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A generic method for finding central configurations in the general four body coplanar problem with four different masses is derived. New families of non-symmetrical central configurations with non-equal masses are found and all known symmetrical four body non-collinear central configurations are shown to be derivable subsets using this generic model. In this model, three masses are distributed on the vertices of a triangle. Then the fourth mass is allowed to be at any other point in the plane forming either convex or concave central configurations. In this general setting, a necessary condition is derived for the existence of central configurations and for requiring the masses to be positive. Using both analytical and numerical techniques, regions of possible central configurations can be derived. The special cases of four body central configurations investigated include isosceles trapezoids, right trapezoids, right quadrilaterals, convex kites and concave kites. In most of these special cases, the number of parameters of the problem can be reduced from four to two and hence the regions of existence can be shown graphically. A number of cases both convex and concave with no symmetry restrictions or equality of masses were also investigated to demonstrate the generality of the method for finding central configurations in the general four body coplanar problem with four different masses.

Species' responses to broad-scale environmental or spatial gradients are typically unimodal. Current models of species' responses along gradients tend to be overly simplistic (e.g., linear, quadratic or Gaussian GLMs), or are suitably flexible (e.g., splines, GAMs) but lack direct ecologically interpretable parameters. We describe a parametric framework for species-environment non-linear modelling ('senlm'). The framework has two components: (i) a non-linear parametric mathematical function to model the mean species response along a gradient that allows asymmetry, flattening/peakedness or bimodality; and (ii) a statistical error distribution tailored for ecological data types, allowing intrinsic mean-variance relationships and zero-inflation. We demonstrate the utility of this model framework, highlighting the flexibility of a range of possible mean functions and a broad range of potential error distributions, in analyses of fish species' abundances along a depth gradient, and how they change over time and at different latitudes.

We consider four- and five-body problems with symmetrical masses (Caledonian problems). Families of periodic orbits originate from the collinear Schubart orbits. We present and discuss some of these periodic orbits.

This paper describes the implementation of a class of IRKS methods (Wright 2002). These GLM algorithms are practical with reliable error estimators (Butcher and Podhaisky, Appl. Numer. Math. 56, 345–357 2006). The current robust ODE solvers in variable stepsize as well as in variable-order mode are based upon heuristics. In this paper, we examine an optimisation approach, based on Euler-Lagrange theory (Butcher, IMA J. Numer. Anal. 6, 433–438 1986), (Butcher, Computing 44, 209–220 1990), to control the stepsize as well as the order and implement the GLMs in an efficient manner. A set of nonstiff to mildly stiff problems have been used to investigate this approach in fixed-order and variable-order modes.

In this paper, we develop an analytical stability criterion for a five-body symmetrical system, called the Caledonian Symmetric Five-Body Problem (CS5BP), which has two pairs of equal masses and a fifth mass located at the centre of mass. The CS5BP is a planar problem that is configured to utilise past–future symmetry and dynamical symmetry. The introduction of symmetries greatly reduces the dimensions of the five-body problem. Sundman’s inequality is applied to derive boundary surfaces to the allowed real motion of the system. This enables the derivation of a stability criterion valid for all time for the hierarchical stability of the CS5BP. We show that the hierarchical stability depends solely on the Szebehely constant $$C_0$$ C 0 which is a dimensionless function involving the total energy and angular momentum. We then explore the effect on the stability of the whole system of varying the relative sizes of the masses. The CS5BP is hierarchically stable for $$C_0 > 0.065946$$ C 0 > 0.065946 . This criterion can be applied in the investigation of the stability of quintuple hierarchical stellar systems and symmetrical planetary systems.

This paper presents mathematical models that can simulate the cardiovascular system of a healthy sheep under normal resting conditions in which the heart rate changes significantly. The models include several new modelling features that are introduced progressively. The contraction of the cardiac chambers is modelled using a time-dependent muscle force with constant elasticity instead of time dependent elasticity. A new hypothesis about the mechanical contraction of the atria generates realistic pressure volume (PV) loops. The inter-ventricular interaction is modelled as well. Additionally, hysteresis is incorporated in the aortic valve to produce an end-systolic reverse (negative) flow. Most of the model parameter values are based on previous literature data while time periods of delay, atrial and ventricular contraction are derived using experimental data from 14 sheep. We provide new relationships between contraction time and delay as a function of heart period. The effects of different aspects of our modelling on the mean cardiac output, stroke volume, ejection time, ejection fraction and PV loops are studied. Model outputs are compared with published experimental results where possible, and are within a wide range of physiological observations.

