The existence of polynomial-size (monotone) formulae is known to be equivalent to the existence of logarithmic-depth (monotone) circuits of bounded fan-in. Anyhow, we shall prove the existence of logarithmic-depth monotone formulae (of bounded fan-in) for majority. Actually, two radically different proofs are known: The first proof uses a rather complicated construction of sorting networks of l...

We develop in this paper an improvement of the method given by S. Bobkov and M. Ledoux in [BL00]. Using the Prékopa-Leindler inequality, we prove a modified logarithmic Sobolev inequality adapted for all measures on Rn, with a strictly convex and super-linear potential. This inequality implies modified logarithmic Sobolev inequality, developed in [GGM05, GGM07], for all uniformly strictly conve...

In this short note, we present a refinement of the logarithmic-geometric mean inequality. As an application of our result, we obtain an operator inequality associated with geometric and logarithmic means.

In this paper, we show that the moving directions of the primal-affine scaling method (with logarithmic barrier function), the dual-affine scaling method (with logarithmic barrier function), and the primal-dual interior point method are merely the Newton directions along three different algebraic "paths" that lead to a solution of the Karush-Kuhn-Tucker conditions of a given linear programming ...

Since its introduction in 1984, R-tree has been proven to be one of the most practical and well-behaved data structures for accommodating dynamic massive sets of geometric objects and conducting a very diverse set of queries on such datasets in real-world applications. This success has led to a variety of versions, each one trying to tune the performance parameters of the original proposal. Amo...