In this paper, we consider representatives of each element of Artin groups of finite type in terms of the standard Artin generator system $\Sigma$ and give a necessary condition for representatives of elements of Artin groups of finite type to be geodesic.

In this paper, we define a class of cross-validatory model selection criteria as an estimator of the predictive risk function based on a discrepancy between a candidate model and the true model. For a vector of unknown parameters, $n$ estimators are required for the definition of the class, where $n$ is the sample size. The $i$th estimator $(i=1,\dots,n)$ is obtained by minimizing a weighted discrepancy function in which the $i$th observation has a weight of $1-\lambda$ and others have weight of $1$. Cross-validatory model selection criteria in the class are specified by the individual $\lambda$. The sample discrepancy function and the ordinary cross-validation (CV) criterion are special cases of the class. One may choose $\lambda$ to minimize the biases. The optimal $\lambda$ makes the bias-corrected CV (CCV) criterion a second-order unbiased estimator for the risk function, while the ordinary CV criterion is a first-order unbiased estimator of the risk function.

We consider a three-species competition-diffusion system, in order to discuss the problem of competitor-mediated coexistence in situations where one exotic competing species invades a system that already contains two strongly competing species. It is numerically shown that, under some conditions, there exist stable non-constant equilibrium solutions that indicate the coexistence of two strongly competing species. This result motivates us to develop a semi-exact representation for finding these equilibrium solutions from an analytical viewpoint.

We shall define the spectral projection on the homogeneous tree $\mathfrak X$, which is an analogue of the one given by Bray for semisimple Lie groups. We shall prove the Paley--Wiener theorem for the spectral projection on $\mathfrak X$. As an application, we present an elementary proof of the Paley--Wiener theorem for the Helgason--Fourier transform on $\mathfrak X$, which was obtained by Cowling and Setti.

We classify real hypersurfaces in a complex space form whose structural reflections are isometries. We also determine real hypersurfaces in a complex space form whose transversal Jacobi operators have constant eigenvalues and at the same time their eigenspaces are parallel (along transversal geodesics).

Riley "defined'' the Heckoid groups for $2$-bridge links as Kleinian groups, with nontrivial torsion, generated by two parabolic transformations, and he constructed an infinite family of epimorphisms from $2$-bridge link groups onto Heckoid groups. In this paper, we make Riley's definition explicit, and give a systematic construction of epimorphisms from $2$-bridge link groups onto Heckoid groups, generalizing Riley's construction.

In Part I of this series of papers, we made Riley's definition of Heckoid groups for $2$-bridge links explicit, and gave a systematic construction of epimorphisms from $2$-bridge link groups onto Heckoid groups, generalizing Riley's construction. In this paper, we give a complete characterization of upper-meridian-pair-preserving epimorphisms from $2$-bridge link groups onto even Heckoid groups, by proving that they are exactly the epimorphisms obtained by the systematic construction.