The current paper examines some novel and interesting aspects of the mathematical modelling of fluid flow and forced convection heat transfer from a plate stack embedded in a homogeneous porous media. Time-dependent forced convection heat transfer from some cylinders embedded in groundwater are explored numerically using finite difference methods. An algorithm was developed about rectangular heated solid objects with the same size placed in an in-line arrangement in homogeneous porous media. Central and one-sided finite difference formulae are used to discretize the domain. Applying the mathematical model, continuity, momentum, and energy equations are solved by the finite difference methods. Generally, analytical solutions are not possible for such problems, so most of the work is done numerically. The numerical results for the two-dimensional velocity potential and stream function have been calculated and the resulting fluid and temperature contours are represented graphically for various values of the parameters involved (e.g. length of the entrance, the fluid flow rates, coefficient of thermal diffusion and advection in the observed long-time plume behaviour, effect of time and temperature on heat transfer, etc.). It is observed that fluid and heat fluxes and fluid velocity depend upon the size, shape, burial depth, position of heated objects, and positions of the entrance and exit of the homogeneous porous media. Moreover, the hydraulic conductivity of the porous medium affects the pressure drops, fluid velocity as well as the heat transfer rates.

We locate members of a one-parameter family of equal-mass four-body periodic orbits in the plane. The family begins and ends with the rectilinear four-body equal-mass Schubart interplay orbit and passes through a double choreography orbit. The first-order stability of these orbits is computed. Some members of this symmetric family are stable to symmetric perturbations; however, they are unstable when all perturbations are allowed.

We present an approach for modelling the transport of pollution within a groundwater aquifer. The model is based on vertical discretisation of the aquifer into a number of horizontal layers. The approach can be used in a variety of situations. In particular, aquifers with lenses and phreatic surfaces are discussed. The use of the model is illustrated with examples including a comparison with experimental field data and the use of a second transported substance for remediation of the pollution. References A. Ali (2015) Simplified modelling of pollutant transport in naturally-layered aquifers , PhD Thesis, Massey University, NZ. https://mro.massey.ac.nz/handle/10179/6782 A. Ali, R. McKibbin, W.L. Sweatman (2012) Fluid flow and solute transport in unevenly-stratified groundwater aquifers , AIP Conference Proceedings, 1453, 11–16. doi:10.1063/1.4711146 A. Ali, R. McKibbin, W.L. Sweatman (2012) Simplified modelling of pollutant transport in stratified groundwater aquifers , Proceedings of IEMSs 2012, 2739–2746. http://www.iemss.org/sites/iemss2012/proceedings/I1_0675_Ali_et_al.pdf A. Ali, R. McKibbin, W.L. Sweatman (2012) Tracer transport: Appearing or disappearing sub-layer in a groundwater aquifer , In: G. Mallinson, ed., Proceedings of the 23rd International Symposium on Transport Phenomena (ISTP), The University of Auckland, NZ. A. Ali, R. McKibbin, W.L. Sweatman (2014) A simplified model for transport in aquifers , Proceedings of ICPM5, ECI Symposium Series. http://dc.engconfintl.org/porous_media_V/56 A. Ali, W.L. Sweatman, R. McKibbin (2014) Pollutant transport and its alleviation in groundwater aquifers , Mathematics in Industry, 19, 53–58. doi:10.1007/978-3-319-05365-3_8 M.P. Anderson, J.A. Cherry (1979) Using models to simulate the movement of contaminants through groundwater flow systems , CRC Critical Reviews in Environmental Control, 9(2):97–156. doi:10.1080/10643387909381669 , J. Bear (1972) Dynamics of fluid in porous media Dover, New York. ISBN 10 0486656756 J. Bear, Y. Bachmat (1991) Introduction to modeling of transport phenomena in porous media , Kluwer. ISBN 978-94-009-1926-6 J. Bear, A. Verruijt (1978) Modeling groundwater flow and pollution D. Reidel, Dordrecht, Holland. H.P.G. Darcy (1856) Les fontaines publiques de la ville de dijon , Technical report, Victor Dalmont, Paris. J. Dupuit (1863) Etudes theoriques et pratiques sur le mouvement des eaux dans les canaux decouverts et a travers les terrains permeables , Dunod, Paris, 2nd edition. D.L. Freyberg (1986) A natural gradient experiment on solute transport in a sand aquifer: 2. Spatial moments and the advection and dispersion of nonreactive tracers , Water Resources Research, 22, 2031–2046. doi:10.1029/WR022i013p02031 D.M. Mackay, D.L. Freyberg, P.V. Roberts, J.A. Cherry (1986) A natural gradient experiment on solute transport in a sand aquifer: 1. Approach and overview of plume movement , Water Resources Research, 22, 2017–2029. doi:10.1029/WR022i013p02017 R. McKibbin (2009) Groundwater pollutant transport: transforming layered models to dynamical systems , An. St. Univ. Ovidius Constanta, Ser. Mat., 17(3), 183–196. http://www.emis.ams.org/journals/ASUO/mathematics/pdf19/McKibbin.pdf R. McKibbin (2010) Some aspects of modelling pollution transport in groundwater aquifers , In: L.H. Weiranto, S. Pudjaprasetya, editors, Proceedings of CIAM 2010, 9–16, ITB, Indonesia.

