A probabilistic method for gradient estimates of some geometric flows

In general, gradient estimates are very important and necessary for deriving convergence results in different geometric flows, and most of them are obtained by analytic methods. In this paper, we will apply a stochastic approach to systematically give gradient estimates for some important geometric quantities under the Ricci flow, the mean curvature flow, the forced mean curvature flow and the Yamabe flow respectively. Our conclusion gives another example that probabilistic tools can be used to simplify proofs for some problems in geometric analysis.Comment: 22 pages. Minor revision to v1. Accepted for publication in Stochastic Processes and their Application

A Levinasian Reconstruction of the Political Significance of Vulnerability

The concept of vulnerability has been renewed in meaning and importance over recent decades. Scholars such as Judith Butler, Martha Fineman and Pamela Sue Anderson have endeavored to redeem vulnerability from its traditional signification as a negative individual condition, and to reveal the positive meaning of vulnerability as a transformative call for solidarity, equality and love. In this paper we examine the newly constructed positive understanding of vulnerability, and argue that the current way of pursuing this positive understanding affirms a merely functional positivity. In the recent accounts, vulnerability is still a status to avoid, yet functions positively as a corrective force to the environment that produces vulnerability. We will try to find an essential way of designating the positivity of vulnerability by revisiting the philosophical discussion on vulnerability in Emmanuel Levinas. We will argue that Levinas’s notion of vulnerability is positive in a sense which goes beyond the purely functional; it is seen as essential for his definition of humanity. Yet compared to the contemporary discussion, Levinas’s notion of vulnerability lacks a direct social political meaning. We will tentatively explore how the essential positivity drawn from Levinas can provide a new way to construct the political significance of vulnerability

On Optimal Service Differentiation in Congested Network Markets

As Internet applications have become more diverse in recent years, users having heavy demand for online video services are more willing to pay higher prices for better services than light users that mainly use e-mails and instant messages. This encourages the Internet Service Providers (ISPs) to explore service differentiations so as to optimize their profits and allocation of network resources. Much prior work has focused on the viability of network service differentiation by comparing with the case of a single-class service. However, the optimal service differentiation for an ISP subject to resource constraints has remained unsolved. In this work, we establish an optimal control framework to derive the analytical solution to an ISP's optimal service differentiation, i.e. the optimal service qualities and associated prices. By analyzing the structures of the solution, we reveal how an ISP should adjust the service qualities and prices in order to meet varying capacity constraints and users' characteristics. We also obtain the conditions under which ISPs have strong incentives to implement service differentiation and whether regulators should encourage such practices