We study a polyhedron with $n$ vertices of fixed volume having the minimum surface area. Completing the proof of Fejes Tóth, we show that all faces of a minimum polyhedron are triangles, and further prove that a minimum polyhedron does not allow deformation of a single vertex. We also present possible minimum shapes for $n ≤ 12$. Some of them are quite unexpected, in particular $n = 8$.

We show that if three meromorphic functions share three two-point sets CM, then there exist two of the meromorphic functions such that one of them is a Möbius transform of the other.

We introduce the concept CFA modules and their applications in investigation the coassociated primes of local homology modules. The main result of this paper says that if $M$ is a CFA linearly compact $R$-module and $t$ is a non-negative integer such that $H_i^I(M)$ is CFA for all $i < t$, then $R/I{\otimes }_R{H}_t^I(M)$ is CFA. Hence, the set $\mathrm{Coass}_R$ $H_t^I(M)$ is finite.

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. Multiplicities and order sequences at Galois points are also described in detail.

A multivariate generalized ridge (MGR) regression provides a shrinkage estimator of the multivariate linear regression by multiple ridge parameters. Since the ridge parameters which adjust the amount of shrinkage of the estimator are unknown, their optimization is an important task to obtain a better estimator. For the univariate case, a fast algorithm has been proposed for optimizing ridge parameters based on minimizing a model selection criterion (MSC) and the algorithm can be applied to various MSCs. In this paper, we extend this algorithm to MGR regression. We also describe the relationship between the MGR estimator which is not sparse and a multivariate adaptive group Lasso estimator which is sparse, under orthogonal explanatory variables.