Busemann functions and barrier functions

We show that Busemann functions on a smooth, non-compact, complete, boundaryless, connected Riemannian manifold are viscosity solutions with respect to the Hamilton-Jacobi equation determined by the Riemannian metric and consequently they are locally semi-concave with linear modulus. We also analysis the structure of singularity sets of Busemann functions. Moreover we study barrier functions, which are analogues to Mather's barrier functions in Mather theory, and provide some fundamental properties. Based on barrier functions, we could define some relations on the set of lines and thus classify them. We also discuss some initial relations with the ideal boundary of the Riemannian manifold.Comment: comments are welcome

Study on the Performance of TCP over 10Gbps High Speed Networks

Internet traffic is expected to grow phenomenally over the next five to ten years. To cope with such large traffic volumes, high-speed networks are expected to scale to capacities of terabits-per-second and beyond. Increasing the role of optics for packet forwarding and transmission inside the high-speed networks seems to be the most promising way to accomplish this capacity scaling. Unfortunately, unlike electronic memory, it remains a formidable challenge to build even a few dozen packets of integrated all-optical buffers. On the other hand, many high-speed networks depend on the TCP/IP protocol for reliability which is typically implemented in software and is sensitive to buffer size. For example, TCP requires a buffer size of bandwidth delay product in switches/routers to maintain nearly 100\% link utilization. Otherwise, the performance will be much downgraded. But such large buffer will challenge hardware design and power consumption, and will generate queuing delay and jitter which again cause problems. Therefore, improve TCP performance over tiny buffered high-speed networks is a top priority. This dissertation studies the TCP performance in 10Gbps high-speed networks. First, a 10Gbps reconfigurable optical networking testbed is developed as a research environment. Second, a 10Gbps traffic sniffing tool is developed for measuring and analyzing TCP performance. New expressions for evaluating TCP loss synchronization are presented by carefully examining the congestion events of TCP. Based on observation, two basic reasons that cause performance problems are studied. We find that minimize TCP loss synchronization and reduce flow burstiness impact are critical keys to improve TCP performance in tiny buffered networks. Finally, we present a new TCP protocol called Multi-Channel TCP and a new congestion control algorithm called Desynchronized Multi-Channel TCP (DMCTCP). Our algorithm implementation takes advantage of a potential parallelism from the Multi-Path TCP in Linux. Over an emulated 10Gbps network ruled by routers with only a few dozen packets of buffers, our experimental results confirm that bottleneck link utilization can be much better improved by DMCTCP than by many other TCP variants. Our study is a new step towards the deployment of optical packet switching/routing networks

Testing the Universal Structured Jet Models of Gamma-Ray Bursts by BATSE Observations

Assuming that the observed gamma-ray burst (GRB) rate as a function of redshift is proportional to a corrected star formation rate, we derive the empirical distribution of the viewing angles of long BATSE GRBs, $P^{\rm em}(\theta)$, and the distribution of these bursts in the plane of $\theta$ against redshift, $P^{\rm em}(\theta, z)$, by using a tight correlation between $E_{\gamma}$) and $E_{\rm p}^{'}$). Our results show that $P^{\rm em}(\theta)$ is well fitted by a log-normal distribution centering at $\log \theta/{\rm rad}=-0.76$ with a width of $\sigma_{\log \theta}=0.57$. We test different universal structured jet models by comparing model predictions with our empirical results. To make the comparisons reasonable, an "effective" threshold, which corresponds to the sample selection criteria of the long GRB sample, is used. We find that the predictions of a two-Gaussian jet model are roughly consistent with our empirical results. A brief discussion shows that cosmological effect on the $E_{\gamma}-E_{\rm p}^{'}$ relation does not significantly affect our results, but sample selection effects on this relationship might significantly influence our results.Comment: 5 pages, 6 figures, accepted for publication in A