Cuspidal edges and swallowtails are typical non-degenerate singular points on wave fronts in the Euclidean 3-space. Their first fundamental forms belong to a class of positive semi-definite metrics called “Kossowski metrics”. A point where a Kossowski metric is not positive definite is called a singular point or a semi-definite point of the metric. Kossowski proved that real analytic Kossowski metric germs at their non-parabolic singular points (the definition of “non-parabolic singular point” is stated in the introduction here) can be realized as wave front germs (Kossowski’s realization theorem).

On the other hand, in a previous work with K. Saji, the third and the fourth authors introduced the notion of “coherent tangent bundle”. Moreover, the authors, with M. Hasegawa and K. Saji, proved that a Kossowski metric canonically induces an associated coherent tangent bundle.

In this paper, we shall explain Kossowski’s realization theorem from the viewpoint of coherent tangent bundles. Moreover, as refinements of it, we give a criterion that a given Kossowski metric can be realized as the induced metric of a germ of cuspidal edge (resp. swallowtail or cuspidal cross cap). Several applications of these criteria are given. Also, some remaining problems on isometric deformations of singularities of analytic maps are given at the end of this paper.

This paper is concerned with computable error bounds for asymptotic approximations of the expected probabilities of misclassification (EPMC) of the quadratic discriminant function $Q$. A location and scale mixture expression for $Q$ is given as a special case of a general discriminant function including the linear and quadratic discriminant functions. Using the result, we provide computable error bounds for asymptotic approximations of the EPMC of $Q$ when both the sample size and the dimensionality are large. The bounds are numerically explored. Similar results are given for a quadratic discriminant function $Q_0$ when the covariance matrix is known.

We study a prescribed mean curvature problem where we seek a surface whose mean curvature vector coincides with the normal component of a given vector field. We prove that the problem has a solution near a graphical minimal surface if the prescribed vector field is sufficiently small in a dimensionally sharp Sobolev norm.

This paper is concerned with the selection of explanatory variables in multivariate linear regression. The Akaike’s information criterion and the $C_p$ criterion cannot perform in high-dimensional situations such that the dimension of a vector stacked with response variables exceeds the sample size. To overcome this, we consider two variable selection criteria based on an $L_2$ squared distance with a weighted matrix, namely the scalar-type generalized $C_p$ criterion and the ridge-type generalized $C_p$ criterion. We clarify conditions for their consistency under a hybrid-ultra-highdimensional asymptotic framework such that the sample size always goes to infinity but the number of response variables may not go to infinity. Numerical experiments show that the probabilities of selecting the true subset by criteria satisfying consistency conditions are high even when the dimension is larger than the sample size. Finally, we illuminate the practical utility of these criteria using empirical data.

For a finite subgroup $G$ of $\mathrm {GL}(2, \mathbb C)$, we consider the moduli space $\mathscr M _\theta$ of $G$-constellations. It depends on the stability parameter $\theta$ and if $\theta$ is generic it is a resolution of singularities of $\mathbb C ^2/G$. In this paper, we show that a resolution $Y$ of $\mathbb C ^2/G$ is isomorphic to $\mathscr M _\theta$ for some generic $\theta$ if and only if $Y$ is dominated by the maximal resolution under the assumption that $G$ is abelian or small.