The matching complex of a simple graph $G$ is the simplicial complex consisting of the matchings on $G$. Jelić Milutinović et al. [6] studied the matching complexes of the polygonal line tilings, and they gave a lower bound for the connectivity of the matching complexes of polygonal line tilings. In this paper, we determine the homotopy types of the matching complexes of polygonal line tilings recursively, and determine their connectivities.

In [7] (and [8]) M. Harada studied a left artinian ring $R$ such that every non-small left $R$-module contains a non-zero injective submodule. And in [12] K. Oshiro called the ring a left Harada ring (abbreviated left $H$-ring) and further studied the ring in [13], [14], [15], $\ldots$. We can see many results on left Harada rings in [4] and many equivalent conditions in [3, Theorem B]. In [5] we give new concepts “co-$H$-sequence” and “$H$-epimorphism” to study two-sided Harada rings. In this paper, we intruduce another new concept “weak co-$H$-sequence”. And we further study two-sided Harada rings.

Let $E$ be a smooth cubic. A plane curve $D$ is said to be an $n$-contact curve to $E$ if the intersection multiplicities at each intersection point between $E$ and $D$ is $n$. In this note, we give an algorithm to produce possible candidates for $n$-contact curves to $E$ and consider its application.

Based on the behavior of the restrictions, inductions, Frobenius reciprocity in character theory of finite groups, we construct a framework of quiver representations of extended subgroup posets of finite groups. As applications, we provide a new congruence concerning character values, give an alternative proof of a known result, and also deal with two kinds of generalizations of Cartan matrices.

For a map $\varphi : \Gamma \rightarrow \Gamma'$ between tropical curves and an isometric action on $\Gamma$ of a finite group $K$, $\varphi$ is a $K$-Galois covering on $\Gamma'$ if $\varphi$ is a harmonic morphism, the degree of $\varphi$ coincides with the order of $K$ and the action of $K$ induces a transitive action on every fibre. We prove that for a tropical curve $\Gamma$ with an isometric action of a finite group $K$, there exists a rational map, from $\Gamma$ to a tropical projective space, which induces a $K$-Galois covering on the image with proper edge-multiplicities. As an application, we also prove that for a hyperelliptic tropical curve without one valent points and of genus at least two, the invariant linear system of the hyperelliptic involution $\iota$ of the canonical linear system, the complete linear system associated with the canonical divisor, induces an $\langle \iota \rangle$-Galois covering on a tree. This is an analogy of the fact that a compact Riemann surface is hyperelliptic if and only if the canonical map, the rational map induced by the canonical linear system, is a double covering on a projective line $\boldsymbol{P}^1$.

The present paper continues our recent works related to the moduli theory of triply periodic minimal surfaces of genus three in terms of three geometric invariants, namely, the Morse index, the nullity, and the signature of a minimal surface. In this paper, we consider mPCLP/mDCLP family, which is listed in Fogden-Hyde [9] and also contains other families in the same paper as its special cases. In fact, the family is a three-parameter family of compact oriented embedded minimal surfaces of genus three in flat three-tori. By numerical arguments, we will compute the three quantities for mPCLP/mDCLP family. Our previous works only treated one-parameter families, and thus the family is of higher dimensional and has richer properties in the moduli space of triply periodic minimal surfaces of genus three.