A new generalization of the Takagi function

We consider a one-parameter family of functions $\{F(t,x)\}_{t}$ on $[0,1]$ and partial derivatives $\partial_{t}^{k} F(t, x)$ with respect to the parameter $t$. Each function of the class is defined by a certain pair of two square matrices of order two. The class includes the Lebesgue singular functions and other singular functions. Our approach to the Takagi function is similar to Hata and Yamaguti. The class of partial derivatives $\partial_{t}^{k} F(t, x)$ includes the original Takagi function and some generalizations. We consider real-analytic properties of $\partial_{t}^{k} F(t, x)$ as a function of $x$, specifically, differentiability, the Hausdorff dimension of the graph, the asymptotic around dyadic rationals, variation, a question of local monotonicity and a modulus of continuity. Our results are extensions of some results for the original Takagi function and some generalizations.Comment: 22 pages, 2 figures. The structure of paper has been changed significantl

Effect of the smaller mass-squared difference for the long base-line neutrino experiments

Usually, neutrino oscillation experiments are analyzed within the two-flavor framework which is governed by 1 mass-squared difference and 1 mixing angle. But there are 6 parameters,2 mass-squared differences, 3 mixing angles, and 1 CP phase within the three-flavor framework.In this article,we estimate the effect from the smaller mass-squared difference,the other mixing angles, and the CP phase, which we call three-flavor effect, for the determination of the mass-squared difference and the mixing angle from the nu_mu's survival and transition probability with the two-flavor analysis.It is found that the mass-squared difference from the two-flavor analysis is slightly shifted from the larger mass-squared difference by the three-flavor effect.The order of magnitude of the three-flavor effect for the mass-squared difference is comparable with that of the expected error for the mass-squared difference of the two-flavor analysis in the future long base-line neutrino oscillation experiments.The CP phase dependence of the nu_mu to nu_etransition probability is also shown.Comment: 13 pages, no figure new reference added, correct typos, Accepted by Progress of Theoretical Physic