Tensor Products, Positive Linear Operators, and Delay-Differential Equations

We develop the theory of compound functional differential equations, which are tensor and exterior products of linear functional differential equations. Of particular interest is the equation $\dot x(t)=-\alpha(t)x(t)-\beta(t)x(t-1)$ with a single delay, where the delay coefficient is of one sign, say $\delta\beta(t)\ge 0$ with $\delta\in{-1,1}$. Positivity properties are studied, with the result that if $(-1)^k=\delta$ then the $k$-fold exterior product of the above system generates a linear process which is positive with respect to a certain cone in the phase space. Additionally, if the coefficients $\alpha(t)$ and $\beta(t)$ are periodic of the same period, and $\beta(t)$ satisfies a uniform sign condition, then there is an infinite set of Floquet multipliers which are complete with respect to an associated lap number. Finally, the concept of $u_0$-positivity of the exterior product is investigated when $\beta(t)$ satisfies a uniform sign condition.Comment: 84 page

John Mallet: 'This is the way that life happened'

Article for festschrift publication in honour of V&A former Keeper of Ceramics, John Mallet, drawing on extracts of his life history recording with Linda Sandino for the V&A Curators Lives oral history project

Snakes: Oriented families of periodic orbits, their sources, sinks, and continuation

AbstractPoincaré observed that for a differential equation x′ = ƒ(x, α) depending on a parameter α, each periodic orbit generally lies in a connected family of orbits in (x, α)-space. In order to investigate certain large connected sets (denoted Q) of orbits containing a given orbit, we introduce two indices: an orbit index φ and a “center” index defined at certain stationary points. We show that genetically there are two types of Hopf bifurcation, those we call “sources” ( = 1) and “sinks” ( = −1). Generically if the set Q is bounded in (x, α)-space, and if there is an upper bound for periods of the orbits in Q, then Q must have as many source Hopf bifurcations as sink Hopf bifurcations and each source is connected to a sink by an oriented one-parameter “snake” of orbits. A “snake” is a maximal path of orbits that contains no orbits whose orbit index is 0. See Fig. 1.1

Traveling Wave Solutions for Systems of ODEs on a Two-Dimensional Spatial Lattice

This is the published version, also available here: http://dx.doi.org/10.1137/S0036139996312703.We consider infinite systems of ODEs on the two-dimensional integer lattice, given by a bistable scalar ODE at each point, with a nearest neighbor coupling between lattice points. For a class of ideal nonlinearities, we obtain traveling wave solutions in each direction $e^{i\theta}$, and we explore the relation between the wave speed c, the angle $\theta$, and the detuning parameter a of the nonlinearity. Of particular interest is the phenomenon of "propagation failure," and we study how the critical value $a=a^*(\theta)$ depends on $\theta$, where $a^*(\theta)$ is defined as the value of the parameter a at which propagation failure (that is, wave speed c=0) occurs. We show that $a^*:\Bbb{R}\to\Bbb{R} is continuous at each point$\theta$where$\tan\theta$is irrational, and is discontinuous where$\tan\theta$ is rational or infinite

A model of workplace safety incorporating worker interactions and simple interventions

