In this paper we derive some elements of the rings of integers in the cubic fields of the form $\mathbf{Q}(\sqrt[3]{d})$, where $d$ is even, which cannot be written as a difference of two squares in the considered ring. We show that corresponding Diophantine quadruples do not exist for such elements, what supports the hypothesis mainly proved for the ring of integers and for certain quadratic fields.

We obtain the best constant in Korn’s inequality for solenoidal vector fields on a periodic slab which vanish on a part of its boundary. To do this we consider the Stokes equations with Dirichlet boundary conditions, following H. Ito [6],[7].

The asymptotic behavior of the spectral function of a one-dimensional second-order differential operator is discussed. We give a necessary and sufficient condition in order that the spectral function varies regularly with index 1. The condition is closely related to the class $\Gamma$ which appears in the de Haan theory.

Let $X$ be a complex manifold. The classical Riemann-Hilbert correspondence associates to a regular holonomic system $\mathcal{M}$ the $\mathbf{C}$-constructible complex of its holomorphic solutions. Let $t$ be the affine coordinate in the complex projective line. If $\mathcal{M}$ is not necessarily regular, we associate to it the ind-$\mathbf{R}$-constructible complex $G$ of tempered holomorphic solutions to $\mathcal{M}\boxtimes\mathcal{D} e^{t}$. We conjecture that this provides a Riemann-Hilbert correspondence for holonomic systems. We discuss the functoriality of this correspondence, we prove that $\mathcal{M}$ can be reconstructed from $G$ if $\dim X=1$, and we show how the Stokes data are encoded in $G$.