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Xiquan Liang, Piqing Zhao and Ou Bai

Vector Functions and their Differentiation Formulas in 3-dimensional Euclidean Spaces

In this article, we first extend several basic theorems of the operation of vector in 3-dimensional Euclidean spaces. Then three unit vectors: e1, e2, e3 and the definition of vector function in the same spaces are introduced. By dint of unit vector the main operation properties as well as the differentiation formulas of vector function are shown [12].

Open access

Liang Bai and John Y-J. Shyy

Open access

Yaoqi Jia, Guangdong Bai, Prateek Saxena and Zhenkai Liang


The peer-assisted CDN is a new content distribution paradigm supported by CDNs (e.g., Akamai), which enables clients to cache and distribute web content on behalf of a website. Peer-assisted CDNs bring significant bandwidth savings to website operators and reduce network latency for users. In this work, we show that the current designs of peer-assisted CDNs expose clients to privacy-invasive attacks, enabling one client to infer the set of browsed resources of another client. To alleviate this, we propose an anonymous peer-assisted CDN (APAC), which employs content delivery while providing initiator anonymity (i.e., hiding who sends the resource request) and responder anonymity (i.e., hiding who responds to the request) for peers. APAC can be a web service, compatible with current browsers and requiring no client-side changes. Our anonymity analysis shows that our APAC design can preserve a higher level of anonymity than state-of-the-art peer-assisted CDNs. In addition, our evaluation demonstrates that APAC can achieve desired performance gains.