In this paper, we introduce the $k$-th order slant Toeplitz operator on the Lebesgue space $L^2(\mathbb{T}^n)$ for a fixed integer $k\ge2$ and also investigate certain algebraic properties of the $k$-th order slant Toeplitz operator on the Lebesgue space $L^2(\mathbb{T}^n)$. The complete characterization of the $k$-th order slant Toeplitz operator is also given. Finally, norms and the spectral radius of the introduced operator are discussed.

We define $r$-inferior regular partition which is a restriction of partition. Its generating function equals to that of the number of operations in the Glaisher correspondence. Using this result, we prove Mizukawa-Yamada's identity separately. Moreover we extend this identity to $m$-tuple version.

In the present paper, we study profinite groups of PIPSC-type, i.e., abstract profinite groups isomorphic to the extensions determined by outer representations of PIPSC-type. In particular, we establish a “group-theoretic” algorithm for constructing, from a profinite group of PIPSC-type that is noncuspidal, a certain profinite graph.

We introduce a Kähler-like almost Hermitian metric and an almost balanced metric. We prove that on a Kähler-like almost Hermitian manifold, we have an identity between the first derivative of the torsion $(1,0)$-tensor and the Nijenhuis tensor. By applying the identity, then we figure out what the equivalent condition of being almost balanced on a compact Kähler-like almost Hermitian manifold is. We apply the result to a 2-step nilpotent Lie algebra, and also to the almost Hermitian curvature flow (AHCF). We obtain a lower bound for the scalar curvature along (AHCF). Also we have some results on the monotonicity of the volume along (AHCF) by studying the relation between the volume and the scalar curvature.

In this note we prove that the Selberg zeta-function associated to a compact Riemann surface is pseudo-prime and right-prime in the sense of a decomposition.

Let $R$ be a ring and $M$ be an $R$-module. Two modules $A$ and $B$ are called orthogonal, written $A\perp B$, if they do not have non-zero isomorphic submodules. We associate a graph $\Gamma_{\bot}(M)$ to $M$ with vertices $\mathcal{M}_{\perp}=\{(0)\neq A\leq M \mid \exists B\neq (0) \;\mbox{such that}\; A\perp B\}$, and for distinct $A,B\in \mathcal{M}_{\perp}$, the vertices $A$ and $B$ are adjacent if and only if $A\perp B$. The main object of this article is to study the interplay of module-theoretic properties of $M$ with graph-theoretic properties of $\Gamma_{\bot}(M)$. An algorithm is given to generate perpendicular graphs of $\mathbb{Z}_n$.

The Cauchy problem for the semilinear diffusion equation is considered in the de Sitter spacetime with the spatial zero-curvature. Global solutions and their asymptotic behaviors for small initial data are obtained for positive and negative Hubble constants. The effects of the spatial expansion and contraction are studied on the problem.