The extraction of ore from a mine must be scheduled to meet specific order production targets or `builds'. A number of physical, logical, and capacity constraints affect this planning. There is uncertainty in the process due to the imprecision of our knowledge of the mine's content until extracted from the ground. At the 2016 mathematics-in-industry study group workshop in Australia, Schneider Electric presented a project to consider mine scheduling. This paper reports on the assorted modelling approaches: exploration of sample data; considerations of the physically feasible mining sequences; the construction of a mixed integer program; a general heuristic strategy for dealing with different levels of uncertainty; and a build simulation. These provide promising avenues for further research on mine sequencing and related problems. References M. Menabde, G. Froyland, P. Stone, G. Yeates (2005), ``Mining schedule optimisation for conditionally simulated orebodies'', in Dimitrakopoulos R. (ed.) Orebody modelling and strategic mine planning: uncertainty and risk management models , Australasian Inst. Mining and Metallurgy, pp. 343–357, ISBN:1-920806-42-3 M. Ibrahimov, A. Mohais, S. Schellenberg, Z. Michalewicz (2014). ``Scheduling in iron ore open-pit mining'', The International Journal of Advanced Manufacturing Technology , 72(5–8), 1021–1037. doi:10.1007/s00170-014-5619-8 L. Caccetta, S. P. Hill (2003) ``An Application of Branch and Cut to Open Pit Mine Scheduling'' Journal of Global Optimization , 27, 349–365. doi:10.1023/A:1024835022186

We describe an application written for the Android platform for receiving real-time notifications of astrophysical transient events. The key feature of our application is the use of message oriented middleware as a message broker, with the messages in VOEvent format. We describe the design features and implementation details of our application. In particular, it was necessary to implement support for the Simple Text Oriented Messaging Protocol (STOMP) to allow communication with the broker. Our application is designed around VOEvent alerts from the Microlensing Observations in Astrophysics (MOA) project, but could easily be adapted for other surveys that issue VOEvent notices of astrophysical transients.

This study relates to equal-mass four-body orbits close to a quadruple central configuration. Locally, these orbits can be approximated by a perturbation from the homothetic quadruple collision/expansion orbit. Appropriate expressions are derived and the equal-mass four-body Siegel exponents and associated eigenmodes are presented.

Global "second-generation" microlensing surveys aim to discover and characterize extrasolar planets and their frequency, by means of round-the-clock high-cadence monitoring of a large area of the Galactic bulge, in a controlled experiment. We report the discovery of a giant planet in microlensing event MOA-2011-BLG-322. This moderate-magnification event, which displays a clear anomaly induced by a second lensing mass, was inside the footprint of our second-generation microlensing survey, involving MOA, OGLE and the Wise Observatory. The event was observed by the survey groups, without prompting alerts that could have led to dedicated follow-up observations. Fitting a microlensing model to the data, we find that the timescale of the event was t_E=23.2 +/-0.8 days, and the mass ratio between the lens star and its companion is q=0.028 +/-0.001. Finite-source effects are marginally detected, and upper limits on them help break some of the degeneracy in the system parameters. Using a Bayesian analysis that incorporates a Galactic structure model, we estimate the mass of the lens at 0.39 +0.45/-0.19 M_sun, at a distance of 7.56 +/-0.91 kpc. Thus, the companion is likely a planet of mass 11.6 +13.4/-5.6 M_J, at a projected separation of 4.3 +1.5/-1.2 AU, rather far beyond the snow line. This is the first pure-survey planet reported from a second-generation microlensing survey, and shows that survey data alone can be sufficient to characterize a planetary model. With the detection of additional survey-only planets, we will be able to constrain the frequency of extrasolar planets near their systems' snow lines., MNRAS in press, 7 pages, 5 figures, 2 tables

This project considers the retrieval of vanadium and similar valuable by-products when producing steel from iron sands. This recovery should be optimised without compromising steel production. The process is modelled with a system of differential equations. A method is outlined that optimises vanadium recovery once parameters have been identified experimentally. References G. Aylward and T. Findlay. SI Chemical data . Wiley, 1994. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelize and E.F. Mishchenko. The Mathematical Theory of Optimal Processes . Wiley, 1962.

Cheddar cheese undergoes a number of biochemical changes during ripening. These processes were modelled with differential equations in a project at MISG2013 (the 2013 mathematics-in-industry study group) at Queensland University of Technology, Australia. Models could aid in the prediction of cheese quality from initial measurements. The model is presented and the effect of small changes in initial conditions is explored.

In this paper, seaweed cultivation is mathematically modelled. The potential use of the crop to consume by-products from ethanol production is considered with a feasibility study and simple financial model. The growth of seaweed is described using differential equations and considering various factors such as solar radiation. Refereed/Peer-reviewed

The microlensing event OGLE-2008-BLG-510 is characterised by an evident asymmetric shape of the peak, promptly detected by the ARTEMiS system in real time. The skewness of the light curve appears to be compatible both with binary-lens and binary-source models, including the possibility that the lens system consists of an M dwarf orbited by a brown dwarf. The detection of this microlensing anomaly and our analysis demonstrates that: 1) automated real-time detection of weak microlensing anomalies with immediate feedback is feasible, efficient, and sensitive, 2) rather common weak features intrinsically come with ambiguities that are not easily resolved from photometric light curves, 3) a modelling approach that finds all features of parameter space rather than just the `favourite model' is required, and 4) the data quality is most crucial, where systematics can be confused with real features, in particular small higher-order effects such as orbital motion signatures. It moreover becomes apparent that events with weak signatures are a silver mine for statistical studies, although not easy to exploit. Clues about the apparent paucity of both brown-dwarf companions and binary-source microlensing events might hide here.

