Asymptotic Exit Location Distributions in the Stochastic Exit Problem

Consider a two-dimensional continuous-time dynamical system, with an attracting fixed point $S$. If the deterministic dynamics are perturbed by white noise (random perturbations) of strength $\epsilon$, the system state will eventually leave the domain of attraction $\Omega$ of $S$. We analyse the case when, as $\epsilon\to0$, the exit location on the boundary $\partial\Omega$ is increasingly concentrated near a saddle point $H$ of the deterministic dynamics. We show that the asymptotic form of the exit location distribution on $\partial\Omega$ is generically non-Gaussian and asymmetric, and classify the possible limiting distributions. A key role is played by a parameter $\mu$, equal to the ratio $|\lambda_s(H)|/\lambda_u(H)$ of the stable and unstable eigenvalues of the linearized deterministic flow at $H$. If $\mu<1$ then the exit location distribution is generically asymptotic as $\epsilon\to0$ to a Weibull distribution with shape parameter $2/\mu$, on the $O(\epsilon^{\mu/2})$ length scale near $H$. If $\mu>1$ it is generically asymptotic to a distribution on the $O(\epsilon^{1/2})$ length scale, whose moments we compute. The asymmetry of the asymptotic exit location distribution is attributable to the generic presence of a `classically forbidden' region: a wedge-shaped subset of $\Omega$ with $H$ as vertex, which is reached from $S$, in the $\epsilon\to0$ limit, only via `bent' (non-smooth) fluctuational paths that first pass through the vicinity of $H$. We deduce from the presence of this forbidden region that the classical Eyring formula for the small-$\epsilon$ exponential asymptotics of the mean first exit time is generically inapplicable.Comment: This is a 72-page Postscript file, about 600K in length. Hardcopy requests to [email protected] or [email protected]

Supply chain challenges for sustainability: the case of waste textiles as raw materials

Purpose: This paper addresses the growing problem of textile waste in the rapidly developing cities of subSaharan Africa and examines, from a supply chain perspective, the potential for waste textile materials to be transformed into the raw materials for new consumer products. Research Approach: The paper reflects on the outcomes of a field trip to Dar es Salaam in which stakeholders in a hypothesised textile waste supply chain were interviewed and waste textile materials were analysed in order to determine their content and appropriateness for reuse. Findings from the field study have been compared with current literature on logistics and market creation, waste generation, management and recycling in sub-Saharan Africa. Findings and Originality: The findings show that a rudimentary system has been in place for many years to collect and recycle textiles in Dar es Salaam. However, at the same time as textile waste is projected to increase in the city, collection rates are falling. The chief reasons for the falling rates are failures in the ‘modernised mixture’ approach to waste collection employed by Dar es Salaam City Council and market failure for the collected materials. Alternative combinations of ‘modernised mixtures’, incorporating community-based organisations, are likely to increase textile yields from unplanned urban areas but previous high-profile failures in such systems within Dar es Salaam mean there is caution on both sides in entering into such a relationship. The more pressing problem is to identify appropriate end markets for the textile materials, since in a country where recycling is entirely market-driven, failure to do so will undermine any attempt to improve the collection system. Whilst many studies have considered general recycling practices in sub-Saharan Africa, there are few investigations into textile waste. Furthermore, those existing studies do not consider the importance of understanding fibre composition of the materials in order to determine the most appropriate end markets. Research Impact: The research contributes to the growing body of knowledge on ‘bottom of the pyramid’ approaches to sustainable futures. Practical Impact: The work presented considers supply chain problems and offers approaches to tackling the increasing waste management issues of Dar es Salaam and proposes a mechanism for doing so which has the potential to provide income for the poorest sectors of the urban society

Association between MAPT polymorphism but not APOE promoter and elite rugby athlete status

