The Gauss summation theorem for $_2F_1$-series is examined by means of power series expansions. Several infinite series involving harmonic numbers and multinomial coefficients are evaluated in closed forms.

In the present paper, for a pair $(G,N)$ of a group $G$ and its normal subgroup $N$, we consider the mixed commutator length $\mathrm{cl}_{G,N}$ on the mixed commutator subgroup $[G,N]$. We focus on the setting of wreath products: $ (G,N)=(\mathbb{Z}\wr \Gamma, \bigoplus_{\Gamma}\mathbb{Z})$. Then we determine mixed commutator lengths in terms of the general rank in the sense of Malcev. As a byproduct, when an abelian group $\Gamma$ is not locally cyclic, the ordinary commutator length $\mathrm{cl}_G$ does not coincide with $\mathrm{cl}_{G,N}$ on $[G,N]$ for the above pair. On the other hand, we prove that if $\Gamma$ is locally cyclic, then for every pair $(G,N)$ such that $1 \to N \to G \to \Gamma \to 1$ is exact, $\mathrm{cl}_{G}$ and $\mathrm{cl}_{G,N}$ coincide on $[G,N]$. We also study the case of permutational wreath products when the group $\Gamma$ belongs to a certain class related to surface groups.

For all $n \geq 4$, we give a complete classification of the compact $n$-dimensional minimal $C$-totally real submanifolds in the $(2n+1)$-dimensional unit sphere $\mathbb S^{2n+1}(1)$ with non-negative sectional curvature. This generalizes the results of Yamaguchi et al (Proc Amer Math Soc 54: 276-280, 1976) for $n$ = 2 and, Dillen and Vrancken (Math J Okayama Univ 31: 227-242, 1989) for $n$ = 3. Additionally, we show that, as compact minimal $C$-totally real submanifolds, the standard embeddings of the symmetric spaces $\mathrm{SU}(m)/\mathrm{SO}(m)$, $\mathrm{SU}(m)$, $\mathrm{SU}(2m)/\mathrm{Sp}(m)$ for each $m \geq 3$, and $\mathrm{E}_{6}/\mathrm{F}_4$ into $\mathbb S^{2n+1}(1)$ are all Willmore submanifolds, with $n=\frac{1}{2}m(m-1)-1$, $m^2-1$, $2m^2-m-1$ and 26, respectively.

In this note we will provide a gradient estimate for harmonic maps from a complete noncompact Riemannian manifold with compact boundary (which we call "Kasue manifold") into a simply connected complete Riemannian manifold with non-positive sectional curvature. As a consequence, we can obtain a Liouville theorem. We will also show the nonexistence of positive solutions to some linear elliptic equations on Kasue manifolds.

Yotsutani and Zhou gave a sufficient condition for a toric Fano manifold to be relatively K-unstable. In this note, we present a simple obstruction to apply their condition.

In this paper, we study a Wirsing type theorem for points of bounded degree in projective varieties for numerically equivalent ample divisors located in subgeneral position. We also show its application to give some finiteness conditions on decomposable form inequalities with solutions of bounded degrees.