We have observed 7 new transits of the `hot Jupiter' WASP-5b using a 61 cm telescope located in New Zealand, in order to search for transit timing variations (TTVs) which can be induced by additional bodies existing in the system. When combined with other available photometric and radial velocity (RV) data, we find that its transit timings do not match a linear ephemeris; the best fit \chi^2 values is 32.2 with 9 degrees of freedom which corresponds to a confidence level of 99.982 % or 3.7 \sigma. This result indicates that excess variations of transit timings has been observed, due either to unknown systematic effects or possibly to real TTVs. The TTV amplitude is as large as 50 s, and if this is real, it cannot be explained by other effects than that due to an additional body or bodies. From the RV data, we put an upper limit on the RV amplitude caused by the possible secondary body (planet) as 21 m s^{-1}, which corresponds to its mass of 22-70 M_{Earth} over the orbital period ratio of the two planets from 0.2 to 5.0. From the TTVs data, using the numerical simulations, we place more stringent limits down to 2 M_{Earth} near 1:2 and 2:1 mean motion resonances (MMRs) with WASP-5b at the 3 \sigma level, assuming that the two planets are co-planer. We also put an upper limit on excess of Trojan mass as 43 M_{Earth} (3 \sigma) using both RV and photometric data. We also find that if the possible secondary planet has non- or a small eccentricity, its orbit would likely be near low-order MMRs. Further follow-up photometric and spectroscopic observations will be required to confirm the reality of the TTV signal, and results such as these will provide important information for the migration mechanisms of planetary systems., Comment: accepted for publication in PASJ, 30 pages, 8 figures, 8 tables

We report the result of the analysis of the light curve of a caustic-crossing binary-lens microlensing event OGLE-2009-BLG-023/MOA-2009-BLG-028. Even though the event was observed solely by survey experiments, we could uniquely determine the mass of the lens and distance to it by simultaneously measuring the Einstein radius and lens parallax. From this, we find that the lens system is composed of M-type dwarfs with masses $(0.50\pm 0.07) \ M_\odot$ and $(0.15\pm 0.02)\ M_\odot$ located in the Galactic disk with a distance of $\sim 1.8$ kpc toward the Galactic bulge direction. The event demonstrates that physical lens parameters of binary-lens events can be routinely determined from future high-cadence lensing surveys and thus microlensing can provide a new way to study Galactic binaries.

The problem of coating steel by passing it through a molten alloy and then stripping off excess coating using an air jet is considered. The work revisits the analysis of Tuck (Phys Fluids 26:2352, 1983) with the addition of extra shear terms and a consideration of the effect of increased air-jet speeds. A first-order partial differential equation is derived and solved both to obtain the steady-state coating shape and to determine the evolution of any defects that may form.

