The algebra of rewriting for presentations of inverse monoids

We describe a formalism, using groupoids, for the study of rewriting for presentations of inverse monoids, that is based on the Squier complex construction for monoid presentations. We introduce the class of pseudoregular groupoids, an example of which now arises as the fundamental groupoid of our version of the Squier complex. A further key ingredient is the factorisation of the presentation map from a free inverse monoid as the composition of an idempotent pure map and an idempotent separating map. The relation module of a presentation is then defined as the abelianised kernel of this idempotent separating map. We then use the properties of idempotent separating maps to derive a free presentation of the relation module. The construction of its kernel - the module of identities - uses further facts about pseudoregular groupoids.Comment: 22 page

Modelling alternative strategies for delivering hepatitis B vaccine in prisons : the impact on the vaccination coverage of the injecting drug user population

Since 2001 hepatitis B vaccination has been offered to prisoners on reception into prisons in England and Wales. However, short campaigns of vaccinating the entire population of individual prisons have achieved high vaccination coverage for limited periods, suggesting that short campaigns may be a preferable way of vaccinating prisoners. A model is used that describes the flow of prisoners through prisons stratified by injecting status to compare a range of vaccination scenarios that describe vaccination on prison reception or via regular short campaigns. Model results suggest that vaccinating on prison reception can capture a greater proportion of the injecting drug user (IDU) population than the comparable campaign scenarios (63% vs. 55 . 6% respectively). Vaccination on prison reception is also more efficient at capturing IDUs for vaccination than vaccination via a campaign, although vaccination via campaigns may have a role with some infections for overall control

Parafermionic phases with symmetry-breaking and topological order

Parafermions are the simplest generalizations of Majorana fermions that realize topological order. We propose a less restrictive notion of topological order in 1D open chains, which generalizes the seminal work by Fendley [J. Stat. Mech., P11020 (2012)]. The first essential property is that the groundstates are mutually indistinguishable by local, symmetric probes, and the second is a generalized notion of zero edge modes which cyclically permute the groundstates. These two properties are shown to be topologically robust, and applicable to a wider family of topologically-ordered Hamiltonians than has been previously considered. An an application of these edge modes, we formulate a new notion of twisted boundary conditions on a closed chain, which guarantees that the closed-chain groundstate is topological, i.e., it originates from the topological manifold of degenerate states on the open chain. Finally, we generalize these ideas to describe symmetry-breaking phases with a parafermionic order parameter. These exotic phases are condensates of parafermion multiplets, which generalizes Cooper pairing in superconductors. The stability of these condensates are investigated on both open and closed chains.Comment: 27 pages, 9 figure

Theoretical characterization of a model of aragonite crystal orientation in red abalone nacre

Nacre, commonly known as mother-of-pearl, is a remarkable biomineral that in red abalone consists of layers of 400-nm thick aragonite crystalline tablets confined by organic matrix sheets, with the $(001)$ crystal axes of the aragonite tablets oriented to within $\pm$ 12 degrees from the normal to the layer planes. Recent experiments demonstrate that this orientational order develops over a distance of tens of layers from the prismatic boundary at which nacre formation begins. Our previous simulations of a model in which the order develops because of differential tablet growth rates (oriented tablets growing faster than misoriented ones) yield patterns of tablets that agree qualitatively and quantitatively with the experimental measurements. This paper presents an analytical treatment of this model, focusing on how the dynamical development and eventual degree of order depend on model parameters. Dynamical equations for the probability distributions governing tablet orientations are introduced whose form can be determined from symmetry considerations and for which substantial analytic progress can be made. Numerical simulations are performed to relate the parameters used in the analytic theory to those in the microscopic growth model. The analytic theory demonstrates that the dynamical mechanism is able to achieve a much higher degree of order than naive estimates would indicate.Comment: 20 pages, 3 figure

Level density of a Fermi gas and integer partitions: a Gumbel-like finite-size correction

We investigate the many-body level density of gas of non-interacting fermions. We determine its behavior as a function of the temperature and the number of particles. As the temperature increases, and beyond the usual Sommerfeld expansion that describes the degenerate gas behavior, corrections due to a finite number of particles lead to Gumbel-like contributions. We discuss connections with the partition problem in number theory, extreme value statistics as well as differences with respect to the Bose gas.Comment: 5 pages, 1 figure, one figure added, accepted for publication in Phys. Rev.

Influence, originality and similarity in directed acyclic graphs

We introduce a framework for network analysis based on random walks on directed acyclic graphs where the probability of passing through a given node is the key ingredient. We illustrate its use in evaluating the mutual influence of nodes and discovering seminal papers in a citation network. We further introduce a new similarity metric and test it in a simple personalized recommendation process. This metric's performance is comparable to that of classical similarity metrics, thus further supporting the validity of our framework.Comment: 6 pages, 4 figure

Test of nuclear level density inputs for Hauser-Feshbach model calculations

The energy spectra of neutrons, protons, and alpha-particles have been measured from the d+59Co and 3He+58Fe reactions leading to the same compound nucleus, 61$Ni. The experimental cross sections have been compared to Hauser-Feshbach model calculations using different input level density models. None of them have been found to agree with experiment. It manifests the serious problem with available level density parameterizations especially those based on neutron resonance spacings and density of discrete levels. New level densities and corresponding Fermi-gas parameters have been obtained for reaction product nuclei such as 60Ni,60Co, and 57Fe