Although there was substantial research into the occupational health and safety sector over the past forty years, this generally focused on statistical analyses of data related to costs and/or fatalities and injuries. There is a lack of mathematical modelling of the interactions between workers and the resulting safety dynamics of the workplace. There is also little work investigating the potential impact of different safety intervention programs prior to their implementation. In this article, we present a fundamental, differential equation-based model of workplace safety that treats worker safety habits similarly to an infectious disease in an epidemic model. Analytical results for the model, derived via phase plane and stability analysis, are discussed. The model is coupled with a model of a generic safety strategy aimed at minimising unsafe work habits, to produce an optimal control problem. The optimal control model is solved using the forward-backward sweep numerical scheme implemented in Matlab. References Australian Bureau of Statistics. Forms of employment. Commonwealth of Australia, 2010. http://www.abs.gov.au/AUSSTATS/[email protected]/DetailsPage/6359.0November%202010?OpenDocument Bahn, S. Power and Influence: Examining the Communication Pathways that Impact on Safety in the Workplace. J. Occup. Health Safety–-Aust. N.Z., 25(3):213–222, 2009. Bird, P. Reducing Manual Handling Workers Compensation Claims in a Public Health Facility. J. Occup. Health Safety–-Aust. N.Z., 25(6):451–459, 2009. Breslin, P. Improving ohs Standards in the Building and Construction Industry through safe design. J. Occup. Health Safety–-Aust. N.Z., 23(4):89–99, 2007. Breslin, P. National Harmonisation: Designers' Duties of Care in the Australian Building and Construction Industry. J. Occup. Health Safety–-Aust. N.Z., 25(6):495–504, 2009 . Driscoll, T., Mitchell, T., Mandryk, J., Healey, S., Hendrie, L. and Hull, B. Trends in Work-Related Fatalities in Australia, 1982 to 1992. J. Occup. Health Safety–-Aust. N.Z., 18(1):21–33, 2002. Driscoll, T. Fatal Injury of young workers in Australia. J. Occup. Health Safety–-Aust. N.Z., 22(2):151–161, 2006. Foley, G., Gale, J. and Gavenlock, L. The Cost of Work-Related Injury and Disease. J. Occup. Health Safety–-Aust. N.Z., 11(2):171–194, 1995. Glendon, I. and Waring, A. Risk management as a framework for occupational health and safety. J. Occup. Health Safety–-Aust. N.Z., 13(6):525–532, 1997. Gunningham, N. and Healy, P. Agricultural ohs Policy: Towards Systemic Reform. J. Occup. Health Safety–-Aust. N.Z., 20(4):311–318, 2004. Hawkins, A., Eather, J. and Fragar, L. Improving Health and Safety in the Farm Workshop. J. Occup. Health Safety–-Aust. N.Z., 24(2):155–160, 2008. Heads of Workers' Compensation Authorities. 2008/09 Australia and New Zealand Return to Work Monitor. http://www.hwca.org.au/documents/Australia%20and%20New%20Zealand%20Return%20to%20Work%20Monitor%202008-2009.pdf Kermack, W. O. and McKendrick, A. G. A Contribution to the Mathematical Theory of Epidemics. Proc. R. Soc. Lond. A., 115(772):700–721, 1927. doi:10.1098/rspa.1927.0118 Lenhart, S. and Workman, J. T. Optimal control applied to biological models. Chapman and Hall CRC Mathematical and Computational Biology Series, 2007. Mallet, D. G., Bagher-Oskouei, M., Farr, A. C., Simpson, D. P. and Sutton, K-J. A mathematical model of Chlamydial infection incorporating movement of Chlamydial particles. B. Math. Biol., 75(11):2257–2270, 2013. doi:10.1007/s11538-013-9891-9 Murray, J. D. Mathematical Biology, I: An Introduction. Springer, 2002. Raggett, G. F. Modelling the Eyam plague. B. I. Math. Appl., 18:221–226, 1982. Safe Work Australia. The cost of work-related injury and illness for australian employers, workers and the community:2008-2009. Commonwealth of Australia, 2012. http://www.safeworkaustralia.gov.au/sites/SWA/about/Publications/Documents/660/Cost%20of%20Work-related%20injury%20and%20disease.pdf Shulgin, B., Stone, L. and Agur, Z. Pulse vaccination strategy in the sir Epidemic Model. B. Math. Biol., 60(6):1123–1148, 1998. doi:10.1006/S0092-8240(98)90005-2 Vanderkruk, R. Workplace health and safety officers: a Queensland success story. J. Occup. Health Safety–-Aust. N.Z., 15(6):557–563, 1999. Winder, C. The development of ohs legislation in Australia. J. Occup. Health Safety–-Aust. N.Z., 25(4):277–287, 2009

Noiseless nonreciprocity in a parametric active device

Nonreciprocal devices such as circulators and isolators belong to an important class of microwave components employed in applications like the measurement of mesoscopic circuits at cryogenic temperatures. The measurement protocols usually involve an amplification chain which relies on circulators to separate input and output channels and to suppress backaction from different stages on the sample under test. In these devices the usual reciprocal symmetry of circuits is broken by the phenomenon of Faraday rotation based on magnetic materials and fields. However, magnets are averse to on-chip integration, and magnetic fields are deleterious to delicate superconducting devices. Here we present a new proposal combining two stages of parametric modulation emulating the action of a circulator. It is devoid of magnetic components and suitable for on-chip integration. As the design is free of any dissipative elements and based on reversible operation, the device operates noiselessly, giving it an important advantage over other nonreciprocal active devices for quantum information processing applications.Comment: 17 pages, 4 figures + 12 pages Supplementary Informatio

Major improvements to the Heliconius melpomene genome assembly used to confirm 10 chromosome fusion events in 6 million years of butterfly evolution

The Heliconius butterflies are a widely studied adaptive radiation of 46 species spread across Central and South America, several of which are known to hybridize in the wild. Here, we present a substantially improved assembly of the Heliconius melpomene genome, developed using novel methods that should be applicable to improving other genome assemblies produced using short read sequencing. First, we whole-genome-sequenced a pedigree to produce a linkage map incorporating 99% of the genome. Second, we incorporated haplotype scaffolds extensively to produce a more complete haploid version of the draft genome. Third, we incorporated ~20x coverage of Pacific Biosciences sequencing, and scaffolded the haploid genome using an assembly of this long-read sequence. These improvements result in a genome of 795 scaffolds, 275 Mb in length, with an N50 length of 2.1 Mb, an N50 number of 34, and with 99% of the genome placed, and 84% anchored on chromosomes. We use the new genome assembly to confirm that the Heliconius genome underwent 10 chromosome fusions since the split with its sister genus Eueides, over a period of about 6 million yr