Invariant-based approach to symmetry class detection

In this paper, the problem of the identification of the symmetry class of a given tensor is asked. Contrary to classical approaches which are based on the spectral properties of the linear operator describing the elasticity, our setting is based on the invariants of the irreducible tensors appearing in the harmonic decomposition of the elasticity tensor [Forte-Vianello, 1996]. To that aim we first introduce a geometrical description of the space of elasticity tensors. This framework is used to derive invariant-based conditions that characterize symmetry classes. For low order symmetry classes, such conditions are given on a triplet of quadratic forms extracted from the harmonic decomposition of the elasticity tensor $C$, meanwhile for higher-order classes conditions are provided in terms of elements of $H^{4}$, the higher irreducible space in the decomposition of $C$. Proceeding in such a way some well known conditions appearing in the Mehrabadi-Cowin theorem for the existence of a symmetry plane are retrieved, and a set of algebraic relations on polynomial invariants characterizing the orthotropic, trigonal, tetragonal, transverse isotropic and cubic symmetry classes are provided. Using a genericity assumption on the elasticity tensor under study, an algorithm to identify the symmetry class of a large set of tensors is finally provided.Comment: 32 page

A Modular Curve of Level 9 and the Class Number One Problem

In this note we give an explicit parametrization of the modular curve associated to the normalizer of a non-split Cartan subgroup of level 9. We determine all integral points of this modular curve. As an application, we give an alternative solution to the class number one problem.Comment: 18 page