Integer colorings with forbidden rainbow sums

For a set of positive integers $A \subseteq [n]$, an $r$-coloring of $A$ is rainbow sum-free if it contains no rainbow Schur triple. In this paper we initiate the study of the rainbow Erd\H{o}s-Rothchild problem in the context of sum-free sets, which asks for the subsets of $[n]$ with the maximum number of rainbow sum-free $r$-colorings. We show that for $r=3$, the interval $[n]$ is optimal, while for $r\geq8$, the set $[\lfloor n/2 \rfloor, n]$ is optimal. We also prove a stability theorem for $r\geq4$. The proofs rely on the hypergraph container method, and some ad-hoc stability analysis.Comment: 20 page

Influences of Suspended Equipment under Car Body on High-speed Train Ride Quality

AbstractA vertical dynamics model of rigid-flexible coupling railway vehicle including suspended device under chassis is built. Using this model, the influences of devices suspending methods on ride quality of high-speed train are analyzed with the covariance method. Results show that the reasonable parameters of the equipments, which are used to suspend devices under car body, can effectively suppress the vibration of the flexible car body. And the suspending equipment stiffness dominates in the anti-vibration effects. Devices with elastic suspending would better be close to the centre of car body and the bigger the device mass is, the better the ride quality of the car body centre will be achieved. The optimal static deflection point of elastic suspending equipment will change and fluctuate with the change of the vehicle running speed. With the static deflection of about 6mm, the high-speed train in this paper will achieve better ride quality and the vibration of suspended device will be acceptable

BGP-Multipath Routing in the Internet

BGP-Multipath (BGP-M) is a multipath routing technique for load balancing. Distinct from other techniques deployed at a router inside an Autonomous System (AS), BGP-M is deployed at a border router that has installed multiple inter-domain border links to a neighbour AS. It uses the equal-cost multi-path (ECMP) function of a border router to share traffic to a destination prefix on different border links. Despite recent research interests in multipath routing, there is little study on BGP-M. Here we provide the first measurement and a comprehensive analysis of BGP-M routing in the Internet. We extracted information on BGP-M from query data collected from Looking Glass (LG) servers. We revealed that BGP-M has already been extensively deployed and used in the Internet. A particular example is Hurricane Electric (AS6939), a Tier-1 network operator, which has implemented >1,000 cases of BGP-M at 69 of its border routers to prefixes in 611 of its neighbour ASes, including many hyper-giant ASes and large content providers, on both IPv4 and IPv6 Internet. We examined the distribution and operation of BGP-M. We also ran traceroute using RIPE Atlas to infer the routing paths, the schemes of traffic allocation, and the delay on border links. This study provided the state-of-the-art knowledge on BGP-M with novel insights into the unique features and the distinct advantages of BGP-M as an effective and readily available technique for load balancing.Comment: 38 pages, 8 figures, 8 table