Classification of low-rank odd-dimensional modular categories

We prove that any odd-dimensional modular category of rank at most 23 is pointed. We also show that an odd-dimensional modular category of rank 25 is either pointed, perfect, or equivalent to $\operatorname{Rep}(D^\omega(\mathbb Z_7\rtimes \mathbb Z_3))$. Finally, we give partial classification results for modular categories of rank up to 73.Comment: Final version, to appear in Journal of Algebr

On the Hamilton-Waterloo problem for bipartite 2-factors

Given two 2-regular graphs F1 and F2, both of order n, the Hamilton-Waterloo Problem for F1 and F2 asks for a factorization of the complete graph Kn into a1 copies of F1, a2 copies of F2, and a 1-factor if n is even, for all nonnegative integers a1 and a2 satisfying a1+a2=?n-12?. We settle the Hamilton-Waterloo Problem for all bipartite 2-regular graphs F1 and F2 where F1 can be obtained from F2 by replacing each cycle with a bipartite 2-regular graph of the same order

Gene–environment interactions in Leber hereditary optic neuropathy

Leber hereditary optic neuropathy (LHON) is a genetic disorder primarily due to mutations of mitochondrial DNA (mtDNA). Environmental factors are thought to precipitate the visual failure and explain the marked incomplete penetrance of LHON, but previous small studies have failed to confirm this to be the case. LHON has no treatment, so identifying environmental triggers is the key to disease prevention, whilst potentially revealing new mechanisms amenable to therapeutic manipulation. To address this issue, we conducted a large, multicentre epidemiological study of 196 affected and 206 unaffected carriers from 125 LHON pedigrees known to harbour one of the three primary pathogenic mtDNA mutations: m.3460G>A, m.11778G>A and m.14484T>C. A comprehensive history of exposure to smoking, alcohol and other putative environmental insults was collected using a structured questionnaire. We identified a strong and consistent association between visual loss and smoking, independent of gender and alcohol intake, leading to a clinical penetrance of 93% in men who smoked. There was a trend towards increased visual failure with alcohol, but only with a heavy intake. Based on these findings, asymptomatic carriers of a LHON mtDNA mutation should be strongly advised not to smoke and to moderate their alcohol intake

On the Oberwolfach Problem for cycles with multiple lengths

The complete graph of order n, denoted Kn, is the graph consisting of n vertices and edges, one for each unordered pair of vertices. For even n, the graph Kn − I is � n 2 obtained from Kn by removal of n/2 pairwise disjoint edges. A 2-factor of a graph G = (V, E) is a subset F ⊆ E such that each vertex of G is incident with exactly 2 edges in F. Each connected component of the subgraph induced by F is a cycle of length at least three in G. The shape of F is the multiset (a n1 1,..., a nk k) of lengths of these cycles. (The symbol a ni i expresses that F has exactly ni cycles of length ai.) The Oberwolfach problem OP (a n