We are concerned with Stokes equations in the half-space or in an exterior domain of $\mathbb{R}^n$ when slip conditions are imposed on the boundary. We present a mixed velocity-pressure formulation and we show its well posedness. A weighted variant of Korn’s inequality in unbounded domains is the cornerstone of our approach.

A strongly nonperiodic tiling is defined as a tiling that does not admit infinite cyclic symmetry. The purpose of this article is to construct, up to isomorphism, uncountably many strongly nonperiodic hyperbolic tilings with a single vertex configuration by a hyperbolic rhombus tile. We use a tile found by Margulis and Mozes [5], which admits tilings, but no tiling with a compact fundamental domain.

We study Besov and Triebel–Lizorkin space estimates for fractional diffusion. We measure the smoothing effect of the fractional heat flow in terms of the Besov and Triebel–Lizorkin scale. These estimates have many applications to various partial differential equations.

Let $X$ be a complex smooth projective variety of dimension 3, and let $L_1$ and $L_2$ be ample line bundles on $X$. In this paper we classify $(X, L_{1}, L_{2})$ with $h^{0}(K_{X}+L_{1}+L_{2})=1$.

In this paper, we prove that a localization principle for biholomorphic mappings between equidimensional Fock-Bargmann-Hartogs domains holds. As an application of this, we show that any proper holomorphic mapping between two equidimensional Fock-Bargmann-Hartogs domains satisfying some condition is necessarily a biholomorphic mapping.

By means of the polynomial argument, a class of cubic sums of $q$-binomial coefficients are evaluated in closed forms.

This paper considers optimization of the ridge parameters in generalized ridge regression (GRR) by minimizing a model selection criterion. GRR has a major advantage over ridge regression (RR) in that a solution to the minimization problem for one model selection criterion, i.e., Mallows’ $C_p$ criterion, can be obtained explicitly with GRR, but such a solution for any model selection criteria, e.g., $C_p$ criterion, cross-validation (CV) criterion, or generalized CV (GCV) criterion, cannot be obtained explicitly with RR. On the other hand, $C_p$ criterion is at a disadvantage compared to CV and GCV criteria because a good estimate of the error variance is required in order for $C_p$ criterion to work well. In this paper, we show that ridge parameters optimized by minimizing GCV criterion can also be obtained by closed forms in GRR. We can overcome one disadvantage of GRR by using GCV criterion for the optimization of ridge parameters. By using the result, we propose a principle component regression hybridized with the GRR that is a new method for a linear regression with highdimensional explanatory variables.

We give a good reduction criterion for proper polycurves with sections, i.e., successive extensions of family of curves with section, under a mild assumption. This criterion is a higher dimensional version of the good reduction criterion for hyperbolic curves given by Oda and Tamagawa.