INTRODUCTION: Incidence and outcomes of concussions have been hypothesised to be genetically influenced. The APOE Promoter G219T (rs405509) polymorphism has been associated with differential promoter activity and unfavourable outcomes after traumatic brain injury. The TT genotype is associated with a 3-fold greater risk of multiple concussions. The TT genotype of MAPT (rs10445337) has also been associated with poorer outcomes after concussion. Rugby has one of the highest incidences of concussion in sport, so it was hypothesised that APOE Promoter TT and MAPT TT genotypes would be less prevalent in elite rugby athletes because those genotypes, previously associated with increased risk, would be less compatible with achieving elite athlete status. METHODS: Participants were from the RugbyGene project, comprising elite Caucasian male rugby athletes (n = 528; mean (standard deviation) height 1.85 (0.07) m, mass 101 (14) kg, age 29 (7) yr), including 420 rugby union (RU) athletes that for some analyses were divided into forwards and backs and 108 rugby league (RL) athletes. Non-athletes were 592 Caucasian men and women (57% male, height 1.72 (0.10) m, mass 74 (14) kg, age 31 (7) yr). PCR of genomic DNA was used to determine genotypes using TaqMan probes, then groups were compared using χ2 and odds ratio (OR) statistics. RESULTS: All genotype data were in Hardy-Weinberg equilibrium. For MAPT (rs10445337), the risk genotype (TT) was underrepresented in rugby athletes (60%) compared to non-athletes (66%), CT more common in rugby athletes (34%) than non-athletes (29%) and little difference in CC genotype frequencies (χ2 = 7.092, P = 0.029; TT genotype frequency OR = 0.80, 95% confidence intervals (CI) = 0.62-1.02). There were no differences in MAPT (rs10445337) genotype frequencies between RU forwards and backs. For APOE Promoter G219T (rs405509), there were no differences in genotype frequencies between all athletes (RU and RL) and non-athletes (27% TT genotype in players and non-athletes), nor between RU forwards and backs. CONCLUSION: The MAPT (rs10445337) TT genotype is 6% less common in elite rugby athletes than non-athletes. Therefore, carrying at least one rs10445337 C allele appears to increase the probability of sustained career success in the high-risk concussion environment of elite rugby, perhaps via a greater ability to recover from concussions.Peer reviewe

Association of MMP3 but not TIMP2 gene variants with elite rugby player status and rugby code

Introduction: Achilles tendon pathology and anterior cruciate ligament rupture are multifactorial conditions for which genetic risk factors have been identified. Single nucleotide polymorphisms (SNPs) within the MMP3 (rs591058, rs679620, rs650108) and TIMP2 (rs4789932) genes have previously been associated with tendon and ligament pathologies. Although not entirely clear, prior literature indicates the risk alleles for Achilles tendon pathology as T (rs591058), G (rs679620) and A (rs650108) for MMP3. However, prior evidence regarding TIMP2 is equivocal. MMP3 is considered an essential regulator of matrix degradation and remodelling within diseased and normal musculoskeletal soft tissues. TIMP2 maintains homeostasis in the extracellular matrix in part by inhibiting MMP function. Given the high incidence and severity of tendon and ligament injuries in elite rugby athletes, we hypothesised that the aforementioned SNPs would be associated with career success. Methods: Participants from the RugbyGene project were elite Caucasian male rugby athletes (n = 566; mean (standard deviation) height 1.85 (0.07) m, mass 101 (14) kg, age 29 (7) yr), including 420 rugby union (RU) athletes that for some analyses were divided into forwards and backs and 120 rugby league (RL) athletes. Non-athletes were 589 Caucasian men and women (n = 589, 57% male, height 1.72 (0.10) m, mass 74 (14) kg, age 31 (7) yr). PCR of genomic DNA was used to determine genotypes using TaqMan probes, then groups were compared using Χ2 and odds ratio (OR) statistics. Results: As hypothesized, the MMP3 rs591058 risk genotype (TT) was less frequent in rugby athletes (28%) compared to non-athletes (33%) (Χ2 = 7.265, P = 0.026; OR = 1.18, 95% confidence intervals (CI) = 0.86-1.63). No differences were found for MMP3 rs679620, rs650108 or TIMP2 rs4789932 between rugby athletes and non-athletes. When RL athletes were compared to non-athletes, the risk genotype (TT) of MMP3 rs591058 was underrepresented in RL athletes (19%) compared to non-athletes (33%). The MMP3 rs679620 ‘protective’ allele (C) was more frequent in RL athletes (55%) compared to non-athletes (48%) (OR = 1.3, 95% CI = 0.98-1.74). However, for MMP3 rs650108 the ‘risk’ allele (A) was overrepresented in RL athletes (32%) compared to non-athletes (26%). There were no genotype differences for any gene variant between RU athletes and non-athletes. The ‘risk’ allele (T) of the MMP3 rs679629 polymorphism and the ‘protective’ allele (G) of the MMP3 rs650108 polymorphism were less common in RL (45%, 68%, respectively) than RU athletes (54%, 76%, respectively). Conclusion: We provide evidence for elite rugby athletes possessing a protective genetic profile regarding tendon and ligament injury risk. Notably, a less frequent rs591058 TT genotype in athletes suggests a lower risk of injury could therefore enhance career success in rugby. Furthermore, RL players appear to have differing genetic characteristics compared to their RU counterparts, which might reflect some differences in physiological demands between codes.Peer reviewedFinal Published versio

