Nathan Clisby’s research while affiliated with Swinburne University of Technology and other places

We extend a binary tree implementation of the pivot algorithm to the face-centered cubic and body-centered cubic lattices, and use it to calculate the growth constant, $\mu$, for self-avoiding walks on these lattices. We find that \mufcc =10.037\, 057\, 85 \pm 0.000\, 000\, 14 and \mubcc = 6.530\, 511\, 501 \pm 0.000\, 000\, 084. In addition, we estimate the amplitudes \Afcc = 1.17119 \pm 0.00003 and \Abcc = 1.17637 \pm 0.00003, and provide convincing numerical evidence in support of the hypothesis that the critical exponent $\gamma$ is a universal quantity.

We have made a systematic numerical study of the 16 Wilf classes of length-5 classical pattern-avoiding permutations from their generating function coefficients. We have extended the number of known coefficients in fourteen of the sixteen classes. Careful analysis, including sequence extension, has allowed us to estimate the growth constant of all classes, and in some cases to estimate the sub-dominant power-law term associated with the exponential growth. In six of the sixteen classes we find the familiar power-law behaviour, so that the coefficients behave like $s_n \sim C \cdot \mu^n \cdot n^g,$ while in the remaining ten cases we find a stretched exponential as the most likely sub-dominant term, so that the coefficients behave like $s_n \sim C \cdot \mu^n \cdot \mu_1^{n^\sigma} \cdot n^g,$ where $0 < \sigma < 1.$ We have also classified the 120 possible permutations into the 16 distinct classes. We give compelling numerical evidence, and in one case a proof, that all 16 Wilf-class generating function coefficients can be represented as moments of a non-negative measure on $[0,\infty)$. Such sequences are known as Stieltjes moment sequences. They have a number of nice properties, such as log-convexity, which can be used to provide quite strong rigorous lower bounds. Stronger bounds still can be established under plausible monotonicity assumptions about the terms in the continued-fraction expansion of the generating functions implied by the Stieltjes property. In this way we provide strong (non-rigorous) lower bounds to the growth constants, which are sometimes within a few percent of the exact value.

The pivot algorithm is the most efficient known method for sampling polymer configurations for self-avoiding walks and related models. Here we introduce two recent improvements to an efficient binary tree implementation of the pivot algorithm: an extension to an off-lattice model, and a parallel implementation.

In this classroom note, we outline a system of assessment used by the authors since 2020 to deliver individualized summative assessments to students from first- and second-year mathematics courses. Our system comprises three modular components allowing a mix-and-match of different technological approaches and mathematical question types. First is a question generation module that generates appropriate variables and question syntax, second is a delivery module to send out the individualized assessment to students, and third is a marking module to generate worked solutions and final answers for markers. The key benefits of these assessments are an ability to incorporate individualized authentic elements into assessments, to allow access to technology that would be impractical for an invigilated exam setting, whilst overall reducing the likelihood – but increasing the ease of detection – of academic misconduct and contract cheating, compared with other non-invigilated assessment protocols.

We have made a systematic numerical study of the 16 Wilf classes of length-5 classical pattern-avoiding permutations from their generating function coefficients. We have extended the number of known coefficients in fourteen of the sixteen classes. Careful analysis, including sequence extension, has allowed us to estimate the growth constant of all classes, and in some cases to estimate the sub-dominant power-law term associated with the exponential growth. In six of the sixteen classes we find the familiar power-law behaviour, so that the coefficients behave like $s_n \sim C \cdot \mu^n \cdot n^g,$ while in the remaining ten cases we find a stretched exponential as the most likely sub-dominant term, so that the coefficients behave like $s_n \sim C \cdot \mu^n \cdot \mu_1^{n^\sigma} \cdot n^g,$ where $0 < \sigma < 1.$ We have also classified the 120 possible permutations into the 16 distinct classes. We give compelling numerical evidence, and in one case a proof, that all 16 Wilf-class generating function coefficients can be represented as moments of a non-negative measure on $[0,\infty).$ Such sequences are known as {\em Stieltjes moment sequences}. They have a number of nice properties, such as log-convexity, which can be used to provide quite strong rigorous lower bounds. Stronger bounds still can be established under plausible monotonicity assumptions about the terms in the continued-fraction expansion of the generating functions implied by the Stieltjes property. In this way we provide strong (non-rigorous) lower bounds to the growth constants, which are sometimes within a few percent of the exact value.

Many metrics for comparing greenhouse gas emissions can be expressed as an instantaneous global warming potential multiplied by the ratio of airborne fractions calculated in various ways. The forcing equivalent index (FEI) provides a specification for equal radiative forcing at all times at the expense of generally precluding point-by-point equivalence over time. The FEI can be expressed in terms of asymptotic airborne fractions for exponentially growing emissions. This provides a reference against which other metrics can be compared. Four other equivalence metrics are evaluated in terms of how closely they match the timescale dependence of FEI, with methane referenced to carbon dioxide used as an example. The 100-year global warming potential overestimates the long-term role of methane, while metrics based on rates of change overestimate the short-term contribution. A recently proposed metric based on differences between methane emissions 20 years apart provides a good compromise. Analysis of the timescale dependence of metrics expressed as Laplace transforms leads to an alternative metric that gives closer agreement with FEI at the expense of considering methane over longer time periods. The short-term behaviour, which is important when metrics are used for emissions trading, is illustrated with simple examples for the four metrics.

We analyze the phase-space compression, characteristic of all deterministic, dissipative systems for an inhomogeneous boundary-driven shear fluid via nonequilibrium molecular dynamics simulations. We find that, although the full system undergoes a phase space contraction, the marginal distribution of the fluid particles is described by a smooth, volume preserving probability density function. This is the case for most thermodynamic states of physical interest. Hence, we show that the models currently employed to investigate inhomogeneous fluids in a nonequilibrium steady state, in which only walls are thermostatted, generate a non-singular distribution for the fluid.