Abstract:
We study nonautonomous discrete dynamical systems with randomly perturbed trajectories. We suppose that such a system is generated by a sequence of continuous maps which converges uniformly to a map $f$. We give conditions, under which a recurrent point of a (standard) autonomous discrete dynamical system generated by the limit function $f$ is also recurrent for the nonautonomous system with randomly perturbed trajectories. We also provide a necessary condition for a nonautonomous discrete dynamical system to be nonchaotic in the sense of Li and Yorke with respect to small random perturbations.

Abstract:
We investigate dynamical properties of chaotic trajectories in mushroom billiards. These billiards present a well-defined simple border between a single regular region and a single chaotic component. We find that the stickiness of chaotic trajectories near the border of the regular region occurs through an infinite number of marginally unstable periodic orbits. These orbits have zero measure, thus not affecting the ergodicity of the chaotic region. Notwithstanding, they govern the main dynamical properties of the system. In particular, we show that the marginally unstable periodic orbits explain the periodicity and the power-law behavior with exponent $\gamma=2$ observed in the distribution of recurrence times.

Abstract:
In [2] the notion of stickiness for stochastic processes was introduced. It was also shown that stickiness implies absense of arbitrage in a market with proportional transaction costs. In this paper, we investigate the notion of stickiness further. In particular, we give examples of processes that are not semimartingales but are sticky.

Abstract:
In this paper we derive local estimates of solutions of the Perturbed Stokes system. This system arises as a reduction of the Stokes system near a curved part of the boundary of the domain if one applies a diffeomorphism flatting the boundary. The estimates obtained in the paper play the crucial role in the investigation of partial regularity of weak solutions to the Navier-Stokes system near a curved part of the boundary of the domain.

Abstract:
摘 要： 以1999―2016年A股制造业上市公司为样本，用多元混合回归模型对企业税负变动进行检验。结果显示，企业税负变动存在粘性，即营业收入上升时，企业税负上升幅度大于营业收入同比下降时企业税负下降幅度。进一步检验企业税负粘性程度，发现自2009年以后，由于实施一系列税制结构改革措施，企业税负粘性程度明显下降，表明税制结构是导致企业税负粘性的重要原因，近年税制改革更深层次的作用还在于降低了企业税负粘性程度；同时，由于具有政治成本和税收征管优势，国有企业税负粘性程度低于非国有企业，我国应继续深化国有企业改革，规范税收征管，缩小企业税负粘性的股权性质异质差异，促进税收公平。 Abstract: The present study, by taking as samples China's listed companies in manufacturing industry, and by employing multivariate mixed regression as analytical model, tests and analyzes the change of their tax burden. The results show that the change of tax burden is sticky, which means that, when revenue rises, the increase in tax burden of the enterprise is larger than the decrease rate when revenue goes down in same ratio. Further tests on the degree of the stickiness show that, owing to the execution of a series of reform measures in tax structure since 2009, the degree of stickiness of tax burden has decreased remarkably, which suggests that the structure of tax system is a vital reason that leads to the stickiness of the industry's tax burden. Moreover, due to advantages in political cost and tax collection and management, the tax stickiness of state-owned enterprises is lower than that of non-state-owned enterprises. China should further deepen the reform of state-owned enterprises, standardize tax collection and management, minimize the heterogeneity of enterprises' tax stickiness due to different ownership, and promote tax fairness

Abstract:
Investigations of low energy transfer trajectories are important for both celestial mechanics and astronautics. Methodologies using the theories from dynamical systems are developed in recent years. This paper investigates the dynamics of the earth--moon system. Low energy transfer trajectories are solved numerically by employing a hybrid strategy: first, a genetic hide and seek method performs a search in large domain to confine the global minimum $f({\eta})$ (objective function) region; then, a deterministic Nelder--Mead method is utilized to refine the minimum quickly. Some transfer trajectories of the spacecraft in the earth--moon system are successfully simulated which verify the desired efficiency and robustness of the method of this paper.

Abstract:
Periodic classical trajectories are of fundamental importance both in classical and quantum physics. Here we develop path integral techniques to investigate such trajectories in an arbitrary, not necessarily energy conserving hamiltonian system. In particular, we present a simple derivation of a lower bound for the number of periodic classical trajectories.

Abstract:
We show that a strongly perturbed quantum system, being a semiclassical system characterized by the Wigner-Kirkwood expansion for the propagator, has the same expansion for the eigenvalues as for the WKB series. The perturbation series is rederived by the duality principle in perturbation theory.

Abstract:
We investigate mushroom billiards, a class of dynamical systems with sharply divided phase space. For typical values of the control parameter of the system $\rho$, an infinite number of marginally unstable periodic orbits (MUPOs) exist making the system sticky in the sense that unstable orbits approach regular regions in phase space and thus exhibit regular behaviour for long periods of time. The problem of finding these MUPOs is expressed as the well known problem of finding optimal rational approximations of a real number, subject to some system-specific constraints. By introducing a generalized mushroom and using properties of continued fractions, we describe a zero measure set of control parameter values $\rho\in(0,1)$ for which all MUPOs are destroyed and therefore the system is less sticky. The open mushroom (billiard with a hole) is then considered in order to quantify the stickiness exhibited and exact leading order expressions for the algebraic decay of the survival probability function $P(t)$ are calculated for mushrooms with triangular and rectangular stems.

Abstract:
We present a variant of a Global Navigation Satellite System called a Relativistic Positioning System (RPS), which is based on emission coordinates. We modelled the RPS dynamics in a space-time around Earth, described by a perturbed Schwarzschild metric, where we included the perturbations due to Earth multipoles (up to the 6th), the Moon, the Sun, Venus, Jupiter, solid tide, ocean tide, and Kerr rotation effect. The exchange of signals between the satellites and a user was calculated using a ray-tracing method in the Schwarzschild space-time. We find that positioning in a perturbed space-time is feasible and is highly accurate already with standard numerical procedures: the positioning algorithms used to transform between the emission and the Schwarzschild coordinates of the user are very accurate and time efficient -- on a laptop it takes 0.04 s to determine the user's spatial and time coordinates with a relative accuracy of $10^{-28}-10^{-26}$ and $10^{-32}-10^{-30}$, respectively.