Just as a residue field can be considered for a point of an algebraic variety, we can also consider a residue field for a point of a Berkovich analytic space. This residue field is a valuation field in the algebraic sense. Then we can consider its residue field as a valuation field. We call it the Berkovich double residue field at the point.

In this paper, we consider a point $x$ of the Berkovich analytification of an algebraic variety and identify the Berkovich double residue field at $x$ with the union of the residue fields at the center of $x$ in birational models. Besides, we concretely compute the Berkovich double residue field for any quasi monomial valuation.

A close relationship between the coding theory and the design theory has been studied by many researchers. The principal concern is directed to the designs formed by minimal weight code-words or very small weight codewords. In the present article we study more designs. We extend the concept of the incidence relation, one of which is a classical one and the other is a dual one to the classical one in a certain sense. In the present article we focus on the binary Golay code of length 24. But the idea will be applied to a wide class of self-dual codes such as self-dual extremal binary codes or self-dual extremal ternary codes.

In 2001, M. Rȩkoś described the analytic behavior for a function connected with the Euler totient function for the upper half-plane H. In this paper, for Im $z>0$ we describe the analytic behavior of the generalized function $f(z,\mathit{F})$, where the function $\mathit{F}$ belongs to a subclass of the Selberg class which has a polynomial Euler product expressions and satisfies some special conditions.

Ollivier and Lin–Lu–Yau established the theory of graph Ricci curvature (LLY curvature) via optimal transport on graphs. Ikeda–Kitabeppu–Takai–Uehara introduced a new distance called the Kantorovich difference on hypergraphs and generalized the LLY curvature to hypergraphs (IKTU curvature). As the LLY curvature can be represented by the graph Laplacian by Münch–Wojciechowski, Ikeda–Kitabeppu–Takai–Uehara conjectured that the IKTU curvature has a similar expression in terms of the hypergraph Laplacian. In this paper, we introduce a variant of the Kantorovich difference inspired by the above conjecture and study the Ricci curvature associated with this distance ($\text{wIKTU}$ curvature). Moreover, for hypergraphs with a specific structure, we analyze a quantity $\mathcal{C}(x,y)$ at two distinct vertices $x,y$ defined by using the hypergraph Laplacian. If the resolvent operator converges uniformly to the identity and the hypergraph Laplacian satisfies a certain property, then $\mathcal{C}(x,y)$ coincides with the $\text{wIKTU}$ curvature along $x,y$.