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We extend the idea of crossvalidation to choose the smoothing parameters of the double-kernel local linear regression for estimating a conditional density. Our selection rule optimises the estimated conditional density function by minimising the integrated squared error. We also discuss three other bandwidth selection rules, an ad hoc method used by Fan et al. (1996), a bootstrap method of Hall et al. (1999) for bandwidth selection in the estimation of conditional distribution functions, modified by Bashtannyk & Hyndman (2001) to cover conditional density functions, and finally a simple approach proposed by Hyndman & Yao (2002). The performance of the new approach is compared with these three methods by simulation studies, and our method performs outstandingly well. The method is illustrated by an application to estimating the transition density and the Value-at-Risk of treasury-bill data.

Detailed studies of four dimensional N= 2 superconformal field theories (SCFT) defined by isolated complete intersection singularities are performed: we compute the Coulomb branch spectrum, Seiberg-Witten solutions and central charges. Most of our theories have exactly marginal deformations and we identify the weakly coupled gauge theory descriptions for many of them, which involve (affine) D and E shaped quiver gauge theories and theories formed from Argyres-Douglas matters. These investigations provide strong evidence for the singularity approach in classifying 4d N= 2 SCFTs.

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We study how to characterize the families of paths between any two nodes s, t in a sensor network with holes. Two paths that can be deformed to one another through local changes are called homotopy equivalent. Two paths that pass around holes in different ways have different homotopy types. With a distributed algorithm we compute an embedding of the network in hyperbolic space by using Ricci flow such that paths of different homotopy types are mapped naturally to paths connecting s with different images of t. Greedy routing to a particular image is guaranteed with success to find a path with a given homotopy type. This leads to simple greedy routing algorithms that are resilient to both local link dynamics and large scale jamming attacks and improve load balancing over previous greedy routing algorithms.