Towards the prediction of critical transitions in spatially extended populations with cubical homology

The prediction of critical transitions, such as extinction events, is vitally important to preserving vulnerable populations in the face of a rapidly changing climate and continuously increasing human resource usage. Predicting such events in spatially distributed populations is challenging because of the high dimensionality of the system and the complexity of the system dynamics. Here, we reduce the dimensionality of the problem by quantifying spatial patterns via Betti numbers ($\beta_0$ and $\beta_1$), which count particular topological features in a topological space. Spatial patterns representing regions occupied by the population are analyzed in a coupled patch population model with Ricker map growth and nearest-neighbors dispersal on a two-dimensional lattice. We illustrate how Betti numbers can be used to characterize spatial patterns by type, which in turn may be used to track spatiotemporal changes via Betti number time series and characterize asymptotic dynamics of the model parameter space. En route to a global extinction event, we find that the Betti number time series of a population exhibits characteristic changes. We hope these preliminary results will be used to aide in the prediction of critical transitions in spatially extended systems. Additional applications of this technique include analysis of spatial data (e.g., GIS) and model validation.Comment: Published in Contemporary Mathematics: Dynamical Systems and Random Processes, Volume 736, 201

Phase diagram of the antiferromagnetic XY model in two dimensions in a magnetic field

The phase diagram of the quasi-two-dimensional easy-plane antiferromagnetic model, with a magnetic field applied in the easy plane, is studied using the self-consistent harmonic approximation. We found a linear dependence of the transition temperature as a function of the field for large values of the field. Our results are in agreement with experimental data for the spin-1 honeycomb compound BaNi_2V_2O_3Comment: 3 page

Stability of pulse-like earthquake ruptures

Pulse-like ruptures arise spontaneously in many elastodynamic rupture simulations and seem to be the dominant rupture mode along crustal faults. Pulse-like ruptures propagating under steady-state conditions can be efficiently analysed theoretically, but it remains unclear how they can arise and how they evolve if perturbed. Using thermal pressurisation as a representative constitutive law, we conduct elastodynamic simulations of pulse-like ruptures and determine the spatio-temporal evolution of slip, slip rate and pulse width perturbations induced by infinitesimal perturbations in background stress. These simulations indicate that steady-state pulses driven by thermal pressurisation are unstable. If the initial stress perturbation is negative, ruptures stop; conversely, if the perturbation is positive, ruptures grow and transition to either self-similar pulses (at low background stress) or expanding cracks (at elevated background stress). Based on a dynamic dislocation model, we develop an elastodynamic equation of motion for slip pulses, and demonstrate that steady-state slip pulses are unstable if their accrued slip $b$ is a decreasing function of the uniform background stress $\tau_\mathrm{b}$. This condition is satisfied by slip pulses driven by thermal pressurisation. The equation of motion also predicts quantitatively the growth rate of perturbations, and provides a generic tool to analyse the propagation of slip pulses. The unstable character of steady-state slip pulses implies that this rupture mode is a key one determining the minimum stress conditions for sustainable ruptures along faults, i.e., their ``strength''. Furthermore, slip pulse instabilities can produce a remarkable complexity of rupture dynamics, even under uniform background stress conditions and material properties