On the blow-up structure for the generalized periodic Camassa-Holm and Degasperis-Procesi equations

Considered herein are the generalized Camassa-Holm and Degasperis-Procesi equations in the spatially periodic setting. The precise blow-up scenarios of strong solutions are derived for both of equations. Several conditions on the initial data guaranteeing the development of singularities in finite time for strong solutions of these two equations are established. The exact blow-up rates are also determined. Finally, geometric descriptions of these two integrable equations from non-stretching invariant curve flows in centro-equiaffine geometries, pseudo-spherical surfaces and affine surfaces are given.Comment: 26 page

Statistical Constraints on the Error of the Leptonic CP Violation of Neutrinos

A constraint on the error of leptonic CP violation, which require the phase $\delta_{CP}$ to be less than $\pi/4$ for it to be distinguishable on a $2\pi$ cycle, is presented. Under this constraint, the effects of neutrino detector 's distance, beam energy, and energy resolution are discussed with reference to the present values of these parameters in experiments. Although an optimized detector performances can minimize the deviation to yield a larger distinguishable range of the leptonic CP phase on a $2\pi$ cycle, it is not possible to determine an arbitrary leptonic CP phase in the range of $2\pi$ with the statistics from a single detector because of the existence of two singular points. An efficiency factor $\eta$ is defined to characterize the distinguishable range of $\delta_{CP}$. To cover the entire possible $\delta_{CP}$ range, a combined efficiency factor $\eta^*$ corresponding to multiple sets of detection parameters with different neutrino beam energies and distances is proposed. The combined efficiency factors $\eta^*$ of various major experiments are also presented.Comment: 9 pages, 5 figure