In the late stages of steel production in the New Zealand Steel hot strip mill, the sheets pass through a series of four pairs of rollers that squeeze the metal to the required thickness. The gaps between the pairs of rollers are controlled by controllers based upon a combination of models with feedback from measurements. New Zealand Steel brought a project to consider this process to the 2016 Mathematics-in-Industry New Zealand Study Group. The lines of investigation described include mathematical models based on the physics of the process, which have identified the primary parameters affecting steel processing between the rollers; and statistical analysis of data provided, revealing that the present electronic controllers improve target gauge over time. The first few metres of a strip are often outside acceptable thickness limits after going through the rollers. We diagnose likely causes and suggest possible cures. References Anderssen, R., Fowkes, N., Hickson, R., McGuinness, M.: Analysis of coil slumping. In: T. Marchant, M. Edwards, G. Mercer (eds.), Proceedings of the 2009 Mathematics and Statistics in Industry Study Group , pp. 90–108, University of Wollongong, Australia ISBN: 978-1-74128-181-1 (2010) Barry, S.I., Sweatman, W.L.: Modelling heat transfer in steel coils. ANZIAM J. 50, pp. C668–C681 ISSN: 1446-8735 (2009) Fraser, W.B., Macaskill, C., McGuinness, M., Thornton, A.: Strip track-off and buckling between transport rollers. In: T. Marchant, M. Edwards, G. Mercer (eds.), Proceedings of the 2007 Mathematics and Statistics in Industry Study Group , pp. 13–31, University of Wollongong, Australia ISBN 978-0-646-48555-3 (2008) Hocking, G.C., Sweatman, W.L., Fitt A.D., Breward, C.: Deformations during jet-stripping in the galvanizing process. Journal of Engineering Mathematics , 70, 297–306 doi:10.1007/s10665-010-9394-8 (2011) Hocking, G.C., Sweatman, W.L., Fitt A.D., Breward, C.: Deformations arising during air-knife stripping in the galvanisation of steel. In: M. Gunther, A. Bartel, M. Brunk, S. Schops, M. Striebel (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, Springer Berlin Heidelberg, 311–317 doi:10.1007/978-3-642-25100-9_36 (2012) Hocking, G.C., Sweatman, W.L., Roberts, M.E., Fitt A.D.: Coating Deformations in the continuous hot-dipped galvanizing process. In: T. Marchant, M. Edwards, G. Mercer (eds.), Proceedings of the 2009 Mathematics and Statistics in Industry Study Group , pp. 75–89, University of Wollongong, Australia ISBN: 978-1-74128-181-1 (2010) Johnson, K.L.: Contact Mechanics , Cambridge University Press, Cambridge, pp 452 ISBN 0-521-34796-3 (1987) Koiso, M., Ninomiya., Y., Yamamoto, M. (eds.): Study Group Workshop 2016, MI Lecture Note Series , 71, 92–111, ISSN:2188-1200 (2016) Marchant, T., Nickerson, A., Scott, D., Taylor, S.: Development of empirical relationships for metallurgical design of hot-rolled steel products. In: G.C. Wake (ed.), Proceedings of the 2005 Mathematics in Industry Study Group , pp. 53–72, Massey University, New Zealand ISBN 0-473-10423-7 (2005) McGuinness, M., Sweatman, W.L., Baowan, D., Barry, S.I.: Annealing Steel Coils. In: T. Marchant, M. Edwards, G. Mercer (eds.), Proceedings of the 2008 Mathematics and Statistics in Industry Study Group , pp. 61–80, University of Wollongong, Australia ISBN 978-0-646-50544 (2009) McGuinness, M., Taylor, S.: Strip temperature in a metal coating line annealing furnace. In: G.C. Wake (ed.), Proceedings of the 2004 Mathematics in Industry Study Group , pp. 23–45, Massey University, New Zealand ISBN 0-476-01130-2 (2005) Nishii, R., Okada, K., Kajiwara, K., Takagi, T., Wakayama, M., Waki, H., Yamamoto, M. (eds.): Study Group Workshop 2014, MI Lecture Note Series , 59, 92–110, ISSN:2188-1200 (2014) Okada, K., Fujisawa, K., Shirai, T., Wakayama, M., Waki, H., Broadbridge, P., Yamamoto, M. (eds.): Study Group Workshop 2015, MI Lecture Note Series , 66, 29–45, ISSN:2188-1200 (2015) Saeki, O., Wakayama, M., Yamamoto, M. (eds.): Study Group Workshop 2012, MI Lecture Note Series , 42, 93–121, ISSN:1881-4042 (2012) Saeki, O., Okada, K., Takagi, T., Wakayama, M., Yamamoto, M. (eds.): Study Group Workshop 2013, MI Lecture Note Series , 52, 109–125, ISSN:2188-1200 (2013) Scott, D.J., Russell, K., Scheffer, J.: Multi-variable relationships in a batch annealing process. In: G.C. Wake (ed.), Proceedings of the 2006 Mathematics in Industry Study Group , pp. 33–55, Massey University, New Zealand ISBN 0-473-11068-7 (2007) Sweatman, W.L.: Mathematics-in-industry study group (MISG) steel projects from Australia and New Zealand, in M. Wakayama, R.S. Anderssen, J. Cheng, Y. Fukumoto, R. McKibbin, K. Polthier, T. Takagi, K.-C. Toh (eds.), The impact of applications on mathematics, Mathematics for Industry , 1, Springer, 307-322, ISBN 978-4-431-54907-9 (ebook), ISBN 978-4-431-54906-2 (hardcover) (2014) Sweatman, W.L., McGuinness, M. and Barry, S.I.: Heat transfer during annealing of steel coils. In: M. Gunther, A. Bartel, M. Brunk, S. Schops, M. Striebel (eds.), Progress in Industrial Mathematics at ECMI 2010, Mathematics in Industry 17, Springer Berlin Heidelberg, 303–309 doi:10.1007/978-3-642-25100-9_35 (2012) Sweatman, W.L., Wake, G.C., Fullard, L., Bruna, M.: Recovering vanadium during the production of steel from iron sand. ANZIAM J. 53, M1–M21 doi:10.21914/anziamj.v53i0.4674 (2012) von Karman, T.: Beitrag zur Theorie des Walzvorganges, Z. angew Math. Mech. 5, 139 (1925) Wakayama, M., Fukumoto, Y., Takagi, T., Yamamoto, M. (eds.): Study Group Workshop 2010, MI Lecture Note Series , 27, 72–86, ISSN : 1881-4042 (2010) Wakayama, M., Fukumoto, Y., Takagi, T., Yamamoto, M. (eds.): Study Group Workshop 2011, MI Lecture Note Series , 33, 101–118, ISSN:1881-4042 (2011)

Several papers in the last decade have studied the Caledonian symmetric four-body problem (CSFBP), a restricted four-body system with a symmetrically reduced phase space. During these studies, difficulties have arisen when the system approaches a close encounter. These are due to collision singularities causing numerical integration algorithms to fail. In this paper, we give the full details of a regularisation approach that now enables us to study these close encounters and collision events. The resulting equations of motion can be efficiently integrated by a high-order integrator. The results from numerical testing of the algorithm verify that the regularisation is advantageous in preserving numerical stability. The effectiveness of the approach is illustrated for a range of CSFBP orbits. Numerical experiments show that the newly developed regularisation algorithm has excellent energy conservation properties.

