A parallel edge orientation algorithm for quadrilateral meshes

One approach to achieving correct finite element assembly is to ensure that the local orientation of facets relative to each cell in the mesh is consistent with the global orientation of that facet. Rognes et al. have shown how to achieve this for any mesh composed of simplex elements, and deal.II contains a serial algorithm to construct a consistent orientation of any quadrilateral mesh of an orientable manifold. The core contribution of this paper is the extension of this algorithm for distributed memory parallel computers, which facilitates its seamless application as part of a parallel simulation system. Furthermore, our analysis establishes a link between the well-known Union-Find algorithm and the construction of a consistent orientation of a quadrilateral mesh. As a result, existing work on the parallelisation of the Union-Find algorithm can be easily adapted to construct further parallel algorithms for mesh orientations.Comment: Second revision: minor change

Principal and syntactic congruences in congruence-distributive and congruence-permutable varieties

We give a new proof that a finitely generated congruence-distributive variety has finitely determined syntactic congruences (or, equivalently, term finite principal congruences), and show that the same does not hold for finitely generated congruence-permutable varieties, even under the additional assumption that the variety is residually very finite

Finitely forcible graphons with an almost arbitrary structure

Graphons are analytic objects representing convergent sequences of large graphs. A graphon is said to be finitely forcible if it is determined by finitely many subgraph densities, i.e., if the asymptotic structure of graphs represented by such a graphon depends only on finitely many density constraints. Such graphons appear in various scenarios, particularly in extremal combinatorics. Lovasz and Szegedy conjectured that all finitely forcible graphons possess a simple structure. This was disproved in a strong sense by Cooper, Kral and Martins, who showed that any graphon is a subgraphon of a finitely forcible graphon. We strenghten this result by showing for every $\varepsilon>0$ that any graphon spans a $1-\varepsilon$ proportion of a finitely forcible graphon

RPA Green's Functions of the Anisotropic Heisenberg Model

We solve in random-phase approximation the anisotropic Heisenberg model, including nearest and next-nearest neighbour interactions by calculating all Green's functions and pair correlation functions in a cumulant decoupling scheme. The general exposition is pedagogic in tone and is intended to be accessible to any graduate student or physicist who is not an expert in the field.Comment: 26 pages, 4 figure