Ghosts in modular representation theory

A ghost over a finite p-group G is a map between modular representations of G which is invisible in Tate cohomology. Motivated by the failure of the generating hypothesis---the statement that ghosts between finite-dimensional G-representations factor through a projective---we define the ghost number of kG to be the smallest integer l such that the composition of any l ghosts between finite-dimensional G-representations factors through a projective. In this paper we study ghosts and the ghost numbers of p-groups. We begin by showing that a weaker version of the generating hypothesis, where the target of the ghost is fixed to be the trivial representation k, holds for all p-groups. We then compute the ghost numbers of all cyclic p-groups and all abelian 2-groups with C_2 as a summand. We obtain bounds on the ghost numbers for abelian p-groups and for all 2-groups which have a cyclic subgroup of index 2. Using these bounds we determine the finite abelian groups which have ghost number at most 2. Our methods involve techniques from group theory, representation theory, triangulated category theory, and constructions motivated from homotopy theory.Comment: 15 pages, final version, to appear in Advances in Mathematics. v4 only makes changes to arxiv meta-data, correcting the abstract and adding a do

Pattern formation in reaction diffusion models with spatially inhomogeneous diffusion coefficients

Reaction-diffusion models for biological pattern formation have been studied extensively in a variety of embryonic and ecological contexts. However, despite experimental evidence pointing to the existence of spatial inhomogeneities in various biological systems, most models have only been considered in a spatially homogeneous environment. The authors consider a two-chemical reaction-diffusion mechanism in one space dimension in which one of the diffusion coefficients depends explicitly on the spatial variable. The model is analysed in the case of a step function diffusion coefficient and the insight gained for this special case is used to discuss pattern generation for smoothly varying diffusion coefficients. The results show that spatial inhomogeneity may be an important biological pattern regulator, and possible applications of the model to chondrogenesis in the vertebrate limb are suggested

Unravelling the Turing bifurcation using spatially varying diffusion coefficients

The Turing bifurcation is the basic bifurcation generating spatial pattern, and lies at the heart of almost all mathematical models for patterning in biology and chemistry. In this paper the authors determine the structure of this bifurcation for two coupled reaction diffusion equations on a two-dimensional square spatial domain when the diffusion coefficients have a small explicit variation in space across the domain. In the case of homogeneous diffusivities, the Turing bifurcation is highly degenerate. Using a two variable perturbation method, the authors show that the small explicit spatial inhomogeneity splits the bifurcation into two separate primary and two separate secondary bifurcations, with all solution branches distinct. This splitting of the bifurcation is more effective than that given by making the domain slightly rectangular, and shows clearly the structure of the Turing bifurcation and the way in which the! var ious solution branches collapse together as the spatial variation is reduced. The authors determine the stability of the solution branches, which indicates that several new phenomena are introduced by the spatial variation, including stable subcritical striped patterns, and the possibility that stable stripes lose stability supercritically to give stable spotted patterns

The generating hypothesis for the stable module category of a pp-group

Freyd's generating hypothesis, interpreted in the stable module category of a finite p-group G, is the statement that a map between finite-dimensional kG-modules factors through a projective if the induced map on Tate cohomology is trivial. We show that Freyd's generating hypothesis holds for a non-trivial finite p-group G if and only if G is either C_2 or C_3. We also give various conditions which are equivalent to the generating hypothesis.Comment: 6 pages, fixed minor typos, to appear in J. Algebr

Topologically Alice Strings and Monopoles

Symmetry breaking can produce ``Alice'' strings, which alter scattered charges and carry monopole number and charge when twisted into loops. Alice behavior arises algebraically, when strings obstruct unbroken symmetries -- a fragile criterion. We give a topological criterion, compelling Alice behavior or deforming it away. Our criterion, that \pi_o(H) acts nontrivially on \pi_1(H), links topologically Alice strings to topological monopoles. We twist topologically Alice loops to form monopoles. We show that Alice strings of condensed matter systems (nematic liquid crystals, helium 3A, and related non-chiral Bose condensates and amorphous chiral superconductors) are topologically Alice, and support fundamental monopole charge when twisted into loops. Thus they might be observed indirectly, not as strings, but as loop-like point defects. We describe other models, showing Alice strings failing our topological criterion; and twisted Alice loops supporting deposited, but not fundamental, monopole number.Comment: 2 figures; this paper consolidates preprints hep-th/0304161 and hep-th/0304162, to appear in Phys. Rev.

