A rational Diophantine $m$-tuple is a set of $m$ nonzero rationals such that the product of any two of them is one less than a perfect square. Recently Gibbs constructed several examples of rational Diophantine sextuples with positive elements. In this note, we construct examples of rational Diophantine sextuples with mixed signs. Indeed, we show that such examples exist for all possible combinations of signs.

The aim of this paper is to study, in the framework of the $\nabla\varphi$ interface model, the macroscopic behavior of microscopic interfaces under the finite volume Gibbs measures with self potentials, especially by establishing the large deviation principle. We consider the case where the self potential depends on the position and on both macroscopic and microscopic heights of the interfaces as well. The assumption on the upper bound for the self potential required by [10] is relaxed.

Some \textit{a priori} estimates of the pressure for the Stokes equations in an infinite sector with the slip and the stress conditions on the boundary are established in weighted Sobolev spaces. The estimates for its higher order derivatives are obtained by the general scheme of Kondrat’ev. Instead the estimates for the lower order ones are derived by making use of the Mellin transformation and the explicit representation formula of the pressure.

The aim of the present note is to discuss the explicit determination of the deformation space of the action of a discontinuous subgroup for a homogeneous space in the setting where the basis group is the Heisenberg group.

We denote by $h_{D}$ the class number and by $p_{D}$ the Ono number of the imaginary quadratic fields $\mathbf{Q}(\sqrt{-D})$. Sairaiji-Shimizu [2] showed that there are infinitely many imaginary quadratic fields such that the inequality $h_{D} >c^{ p_{D}}$ holds for any real number. On the other hand we have the possibility that $h_{D} \leqq c^{ p_{D}}$ holds for infinitely many imaginary quadratic fields for the same real number $c$. In this paper, given a real number $c$, we consider whether $h_{D} \leqq c^{ p_{D}}$ holds for infinitely many imaginary quadratic fields or not.

We show six Legendrian dualities between pseudo-spheres in semi-Euclidean space which are basic tools for the study of extrinsic differential geometry of submanifolds in these pseudo-spheres.

We show how to invert the multiplication-by-2 map in Jacobians of genus 2 curves $\mathrm{C}$ over finite fields $\mathbf{F}_{q}$ of odd characteristic. For any divisor $D\in \mathrm{Jac}(\mathrm{C})(\mathbf{F}_{q})$ we provide a method to construct the coordinates of all divisors $D'\in \mathrm{Jac}(\mathrm{C})(\mathbf{F}_{q})$ such that $2D'=D$.