Addressing the Node Discovery Problem in Fog Computing

In recent years, the Internet of Things (IoT) has gained a lot of attention due to connecting various sensor devices with the cloud, in order to enable smart applications such as: smart traffic management, smart houses, and smart grids, among others. Due to the growing popularity of the IoT, the number of Internet-connected devices has increased significantly. As a result, these devices generate a huge amount of network traffic which may lead to bottlenecks, and eventually increase the communication latency with the cloud. To cope with such issues, a new computing paradigm has emerged, namely: fog computing. Fog computing enables computing that spans from the cloud to the edge of the network in order to distribute the computations of the IoT data, and to reduce the communication latency. However, fog computing is still in its infancy, and there are still related open problems. In this paper, we focus on the node discovery problem, i.e., how to add new compute nodes to a fog computing system. Moreover, we discuss how addressing this problem can have a positive impact on various aspects of fog computing, such as fault tolerance, resource heterogeneity, proximity awareness, and scalability. Finally, based on the experimental results that we produce by simulating various distributed compute nodes, we show how addressing the node discovery problem can improve the fault tolerance of a fog computing system

Tensor decomposition and homotopy continuation

A computationally challenging classical elimination theory problem is to compute polynomials which vanish on the set of tensors of a given rank. By moving away from computing polynomials via elimination theory to computing pseudowitness sets via numerical elimination theory, we develop computational methods for computing ranks and border ranks of tensors along with decompositions. More generally, we present our approach using joins of any collection of irreducible and nondegenerate projective varieties $X_1,\ldots,X_k\subset\mathbb{P}^N$ defined over $\mathbb{C}$. After computing ranks over $\mathbb{C}$, we also explore computing real ranks. Various examples are included to demonstrate this numerical algebraic geometric approach.Comment: We have added two examples: A Coppersmith-Winograd tensor, Matrix multiplication with zeros. (26 pages, 1 figure