The best constant of $L^p$ Sobolev inequality for a function with Neumann boundary condition is obtained. The best constant is expressed by $L^q$ norm of $M$-th order Bernoulli polynomial. For $L^p$ Sobolev inequality, the equality holds for a function which is written by Green function with Neumann boundary value problem for $(-1)^M(d/dx)^{2M}$.

Generalized ridge (GR) regression for an univariate linear model was proposed simultaneously with ridge regression by Hoerl and Kennard (1970). In this paper, we deal with a GR regression for a multivariate linear model, referred to as a multivariate GR (MGR) regression. From the viewpoint of reducing the mean squared error (MSE) of a predicted value, many authors have proposed several GR estimators consisting of ridge parameters optimized by non-iterative methods. By expanding their optimizations of ridge parameters to the multiple response case, we derive some MGR estimators with ridge parameters optimized by the plug-in method. We analytically compare obtained MGR estimators with existing MGR estimators, and numerical studies are also given for an illustration.

We establish the Artin-Schreier-Witt theory in connection with the unit group scheme of a group algebra, following a method presented by Serre in Groupes algébriques et corps de classes. The argument is developed not only over a field but also over a ring, as generally as possible.

In this paper we give the Weierstrass equations for Jacobian fibrations on the $K3$ surface that is the minimal resolution of the double covering of $\mathbb{P}^2$ ramified along generic six lines.

We establish new oscillation and nonoscillation theorems for the second order quasilinear dynamic equation

(r(t)|y^{\Delta}(t)|^{\alpha-1}y^{\Delta}(t))^{\Delta}+f(t,y^{\sigma}(t))=0

on a time scale $\T$. Our results not only extend the results given in J. Wang, "On second order quasilinear oscillations," Funkcialaj Ekvacioj, 41 (1998), 25-54, but also unify the oscillation and nonoscillation criteria for second order quasilinear differential equations and difference equations.

The second author and M. Sakuma gave a complete characterization of those essential simple loops on a 2-bridge sphere of a 2-bridge link which are nullhomotopic in the link complement. In this paper, we give an alternative proof to this result, by giving a simple proof for the small cancellation conditions of the upper presentations of 2-bridge link groups, which holds the key to the proof the result.