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Let M be a compact n-dimensional Riemannian manifold with nonnegative Ricci curvature and mean convex boundary ∂M. Assume that the mean curvature H of the boundary ∂M satisfies H≥(n−1)k>0 for some positive constant k. In this paper, we prove that the distance function d to the boundary ∂M is bounded from above by 1/𝑘 and the upper bound is achieved if and only if M is isometric to an n-dimensional Euclidean ball of radius 1/𝑘.

In this paper we prove the equidistribution of the restriction of the mass of automorphic newforms to a nonsplit torus in the depth aspect. This result is better than the current known results on the similar problem in the eigenvalue aspect. The method is relatively elementary and makes use of the known effective QUE result in the depth aspect.

MutantXL is an algorithm for solving systems of polynomial equations that was proposed at SCC 2008 and improved in PQC 2008. This article gives an overview over the MutantXL algorithm. It also presents experimental results comparing the behavior of the MutantXL algorithm to the $F_4$ algorithm on HFE and randomly generated multivariate systems. In both cases MutantXL is faster and uses less memory than the Magma's implementation of $F_4$.

In this short survey paper, we shall discuss certain recent results in classical gravity. Our main attention will be restricted to two topics in which we have been involved; the positive mass conjecture and its extensions to the case with horizons, including the Penrose conjecture (Part I), and the interaction of gravity with other force fields and quantum-mechanical particles (Part II).