Progress on Hardy-type Inequalities

This paper surveys some of our recent progress on Hardy-type inequa\-lities which consist of a well-known topic in Harmonic Analysis. In the first section, we recall the original probabilistic motivation dealing with the stability speed in terms of the $L^2$-theory. A crucial application of a result by Fukushima and Uemura (2003) is included. In the second section, the non-linear case (a general Hardy-type inequality) is handled with a direct and analytic proof. In the last section, it is illustrated that the basic estimates presented in the first two sections can still be improved considerably.Comment: 13 pages, 5 figures, Festschrift Masatoshi Fukushima: In Honor of Masatoshi Fukushima's Sanju. World Scientific, 201

Models of Neutrino Masses and Mixing

Neutrino physics has entered an era of precision measurements. With these precise measurements, we may be able to distinguish different models that have been constructed to explain the small neutrino masses and the large mixing among them. In this talk, I review some of the existing theoretical models and their predictions for neutrino oscillations.Comment: Talk presented at the 2nd International Colliders to Cosmic Rays Conference (C2CR07), Lake Tahoe, CA, February 25 - March 1, 2007; 8 pages; 2 figure

General estimate of the first eigenvalue on manifolds

Ten sharp lower estimates of the first non-trivial eigenvalue of Laplacian on compact Riemannian manifolds are reviewed and compared. An improved variational formula, a general common estimate, and a new sharp one are added. The best lower estimates are now updated. The new estimates provide a global picture of what one can expect by our approach.Comment: 19 page