In this chapter, we give a selective review of the nonparametric modeling methods using Cox's type of models in survival analysis. We first introduce Cox's model (Cox 1972) and then study its variants in the direction of smoothing. The model fitting, variable selection, and hypothesis testing problems are addressed. A number of topics worthy of further study are given throughout this chapter.

This paper is about the propagation of the singularities in the solutions to the Cauchy problem of the spatially inhomogeneous Boltzmann equation with angular cutoff assumption. It is motivated by the work of BoudinDesvillettes on the propagation of singularities in solutions near vacuum. It shows that for the solution near a global Maxwellian, singularities in the initial data propagate like the free transportation. Precisely, the solution is the sum of two parts in which one keeps the singularities of the initial data and the other one is regular with locally bounded derivatives of fractional order in some Sobolev space. In addition, the dependence of the regularity on the cross-section is also given.

Hidden field equation (HFE) multivariable cryptosystems were first suggested by Patarin. Kipnis and Shamir showed that to make the cryptosystem secure, a special parameter D of any HFE cryptosystem can not be too small. Consequently Kipnis, Patarin and Goubin proposed an enhanced variant of the HFE cryptosystem by combining the idea of Oil and Vinegar construction with the HFE construction. Essentially they “perturb” the HFE system with some external variables. In this paper, we will first present a new cryptanalysis method for the HFEv schemes. We then use the idea of internal perturbation to build a new cryptosystem, an internally perturbed HFE cryptosystem (IPHFE).

We give an explicit formula for (T-equivariant) 3-pointed genus zero Gromov-Witten invariants for G/B. We derive it by finding an explicit formula for the Pontryagin product on the equivariant homology of the based loop group ΩK.

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