Charge Violation and Alice Behavior in Global and Textured Strings

Spontaneous breaking of global symmetries can produce ``Alice'' strings: line defects which make unbroken symmetries multivalued, induce apparent charge violation via Aharonov-Bohm interactions, and form point defects when twisted into loops. We demonstrate this behavior for both divergent and textured global Alice strings. Both adiabatically scatter charged particles via effective Wilson lines. For textured Alice strings, such Wilson lines occur at all radii, and are multivalued only inside the string. This produces measurable effects, including path-dependent charge violation.Comment: 32 pages, 2 epsfigs, Revte

Compassion fatigue and burnout - the role of Balint groups

Copyright © 2005 Royal Australian College of General Practitioners Copyright to Australian Family Physician. Reproduced with permission. Permission to reproduce must be sought from the publisher, The Royal Australian College of General Practitioners.General practitioners are often the ‘first port of call’ for patients with a range of mental health problems, many of whom have a history of trauma or loss. Exposure to emotionally difficult situations puts them at risk of burnout and compassion fatigue. Balint groups are groups of GPs, usually facilitated by a psychiatrist, who discuss the doctor-patient relationship and provide peer support. Participation in Balint groups, along with other professional and personal activities, has the potential to prevent compassion fatigue and burnout in participants.Jill Benson; Karen Magrait

Mechanisms of fragmentation of Al-W granular composites under dynamic loading

Numerical simulations of Aluminum (Al) and Tungsten (W) granular composite rings under various dynamic loading conditions caused by explosive loading were examined. Three competing mechanisms of fragmentation were observed: a continuum level mechanism generating large macrocracks described by the Grady-Kipp fragmentation mechanism, a mesoscale mechanism generating voids and microcracks near the initially unbonded Al/W interfaces due to tensile strains, and a mesoscale jetting due to the development of large velocity gradients between the W particles and adjacent Al. These mesoscale mechanisms can be used to tailor the size of the fragments by selecting an appropriate initial mesostructure for a given loading condition.Comment: 10 pages, 3 figures, submitted to AP

Increased Dementia Mortality in West Virginia Counties with Mountaintop Removal Mining?

(MTM), a practice that has been ongoing in some counties of West Virginia (WV) USA since the 1970s. PM inhalation has been linked to central nervous system pathophysiology, including cognitive decline and dementia. Here we compared county dementia mortality statistics in MTM vs. non-MTM WV counties over a period spanning 2001–2015. We found significantly elevated age-adjusted vascular or unspecified dementia mortality/100,000 population in WV MTM counties where, after adjusting for socioeconomic variables, dementia mortality was 15.60 (±3.14 Standard Error of the Mean (S.E.M.)) times higher than that of non-MTM counties. Further analyses with satellite imaging data revealed a highly significant positive correlation between the number of distinct mining sites vs. both mean and cumulative vascular and unspecified dementia mortality over the 15 year period. This was in contrast to finding only a weak relationship between dementia mortality rates and the overall square kilometers mined. No effect of living in an MTM county was found for the rate of Alzheimer’s type dementia and possible reasons for this are considered. Based on these results, and the current literature, we hypothesize that inhalation of PM associated with MTM contributes to dementia mortality of the vascular or unspecified types. However, limitations inherent in ecological-type studies such as this, preclude definitive extrapolation to individuals in MTM-counties at this time. We hope these findings will inspire follow-up cohort and case-controlled type studies to determine if specific causative factors associated with living near MTM can be identified. Given the need for caregiving and medical support, increased dementia mortality of the magnitude seen here could, unfortunately, place great demands upon MTM county public health resources in the future