We present a simple mathematical model for river pollution and investigate the effect of aeration on the degradation of pollutant. The model consists of a pair of coupled reaction–diffusion–advection equations for the pollutant and dissolved oxygen concentrations, respectively. The coupling of these equations occurs because of reactions between oxygen and pollutant to produce harmless compounds. Here we consider the steady-state case in one spatial dimension. For simplified cases the model is solved analytically. We also present a numerical approach to the solution in the general case. The extension to the transient spatial model is relatively straightforward. The study is motivated by the crucial problem of water pollution in many countries and specifically within the Tha Chin River in Thailand. For such real situations, simple models can provide decision support for planning restrictions to be imposed on farming and urban practices.

A two-dimensional inverse computing method to detect an internal object using microwave measurements is presented. The modelling of the application system has been directed towards the eventual in vivo detection of breast tumours, in particular. This procedure will enable the non-destructive determination of the internal object size and location. The model procedure has been tested by using complex reflection coefficients measured in a microwave experimental application system. Measured data agree with the theoretical calculations.

We report the detection of an extrasolar planet of mass ratio q ~ 2 x 10^(-4) in microlensing event MOA-2007-BLG-192. The best fit microlensing model shows both the microlensing parallax and finite source effects, and these can be combined to obtain the lens masses of M = 0.060 (+0.028 -0.021) M_sun for the primary and m = 3.3 (+4.9 -1.6) M_earth for the planet. However, the observational coverage of the planetary deviation is sparse and incomplete, and the radius of the source was estimated without the benefit of a source star color measurement. As a result, the 2-sigma limits on the mass ratio and finite source measurements are weak. Nevertheless, the microlensing parallax signal clearly favors a sub-stellar mass planetary host, and the measurement of finite source effects in the light curve supports this conclusion. Adaptive optics images taken with the Very Large Telescope (VLT) NACO instrument are consistent with a lens star that is either a brown dwarf or a star at the bottom of the main sequence. Follow-up VLT and/or Hubble Space Telescope (HST) observations will either confirm that the primary is a brown dwarf or detect the low-mass lens star and enable a precise determination of its mass. In either case, the lens star, MOA-2007-BLG-192L, is the lowest mass primary known to have a companion with a planetary mass ratio, and the planet, MOA-2007-BLG-192Lb, is probably the lowest mass exoplanet found to date, aside from the lowest mass pulsar planet., Accepted for publication in the Astrophysical Journal. Scheduled for the Sept. 1, 2008 issue

Numerous tephra dispersion and sedimentation models rely on some abstraction of the volcanic plume to simplify forecasts of tephra accumulation as a function of the distance from the volcano. Here we present solutions to the commonly used advection-dispersion equation using a variety of source shapes: a point, horizontal and vertical lines, and a circular disk. These may be related to some volcanic plume structure, such as a strong plume (vertical line), umbrella cloud (circular disk), or co-ignimbrite plume (horizontal line), or can be used to build a more complex plume structure such as a series of circular disks to represent a buoyant weak plume. Basing parameters upon eruption data, we find that depositions for the horizontal source shapes are very similar but differ from the vertical line source deposition. We also compare the deposition from a series of stacked circular disk sources of increasing radius above the volcanic vent with that from a vertical line source.

Interactions between pairs of binary stars play an important role in the dynamics of dense stellar systems. One possible outcome of a binary-binary encounter is full ionization: the destruction of both interacting binaries resulting in four separating single stars. This process is only possible for positive total energy, and the cross-section for full ionization falls to zero at zero total energy. In the context of a stellar system, full ionization is most likely to involve the interaction of two soft binaries with low binding energies. We consider the case where the four stars have a similar mass and derive an asymptotic power law relating the cross-section for full ionization with the total energy when this latter quantity is small. Our new theory makes use of the fact that the most dramatic changes in scale and energies of a few-body system occur when its components pass near to a central configuration. Numerical simulations are used to verify the result.

Mathematics in Industry Study Groups (MISG) have been an annual event in Australia and New Zealand since 1984. Projects from the last decade are considered. Among the industries involved are those of steel, electricity and agriculture.

