On the Schr\"odinger equations with isotropic and anisotropic fourth-order dispersion

This paper deals with the Cauchy problem associated to the nonlinear fourth-order Schr\"odinger equation with isotropic and anisotropic mixed dispersion. This model is given by the equation $i\partial _{t}u+\epsilon \Delta u+\delta A u+\lambda|u|^\alpha u=0,$ $x\in \mathbb{R}^{n},$ $t\in \mathbb{R},$ where $A$ represents either the operator $\Delta^2$ (isotropic dispersion) or $\sum_{i=1}^d\partial_{x_ix_ix_ix_i},\ 1\leq d<n$ (anisotropic dispersion), and $\alpha, \epsilon, \lambda$ are given real parameters. We obtain local and global well-posedness results in spaces of initial data with low regularity, such as weak-$L^p$ spaces. Our analysis also includes the biharmonic and anisotropic biharmonic equation $(\epsilon=0)$ for which, the existence of self-similar solutions is obtained as consequence of his scaling invariance. In a second part, we investigate the vanishing second order dispersion limit in the framework of weak-$L^p$ spaces. We also analyze the convergence of the solutions for the nonlinear fourth-order Schr\"odinger equation $i\partial _{t}u+\epsilon \Delta u+\delta \Delta^2 u+\lambda|u|^\alpha u=0$, as $\epsilon$ goes to zero, in $H^2$-norm, to the solutions of the corresponding biharmonic equation $i\partial _{t}u+\delta \Delta^2 u+\lambda|u|^\alpha u=0$

Calculating error bars for neutrino mixing parameters

One goal of contemporary particle physics is to determine the mixing angles and mass-squared differences that constitute the phenomenological constants that describe neutrino oscillations. Of great interest are not only the best fit values of these constants but also their errors. Some of the neutrino oscillation data is statistically poor and cannot be treated by normal (Gaussian) statistics. To extract confidence intervals when the statistics are not normal, one should not utilize the value for chisquare versus confidence level taken from normal statistics. Instead, we propose that one should use the normalized likelihood function as a probability distribution; the relationship between the correct chisquare and a given confidence level can be computed by integrating over the likelihood function. This allows for a definition of confidence level independent of the functional form of the !2 function; it is particularly useful for cases in which the minimum of the !2 function is near a boundary. We present two pedagogic examples and find that the proposed method yields confidence intervals that can differ significantly from those obtained by using the value of chisquare from normal statistics. For example, we find that for the first data release of the T2K experiment the probability that chisquare is not zero, as defined by the maximum confidence level at which the value of zero is not allowed, is 92%. Using the value of chisquare at zero and assigning a confidence level from normal statistics, a common practice, gives the over estimation of 99.5%.Comment: 9 pages, 6 figure