Inspired by the halfspace theorem for minimal surfaces in $\mathbb {R}^3$ of Hoffman–Meeks, the halfspace theorem of Rodriguez–Rosenberg, and the classical cone theorem of Omori in $\mathbb {R}^n$, we derive new non-existence results for proper harmonic maps into perturbed cones in $\mathbb {R}^n$, horospheres in $\mathbb {H}^n$, culminating in a generalization of Omori’s theorem in arbitrary Riemannian manifolds. The technical tool proved here extends the foliated Sampson’s maximum principle, initially developed in the first author’s Ph.D. thesis, to a non-compact setting.

Quantales can be regarded as a combination of complete lattices and semigroups. Unital quantales constitute a significant subclass within quantale theory, which play a crucial role in the theoretical framework of quantale research. It is well known that every complete lattice can support a quantale. However, the question of whether every complete lattice can support a unital quantale has not been considered before. In this article, we first give some counter-examples to indicate that the answer to the above question is negative, and then investigate the complete lattices of supporting unital quantales.

We show that a normed linear space is isometrically isomorphic to an inner product space if and only if it is a strongly n-point homogeneous metric space for any (or every) $n \geqslant 3$. The counterpart for $n=2$ is the Banach–Mazur problem.

The usual division algorithms on ${\mathbb {Z}}$ and ${\mathbb {Z}}[i]$ measure the size of remainders using the algebraic norm. These rings are Euclidean with respect to several functions. The pointwise minimum of all Euclidean functions $f: R \setminus \{0\} \rightarrow {\mathbb {N}}$ on a Euclidean domain R is itself a Euclidean function, called the minimal Euclidean function and denoted by $\phi _R$. To the author’s knowledge, the integers, ${\mathbb {Z}}$ and the Gaussians, ${\mathbb {Z}}[i]$ are the only rings of integers of number fields for which we have a formula to compute their minimal Euclidean functions, $\phi _{{\mathbb {Z}}}$ and $\phi _{{\mathbb {Z}}[i]}$. This article presents the first division algorithm (that the author knows of) for ${\mathbb {Z}}[i]$ relative to $\phi _{{\mathbb {Z}}[i]}$, empowering readers to perform the Euclidean algorithm on ${\mathbb {Z}}[i]$ using its minimal Euclidean function.

In the article “Irreducible modules of modular Lie superalgebras and super version of the first Kac-Weisfeiler conjecture, Canad. Math. Bull. 67 (2024), no. 3, 554–573.” The statement in Theorem 4.7 is improper, which is fixed here. Theorem 4.7 is an isolated result in the article. This correction does not influence any arguments and any main results after that in the original article.