A two end family of solutions for the Inhomogeneous Allen-Cahn equation in R^2

In this work we construct a family of entire bounded solution for the singulary perturbed Inhomogeneous Allen-Cahn Equation \ep^2\div\left(a(x)\nabla u\right)-a(x)F'(u)=0 in $\R^2$, where \ep\to 0. The nodal set of these solutions is close to a "nondegenerate" curve which is asymptotically two non paralell straight lines. Here $F'$ is a double-well potential and $a$ is a smooth positive function. We also provide example of curves and functions $a$ where our result applies. This work is in connection with the results found by Z.Du and B.Lai, Z.Du and C.Gui, and F. Pacard and M. Ritore, in "Transition layers for an inhomogeneus Allen-Cahn equation in Riemannian Manifolds", "Interior layers for an inhomogeneous Allen-Cahn equation", "From the constant mean curvature hypersurfaces to the gradient theory of phase transitions" respectively, where they handle the compact case.Comment: arXiv admin note: text overlap with arXiv:0902.2047 by other author

The Gravity of Globalization

Can changes in the trade of the world’s largest trading countries be considered more global? Or should they be labeled as more regional? We investigated these questions for the G7 countries for the time period from 1980 to 1997. We found that the usual dichotomy of global-regional is not rich enough to answer these questions because globalization can be measured in terms of both physical and cultural distance. Our new taxonomy allows for testing these separate impacts on world trade and suggests that trade changes are best described as regional, though with some qualification. With respect to physical distance, we find that trade is clearly becoming more regional. On the cultural dimension, however, we find conflicting results. These results are robust to a series of tests. We find the same pattern at industry level, except for Paper Products and Motor Vehicles. The regionalization pattern holds for both imports to and exports from the G7, but it is stronger for exports.

Leptonic CP phases near the μ−τ\mu-\tau symmetric limit

The neutrino masses and mixings indicated by current neutrino oscillation experiments suggest that the neutrino mass matrix possesses an approximate $\mu-\tau$ exchange symmetry. In this study, we explore the neutrino parameter space and show that if a small $\mu-\tau$ symmetry breaking is considered, the Majorana $CP$ phases must be unequal and non-zero independently of the neutrino mass scale. Moreover, a small $\mu-\tau$ symmetry breaking favors quasi-degenerate masses. We also show that Majorana phases are strongly correlated with the Dirac $CP$ violating phase. Within this framework, we obtain robust predictions for the values of the Majorana phases when the experimental indications for the Dirac $CP$ phase are used.Comment: 13 pages, 3 figures. Version accepted for publication in PL