $\lambda_R$ and exchange field $M$. Based on a continuum model at valley $K$ or $K'$, we show that there exist two distinct physical origins of QAH effect at two different limits. For $M/\lambda_R\gg1$, the quantization of Hall conductance in the absence of Landau-level quantization can be regarded as a summation of the topological charges carried by Skyrmions from real spin textures and Merons from \emph{AB} sublattice pseudo-spin textures; while for $\lambda_R/M\gg1$, the four-band low-energy model Hamiltonian is reduced to a two-band extended Haldane's model, giving rise to a nonzero Chern number $\mathcal{C}=1$ at either $K$ or $K'$. In the presence of staggered \emph{AB} sublattice potential $U$, a topological phase transition occurs at $U=M$ from a QAH phase to a quantum valley-Hall phase. We further find that the band gap responses at $K$ and $K'$ are different when $\lambda_R$, $M$, and $U$ are simultaneously considered. We also show that the QAH phase is robust against weak intrinsic spin-orbit coupling $\lambda_{SO}$, and it transitions a trivial phase when $\lambda_{SO}>(\sqrt{M^2+\lambda^2_R}+M)/2$. Moreover, we use a tight-binding model to reproduce the ab-initio method obtained band structures through doping magnetic atoms on $3\times3$ and $4\times4$ supercells of graphene, and explain the physical mechanisms of opening a nontrivial bulk gap to realize the QAH effect in different supercells of graphene.Comment: 10pages, ten figure