An extensive English language bibliography on graph theory and its applications, supplement 1

Graph theory and its applications - bibliography, supplement

Towards a dual spin network basis for (3+1)d lattice gauge theories and topological phases

Using a recent strategy to encode the space of flat connections on a three-manifold with string-like defects into the space of flat connections on a so-called 2d Heegaard surface, we propose a novel way to define gauge invariant bases for (3+1)d lattice gauge theories and gauge models of topological phases. In particular, this method reconstructs the spin network basis and yields a novel dual spin network basis. While the spin network basis allows to interpret states in terms of electric excitations, on top of a vacuum sharply peaked on a vanishing electric field, the dual spin network basis describes magnetic (or curvature) excitations, on top of a vacuum sharply peaked on a vanishing magnetic field (or flat connection). This technique is also applicable for manifolds with boundaries. We distinguish in particular a dual pair of boundary conditions, namely of electric type and of magnetic type. This can be used to consider a generalization of Ocneanu's tube algebra in order to reveal the algebraic structure of the excitations associated with certain 3d manifolds.Comment: 45 page

A refinement of Singer's bound for Liouvillian integration. The low-dimensional cases

In a previous article, I refined a bound for Liouvillian integration of Singer (1981). That article of mine was devoted to the generic case, obtaining results for dimension $n\geq 19$ that are completed for any $n$ in the present article. Here I focus on the low-dimensional cases for $n\leq 11$, retrieving the known optimal values of Singer's bound (Cormier, 2001) for $n\leq 5$ in a unified setting based on Collins (2008), and also computing the optimal value $I(n)=3780$ for $6\leq n\leq 11$. For this task, I resort to a previous result of mine on primitive linear groups that simplifies the search of large 1-reducible subgroups, reducing it to each component of quasicomponent separately, so that the computations can be accomplished with the aid of GAP

Orbit structure and (reversing) symmetries of toral endomorphisms on rational lattices

We study various aspects of the dynamics induced by integer matrices on the invariant rational lattices of the torus in dimension 2 and greater. Firstly, we investigate the orbit structure when the toral endomorphism is not invertible on the lattice, characterising the pretails of eventually periodic orbits. Next we study the nature of the symmetries and reversing symmetries of toral automorphisms on a given lattice, which has particular relevance to (quantum) cat maps.Comment: 29 pages, 3 figure

Artin's primitive root conjecture -a survey -

This is an expanded version of a write-up of a talk given in the fall of 2000 in Oberwolfach. A large part of it is intended to be understandable by non-number theorists with a mathematical background. The talk covered some of the history, results and ideas connected with Artin's celebrated primitive root conjecture dating from 1927. In the update several new results established after 2000 are also discussed.Comment: 87 pages, 512 references, to appear in Integer