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This work is concerned with marginal sure independence feature screening for ultrahigh dimensional discriminant analysis. The response variable is categorical in discriminant analysis. This enables us to use the conditional distribution function to construct a new index for feature screening. In this article, we propose a marginal feature screening procedure based on empirical conditional distribution function. We establish the sure screening and ranking consistency properties for the proposed procedure without assuming any moment condition on the predictors. The proposed procedure enjoys several appealing merits. First, it is model-free in that its implementation does not require specification of a regression model. Second, it is robust to heavy-tailed distributions of predictors and the presence of potential outliers. Third, it allows the categorical response having a diverging number of classes in the order of O(n^κ ) with some κ ≥ 0. We assess the finite sample property of the proposed procedure byMonte Carlo simulation studies and numerical comparison.We further illustrate the proposed methodology by empirical analyses of two real-life datasets. Supplementary materials for this article are available online.

We establish the full global non-linear stability of the Kerr–de Sitter family of black holes, as solutions of the initial value problem for the Einstein vacuum equations with positive cosmological constant, for small angular momenta, and without any symmetry assumptions on the initial data. We achieve this by extending the linear and non-linear analysis on black hole spacetimes described in a sequence of earlier papers by the authors: we develop a general framework which enables us to deal systematically with the diffeomorphism invariance of Einstein’s equations. In particular, the iteration scheme used to solve Einstein’s equations automatically finds the parameters of the Kerr–de Sitter black hole that the solution is asymptotic to, the exponentially decaying tail of the solution, and the gauge in which we are able to find the solution; the gauge here is a wave map/DeTurck type gauge, modified by source terms which are treated as unknowns, lying in a suitable finite-dimensional space.

Let W be an extended affine Weyl group. We prove that the minimal length elements W of any conjugacy class W of W satisfy some nice properties, generalizing results of Geck and Pfeiffer [<i>On the irreducible characters of Hecke algebras</i>, Adv. Math. <b>102</b> (1993), 7994] on finite Weyl groups. We also study a special class of conjugacy classes, the straight conjugacy classes. These conjugacy classes are in a natural bijection with the Frobenius-twisted conjugacy classes of some W -adic group and satisfy additional interesting properties. Furthermore, we discuss some applications to the affine Hecke algebra W

In this paper we analyze the Kahrobaei-Lam-Shpilrain (KLS) key exchange protocols that use extensions by endomorpisms of matrices over a Galois field proposed in \cite{Kahrobaei-Lam-Shpilrain:2014}. We show that both protocols are vulnerable to a simple linear algebra attack.