In the present paper, we discuss two categorical characterizations of local fields. We first prove that a certain full subcategory of the category of finite flat coverings of the spectrum of the ring of integers of a local field equipped with coherent modules completely determines the isomorphism class of the local field. Next, we also prove that a certain full subcategory of the category of irreducible schemes which are finite over the spectrum of the ring of integers of a local field completely determines the isomorphism class of the local field.

The Besicovitch covering theorem is well known to be the useful tools in many fields of analysis. Federer extended the result of Besicovitch to a directionally limited metric space. In this paper, we prove the Besicovitch covering theorem for parabolic balls in Euclidean space, although the parabolic metric is not directionally limited.

We begin a global study of information geometry. In this article, we describe the geometry of normal distributions by means of positive and negative contact structures associated to the suspension Anosov flows on $Sol^3$-manifolds.

The present paper considers a model selection criterion in regression models using generalized estimating equation (GEE). Using the prediction mean squared error (PMSE) normalized by the covariance matrix, we propose a new model selection criterion called PMSEG that reflects the correlation between responses. Numerical studies reveal that the PMSEG has better performance than previous other criteria for model selection.

In this paper we show the validity of Stein’s interpolation theorem on variable exponent Morrey spaces.

In this paper, we prove some uniqueness results which improve and generalize several earlier works. Also, we prove a value distribution result concerning $f^{(k)}$ which is related to a conjecture of Fang and Wang [A note on the conjectures of Hayman, Mues and Gol’dberg, Comp. Methods, Funct. Theory (2013) 13, 533–543].

The existence of affine resolvable block designs has been discussed since 1942 in the literature (cf. Bose (1942), Clatworthy (1973), Raghavarao (1988)). Kadowaki and Kageyama (2009, 2010, 2012) obtained a number of results on combinatorics for the existence of an affine resolvable SRGD design. In this paper, a new existence result is shown as a generalization of Theorem 3.3.3 given in Kadowaki and Kageyama (2009, 2010). Furthermore, another existence result is shown as a conditional converse of Theorem 3.3.3 and also a generalization of Theorem 3.3.4, both theorems given in Kadowaki and Kageyama (2009, 2010).

In this paper, we study about estimating the probabilities of misclassification in the high-dimensional data. In many cases, the cross-validation (CV) is often used for estimations of the probabilities of misclassification. CV provides a nearly unbiased estimate, using the original data when the sample sizes are large. On the other hand, the properties of CV are not well-known when the dimension is large as compared to the sample sizes. Therefore, we investigate asymptotic properties of CV when the dimension and the sample sizes tend to be large. Furthermore, we suggest the three methods for correcting the bias by using CV which is usable in the high-dimensional data. We show performances of the estimators in the simulation studies.

In the present paper, we study the pseudo-Hermitian almost CR structure of unit tangent sphere bundle $T_{1}M$ over a Riemannian manifold $M$. Then we prove that if the unit tangent sphere bundle $T_{1}M$ is pseudo-Einstein, that is, the pseudo- Hermitian Ricci tensor is proportional to the Levi form, then the base manifold $M$ is Einstein. Moreover, when $\dim M = 3$ or $4$, we prove that $T_{1}M$ is pseudo-Einstein if and only if $M$ is of constant curvature 1.