The behavior of real exchange rates: the case of Japan

The study examines the convergence rate of mean reversion by contrasting the estimated half-life of real exchange rate (RER). We employ an extensive monthly consumer price index (CPI)-based product price’s panel for Japan (the U.S. as the num´eraire). We find that the disaggregated RERs are persistent due to the cross-sectional dependence problems. By controlling common correlated effects, the estimated half-life for all goods may fall to as low as 2.54 years, below the consensus view of 3 to 5 years summarized by Rogoff (1996). After correcting the small-sample bias, the estimated half-life of deviations from purchasing power parity (PPP) increase by 1.03 year. Our findings also support that the half-life of mean reversion of RER is about 3.55 years for traded goods, about 0.11 year lower than non-traded goods. We also show that traded goods and non-traded goods perform distinct distributions of persistence

Limit of Fractional Power Sobolev Inequalities

We derive the Moser-Trudinger-Onofri inequalities on the 2-sphere and the 4-sphere as the limiting cases of the fractional power Sobolev inequalities on the same spaces, and justify our approach as the dimensional continuation argument initiated by Thomas P. Branson.Comment: 17 page

Some higher order isoperimetric inequalities via the method of optimal transport

In this paper, we establish some sharp inequalities between the volume and the integral of the $k$-th mean curvature for $k+1$-convex domains in the Euclidean space. The results generalize the classical Alexandrov-Fenchel inequalities for convex domains. Our proof utilizes the method of optimal transportation.Comment: 21 page

Some Progress in Conformal Geometry

This is a survey paper of our current research on the theory of partial differential equations in conformal geometry. Our intention is to describe some of our current works in a rather brief and expository fashion. We are not giving a comprehensive survey on the subject and references cited here are not intended to be complete. We introduce a bubble tree structure to study the degeneration of a class of Yamabe metrics on Bach flat manifolds satisfying some global conformal bounds on compact manifolds of dimension 4. As applications, we establish a gap theorem, a finiteness theorem for diffeomorphism type for this class, and diameter bound of the $\sigma_2$-metrics in a class of conformal 4-manifolds. For conformally compact Einstein metrics we introduce an eigenfunction compactification. As a consequence we obtain some topological constraints in terms of renormalized volumes.Comment: This is a contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA