In this paper, we give an arithmetic characterization of Rédei’s dihedral extension over $\mathbf{Q}$ and another simple proof of the reciprocity law of the triple symbol.

For any admissible normal function $\nu$ over any dimensional base, we construct by the method of log geometry a Néron model such that $\nu$ extends to a section of the model over the boundary. The model is Hausdorff and is a relative log manifold.

In this short article, we consider estimates of the ratio \begin{equation*} \|{f}\|_{\mathit{BMO}(w)} / \|{f}\|_{\mathit{BMO}} \end{equation*} from above and below, where $w$ belongs to Muckenhoupt class $A_{\infty}$. The upper bound of the ratio was proved by Hytönen and Pérez in~[6] with the optimal power. We establish the lower bound of the ratio and give two other proofs of the upper bound.

We study a linear semidiscrete-in-time finite difference method for the system of nonlinear Schrödinger equations that is a model of the interaction of non-relativistic particles with different masses. The main aim is to show that the scheme is second-order convergent.

We explain how one can naturally associate a Deitmar scheme (which is a scheme defined over the field with one element, $\mathbf{F}_{1}$) to a so-called “loose graph” (which is a generalization of a graph). Several properties of the Deitmar scheme can be proven easily from the combinatorics of the (loose) graph, and it also appears that known realizations of objects over $\mathbf{F}_{1}$ (such as combinatorial $\mathbf{F}_{1}$-projective and $\mathbf{F}_{1}$-affine spaces) exactly depict the loose graph which corresponds to the associated Deitmar scheme. This idea is then conjecturally generalized so as to describe all Deitmar schemes in a similar synthetic manner.