We present the analysis of MOA-2007-BLG-197Lb, the first brown dwarf companion to a Sun-like star detected through gravitational microlensing. The event was alerted and followed-up photometrically by a network of telescopes from the PLANET, MOA, and uFUN collaborations, and observed at high angular resolution using the NaCo instrument at the VLT. From the modelling of the microlensing light curve, we derived the binary lens separation in Einstein radius units (s~1.13) and a mass ratio of (4.732+/-0.020)x10^{-2}. Annual parallax, lens orbital motion and finite source effects were included in the models. To recover the lens system's physical parameters, we combined the resulting light curve best-fit parameters with (J,H,Ks) magnitudes obtained with VLT NaCo and calibrated using IRSF and 2MASS data. We derived a lens total mass of 0.86+/-0.04 Msun and a lens distance of 4.2+/-0.3 kpc. We find that the companion of MOA-2007-BLG-197L is a brown dwarf of 41+/-2 Mjup observed at a projected separation of 4.3+/-0.1 AU, and orbits a 0.82+/-0.04 Msun G-K dwarf star. We study the statistical properties of this population of brown dwarfs detected by microlensing, transit, radial velocity, and direct imaging (most of these objects orbit solar-type stars), and we performed a two-dimensional, non-parametric probability density distribution fit to the data, which draws a structured brown dwarf landscape. We confirm the existence of a region that is strongly depleted in objects at short periods and intermediate masses (P500 d) and high masses (M>50 Mjup). While these data provide important clues on mechanisms of brown dwarfs formation, more data are needed to establish their relative importance, in particular as a function of host star mass., 16 pages, 12 figures

We locate members of an important category of periodic orbits in the Newtonian four-body problem. These systems perform an interplay motion similar to that of the periodic three-body orbit discovered by Schubart. Such orbits, when stable, have been shown to be a key feature and influence on the dynamics of few-body systems. We consider the restricted case where the masses are collinear and are distributed symmetrically about their centre of mass. A family of orbits is generated from the known (three-dimensionally) unstable equal masses case by varying the mass ratio, whilst maintaining the symmetry. The stability of these orbits to perturbation is studied using linear stability analysis, analytical approximation of limiting cases and nonlinear simulation. We answer the natural question: are there any stable periodic orbits of this kind? Three ranges of the mass ratio are found to have stable orbits and three ranges have unstable orbits for three-dimensional motion. The systems closely resemble their three-body counterparts. Here the family of interplay orbits is simpler requiring just one parameter to characterise the mass ratio. Our results provide a further insight into three-body orbits studied previously.

Numerical simulations of the one-dimensional Newtonian four-body problem have been conducted for the special case in which the bodies are distributed symmetrically about the centre of mass. Simulations show a great similarity between this problem and the one-dimensional Newtonian three-body problem. As in that problem the orbits can be divided into three different categories which form well-defined regions on a Poincare section: there is a region of quasiperiodic orbits about a Schubart-like periodic orbit, there is a region of fast-scattering encounters and in between these two regions there is a chaotic scattering region. The Schubart-like periodic orbit's stability to perturbation is studied. It is apparently stable in one-dimension but is unstable in three-dimensions.

We report the discovery of MOA-2013-BLG-220Lb, which has a super-Jupiter mass ratio $q=3.01\pm 0.02\times 10^{-3}$ relative to its host. The proper motion, $\mu=12.5\pm 1\, {\rm mas}\,{\rm yr}^{-1}$, is one of the highest for microlensing planets yet discovered, implying that it will be possible to separately resolve the host within $\sim 7$ years. Two separate lines of evidence imply that the planet and host are in the Galactic disk. The planet could have been detected and characterized purely with follow-up data, which has important implications for microlensing surveys, both current and into the LSST era., Comment: 2 tables, 2 figures

Dissolved chemicals are transported through groundwater aquifers by a mixture of advection with the underlying fluid flow and dispersion within that fluid. The aquifers can be modelled using a layered structure which simplifies the calculation of vertical transport. This simplified model still allows for the natural stratification which occurs in such systems and the changes in physical properties of the aquifer that occur between different geological layers. Equations are presented to calculate the subsequent concentration of the releases of chemicals into this system. A particular example is considered where an instantaneous release of pollutant occurs and it is subsequently remediated by the downstream release of a suitable pollutant removal agent.

Projects from the steel industry have made a valuable contribution to mathematics-in-industry study groups (MISG). In the eight-year period 2004–2011, there were eight such projects brought to the annual Australia and New Zealand MISG. Three particular projects are reviewed in more detail. The first relates to the extraction of vanadium and other metalloids from molten raw metal sourced from iron sand. The second considers the heating of cold-rolled steel coils during annealing. Finally, the third concerns the continuous hot-dipped galvanising process and coating deformations that may arise therein.

