The present paper is concerned with a constrained matrix optimization problem. The constraint is referred to as the ${\ell }_{2,0}$-norm of the matrix, which is defined as the number of non-zero row vectors of the matrix. We extend the discrete first-order algorithm by Bertsimas et al. [3] to solve the optimization problem. The extended algorithm is useful for selecting variables in multivariate statistical models. Then, the convergence properties of the extended algorithm are established. In numerical experiments, we apply the extended algorithm to the optimization problem for the multivariate linear regression model. Furthermore, we also incorporate selecting variables using information criteria into the optimization problem.

It is proved that a $C^2$-mapping from an open subset in ${ℂ}^q$ to ${ℂ}^k$ is holomorphic if its graph is pseudoconcave of order $q$. This is an extension of Hartogs’ analyticity theorem. The proof is given using the characterization of pseudoconcave sets of order $q$ by local maximum principle for $q$-convex functions.

We investigate a class of second order fully nonlinear elliptic equations containing gradient terms on compact almost Hermitian manifolds and show some a priori estimates for solutions of these equations under some assumptions.

We study nonorientable maximal surfaces in the Lorentz-Minkowski $3$-space ${\mathbb{L}}^3$. We construct nonorientable maximal surfaces containing hypocycloid and maximal surfaces in ${\mathbb{L}}^3$ which are homeomorphic to the real projective plane $ℝ{\mathbb{P}}^2$ minus two points with degree of the Gauss map being equal to $4$.

We show that if three nonconstant meromorphic functions on the complex plane share three one-point or two-point sets CM, then there exist two of the meromorphic functions such that one of them is a Möbius transform of the other. The cases that all three sets are one-point and that all three sets are two-point are obtained by H. Cartan ([C1]) and by the author ([S4]), respectively.

Let $G$ be a group. If an equation $x^n=y^n$ in $G$ implies $x=y$ for any elements $x$ and $y$, then $G$ is called an $R$-group. It is completely understood which knot groups are $R$-groups. Fay and Walls introduced $\overline{R}$-group in which the normalizer and the centralizer of an isolator of $\langle x\rangle$ coincide for any non-trivial element $x$. It is known that $\overline{R}$-groups and $R$-groups share many interesting properties and $\overline{R}$-groups are necessarily $R$-groups. However, in general, the converse does not hold. We will prove that these classes are the same for knot groups. In the course of the proof, we will determine knot groups with generalized torsion of order two.

In this paper, we are concerned with the existence and uniqueness of a generalized solution to a double obstacle problem for Musielak-Orlicz Dirichlet energy integral on metric measure spaces supporting a $\Phi$-Poincaré inequality, as an extension of Farnana (Nonlinear Anal. 73 (2010), pp. 2819–2830).