第七届华山青年学者国际论坛——我院分论坛

1月11日14:00腾讯会议ID：516-216-115

报告题目与摘要  

Quantied conditional variance, skewness, and kurtosis by Cornish-Fisher expansion

张宁宁 博士在读香港大学统计学

The conditional variance, skewness, and kurtosis play a central role in time series analysis. These three conditional moments (CMs) are often studied by some parametric models but with two big issues: the risk of model mis-specification and the instability of model estimation. To avoid the above two issues, this paper proposes a novel method to estimate these three CMs by the so-called quantiled CMs (QCMs). The QCM method first adopts the idea of Cornish-Fisher expansion to construct a linear regression model, based on n different estimated conditional quantiles that can be obtained without assuming any parametric forms of the CMs. Next, it computes the QCMs simply and simultaneously at each fixed timepoint by using the ordinary least squares estimator of this regression model. Under regular conditions, the QCMs are shown to be consistent with the convergence rate n−1/2. Simulation studies indicate that the QCMs perform well under different scenarios of estimated conditional quantiles. In the application, the study of QCMs for eight major stock indexes demonstrates the effectiveness of financial rescue plans during the COVID-19 pandemic outbreak, and unveils a new “non-zero kink” phenomenon in the “news impact curve” function for the conditional kurtosis.

 Matrix-analytic methods for solving  Poisson's equation  with applications to  Markov chains of GI/G/1-type

刘金鹏 博士研究生 中南大学

In this talk, we are devoted to developing matrix-analytic methods for solving Poisson's equation for irreducible and positive recurrent discrete-time Markov chains (DTMCs). Two special solutions, including the deviation matrix D and the expected additive-type functional matrix K, will be considered. The results are applied to Markov chains of GI/G/1-type and MAP/G/1 queues with negative customers. Further extensions to continuous-time Markov chains (CTMCs) are also investigated. This talk is based on a joint with professors Yuanyuan Liu and Yiqiang Q. Zhao (Carleton University).

Symplectically flat connections 

Over a symplectic manifold, we call a connection symplectically flat if its curvature is proportional to the symplectic form and this proportion is covariantly a constant. Such connections can be viewed as a generalization of flat connections. We will go through the relationship between these two types of connections, and give a classification of symplectically flat connections.

Complete positivity of comultiplication and primary criteria for unitary categorification 

In this talk, I will introduce our recent work on quantum Fourier analysis. We provide a family of analytic criteria for fusion rings’ unitary categorification, which are stronger than the Schur product criterion. Various examples of fusion rings will be given. Many variations of the criteria such as localized criteria are introduced. These criteria could also be applied as obstructions for principal graphs of subfactors. This work is joint with Zhengwei Liu, Sebastien Palcoux and Jinsong Wu.

Critical Elliptic Boundary Value Problems with Singular Trudinger-Moser Nonlinearities

付世丘 博士 美国佛罗里达理工大学

In this talk, I will introduce two classes of elliptic problems that are critical with respect to singular Trudinger-Moser embedding. The proofs are based on compactness and regularity arguments.
 

Cascading processes, breather and Fermi-Pasta-Ulam-Tsingou recurrence

殷会敏 博士后 香港大学

Modulation instability, breather formation, and the Fermi-Pasta-Ulam-Tsingou recurrence (FPUT) phenomena are studied. Physically, such nonlinear systems arise when the medium is slightly anisotropic, e.g., optical fibers with weak birefringence where the slowly varying pulse envelopes are governed by these coherently coupled Schrödinger equations. The Darboux transformation is used to calculate a class of breathers. A “cascading mechanism”is utilized to elucidate the initial stages of FPUT. More precisely, higher order nonlinear terms that are exponentially small initially can grow rapidly. A breather is formed when the linear mode and higher order ones attain roughly the same magnitude. The conditions for generating various breathers and connections with modulation instability are elucidated. The growth phase then subsides and the cycle is repeated, leading to FPUT. An analytical formula for the time or distance of breather formation is proposed, based on the disturbance amplitude and instability growth rate. Excellent agreement with numerical simulations is achieved. Furthermore, the roles of modulation instability for FPUT are elucidated with illustrative case studies. In particular, depending on whether the second harmonic falls within the unstable band, FPUT patterns with one single or two distinct wavelength(s) are observed.