A pseudo-Kähler manifold is a natural generalization of a Kähler manifold. It is well-known that any generalized flag manifold has pseudo-Kähler metrics. Moreover, there exists a T-root system corresponding to a generalized flag manifold. In this paper, we investigate the signatures of invariant pseudo-Kähler metrics on a generalized flag manifold of which the T-root system becomes one of the irreducible reduced root systems (in general, a T-root system is not an irreducible reduced root system).

We show that the ring of modular forms with characters for the even unimodular lattice of signature (2,18) is obtained from the invariant ring of $\mathrm {Sym}(\mathrm {Sym}^8(V)\oplus \mathrm {Sym}^{12}(V))$ with respect to the action of $\mathrm{SL}(V)$ by adding a Borcherds product of weight 132 with one relation of weight 264, where $V$ is a 2-dimensional $\mathbb C$-vector space. The proof is based on the study of the moduli space of elliptic K3 surfaces with a section.

A criterion for the existence of a plane model with two non-smooth Galois points for algebraic curves is presented, which is a generalization of Fukasawa's criterion for two smooth Galois points. Owing to this generalized criterion, multiplicities and order sequences at Galois points can be described in detail.

Abstract. Let $\varphi : \mathbb R^n \times [0,\infty) \rightarrow [0,\infty)$ satisfy that $\varphi(x,\cdot)$ is an Orlicz function for any given $x \in \mathbb R^n$, and $\varphi(\cdot,t)$ is a Muckenhoupt $A_\infty$ weight uniformly in $t \in (0,\infty)$. The weak Musielak-Orlicz Hardy space $WH^\varphi(\mathbb R^n)$ is defined to be the set of all tempered distributions such that their grand maximal functions belong to the weak Musielak-Orlicz space $WL^\varphi(\mathbb R^n)$. In this paper, we discuss the boundedness of the Calderon-Zygmund operator with variable kernel from $WH^\varphi(\mathbb R^n)$ to $WL^\varphi(\mathbb R^n)$. These results are new even for the classical weighted weak Hardy space and probably new for the classical weak Hardy space.

In this paper, we consider a model selection criterion using the GEE method including unknown scale and correlation parameters. We propose a model selection criterion for selecting variables and a working correlation structure. Under some regularity conditions, we showed that our criterion is the same as the criterion proposed by Inatsu and Imori [8]. A numerical study reveals that we can reduce the prediction error by selecting both variables and a working correlation structure.

Using the rank of the Mordell-Weil group $E(\mathbb {Q})$ of an elliptic curve $E$ over $\mathbb Q$, we give a lower bound of the class number of the number field $\mathbb {Q}(E[p^{n}])$ generated by $p^n$-division points of $E$ when the curve $E$ does not possess a $p$-adic point of order $p: E(\mathbb {Q}_p)[p]=0$.

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.

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’ Cp criterion, can be obtained explicitly with GRR, but such a solution for any model selection criteria, e.g., Cp criterion, cross-validation (CV) criterion, or generalized CV (GCV) criterion, cannot be obtained explicitly with RR. On the other hand, Cp criterion is at a disadvantage compared to CV and GCV criteria because a good estimate of the error variance is required in order for Cp 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. (Last Modified: May 17, 2013)

For a class of positive homogeneously presented cancellative monoids whose heights are greater than or equal to 2, we will present several explicit calculations of the skew growth functions for them. By the inversion formula, the spherical growth functions for them can be determined. For most of them, the direct calculations are not known. The datum of certain lemmas for proving the cancellativity of the monoids are indispensable to the calculations of the skew growth functions. By improving the technique to show the lemmas, we succeed in the calculations.

In this paper, we formulate a procedure to obtain a generalization of Milnor frames for left-invariant pseudo-Riemannian metrics on a given Lie group. This procedure is an analogue of the recent studies on left-invariant Riemannian metrics, and is based on the moduli space of left-invariant pseudo-Riemannian metrics. As one of applications, we show that any left-invariant pseudo-Riemannian metrics of arbitrary signature on the Lie groups of real hyperbolic spaces have constant sectional curvatures.