This paper completes our previous one in the same journal (vol. 42, pp. 37– 75). Let $\mathscr{C}$ be a covering of the plane by disjoint complete folding curves which satisfies the local isomorphism property. We show that $\mathscr{C}$ is locally isomorphic to an essentially unique covering generated by an $\infty$-folding curve. We prove that $\mathscr{C}$ necessarily consists of 1, 2, 3, 4 or 6 curves. We give examples for each case; the last one is realized if and only if $\mathscr{C}$ is generated by the alternating folding curve or one of its successive antiderivatives. We also extend the results of our previous paper to another class of paperfolding curves introduced by M. Dekking.

A $q$-analogue of a partition identity is presented.

In this paper, we reconstruct Kuperberg’s $G_2$ web space [5, 6]. We introduce a new web diagram (a trivalent graph with only double edges) and new relations between Kuperberg’s web diagrams and the new web diagram. Using the web diagrams, we give crossing formulas for the $R$-matrices associated to some irreducible representations of $U_q(G_2)$ and calculate $G_2$ quantum link invariants for generalized twist links.

In this paper we obtain a higher order asymptotic unbiased estimator for the expected probability of misclassification (EPMC) of the linear discriminant function when both the dimension and the sample size are large. Moreover, we evaluate the mean squared error of our estimator. We also present a numerical comparison between the performance of our estimator and that of the other estimators based on Okamoto (1963, 1968) and Fujikoshi and Seo (1998). It is shown that the bias and the mean squared error of our estimator are less than those of the other estimators.

The purpose of this paper is to investigate the geometric structure of regular overdetermined systems of second order with two independent and one dependent variables from the point of view of the rank two prolongation. Utilizing this prolongation, we characterize the type of overdetermined systems and clarify the specificity for each type. We also give systematic methods for constructing the geometric singular solutions by analyzing a decomposition of this prolongation. As an application, we determine the geometric singular solutions of Cartan’s overdetermined system.

We study complex spatial quartic surfaces with simple singularities up to equisingular deformations; as a first step, give a complete equisingular deformation classification of non-special simple quartic surfaces.