Characterizing a microlensing planet is done from modeling an observed lensing light curve. In this process, it is often confronted that solutions of different lensing parameters result in similar light curves, causing difficulties in uniquely interpreting the lens system, and thus understanding the causes of different types of degeneracy is important. In this work, we show that incomplete coverage of a planetary perturbation can result in degenerate solutions even for events where the planetary signal is detected with a high level of statistical significance. We demonstrate the degeneracy for an actually observed event OGLE-2012-BLG-0455/MOA-2012-BLG-206. The peak of this high-magnification event $(A_{\rm max}\sim400)$ exhibits very strong deviation from a point-lens model with $\Delta\chi^{2}\gtrsim4000$ for data sets with a total number of measurement 6963. From detailed modeling of the light curve, we find that the deviation can be explained by four distinct solutions, i.e., two very different sets of solutions, each with a two-fold degeneracy. While the two-fold (so-called "close/wide") degeneracy is well-understood, the degeneracy between the radically different solutions is not previously known. The model light curves of this degeneracy differ substantially in the parts that were not covered by observation, indicating that the degeneracy is caused by the incomplete coverage of the perturbation. It is expected that the frequency of the degeneracy introduced in this work will be greatly reduced with the improvement of the current lensing survey and follow-up experiments and the advent of new surveys., Comment: 5 pages, 3 figures, ApJ accepted

The mass of the lenses giving rise to Galactic microlensing events can be constrained by measuring the relative lens-source proper motion and lens flux. The flux of the lens can be separated from that of the source, companions to the source, and unrelated nearby stars with high-resolution images taken when the lens and source are spatially resolved. For typical ground-based adaptive optics (AO) or space-based observations, this requires either inordinately long time baselines or high relative proper motions. We provide a list of microlensing events toward the Galactic Bulge with high relative lens-source proper motion that are therefore good candidates for constraining the lens mass with future high-resolution imaging. We investigate all events from 2004 -- 2013 that display detectable finite-source effects, a feature that allows us to measure the proper motion. In total, we present 20 events with mu >~ 8 mas/yr. Of these, 14 were culled from previous analyses while 6 are new, including OGLE-2004-BLG-368, MOA-2005-BLG-36, OGLE-2012-BLG-0211, OGLE-2012-BLG-0456, MOA-2012-BLG-532, and MOA-2013-BLG-029. In, Comment: 10 pages, 8 figures, 6 tables, submitted to ApJ. For a brief video explaining the key results of this paper, please visit: https://www.youtube.com/watch?v=i_dzT8NydJI

Modern domestic clothes dryers contain sensors which attempt to detect when the load is dry, and subsequently, when the drying process should be ceased. However, due to a range of factors, occasionally the dryer may cut-off earlier than is desired. At a recent mathematics-in-industry study group workshop in New Zealand (MINZ-2015), Fisher and Paykel presented the challenge of eliminating these early cut-offs. Using a statistical fitting technique which accounts for stochastic noise and utilises confidence intervals, we outline a more rigorous procedure for detecting the end state of the drying process. We analyse experimental data provided at the workshop and investigate the possibility of estimating drying cycle end time in advance. Finally we interrogate the experimental data to estimate physical drying parameters which may be used in a reaction engineering drying model. References C. Marsh and A. Wilkins, Determining temperature control of wash water in a laundry environment. Proceedings of the 2005 mathematics-in-industry study group 163–208, 2005. C. Marsh, S. Taylor, P. Milliken and G. Senaratne, Developing an understanding of washing machine dynamics. Proceedings of the 2006 mathematics-in-industry study group 133–152, 2007. X. D. Chen, The basics of a reaction engineering approach to modeling air-drying of small droplets or thin-layer materials Drying Technology 26 , 627–639, 2008. doi:10.1080/07373930802045908 X. D. Chen and A. Putranto, Modelling Drying Processes: A Reaction Engineering Approach , Cambridge University Press, 2013. ISBN 978-1-107-01210-3 doi:10.1017/cbo9780511997846 R. B. Keey, Drying of Loose and Particulate Materials , Hemisphere Publishing Corporation, 1992. ISBN 0-89116-878-8 doi:10.1080/07373939208916507 A. K. Haghi, Heat and Mass Transfer in Textiles . WSEAS Press, 2011. ISBN 978-1-61804-025-1 A. Putranto, X. D. Chen, S. Devahastin, Z. Xiao, and P. A. Webley, Application of the reaction engineering approach (rea) for modeling intermittent drying under time-varying humidity and temperature. Chemical Engineering Science , 66(10):2149–2156, 2011. doi:10.1016/j.ces.2011.02.025 A. Putranto, X. D. Chen, and P. A. Webley, Modeling of drying of food materials with thickness of several centimeters by the reaction engineering approach (rea). Drying Technology , 29(8):961–973, 2011. doi:10.1080/07373937.2011.557793 A. Putranto, X. D. Chen, Z. Xiao, and P. A. Webley, Modeling of high-temperature treatment of wood using the reaction engineering approach (rea). Bioresource technology , 102(10):6214–6220, 2011. doi:10.1016/j.biortech.2011.02.053

We present algorithms of the Bulirsch–Stoer–Neville form for calculating the value of the vector rational interpolant. The derivation parallels that of the Bulirsch–Stoer algorithm for the usual scalar rational interpolation algorithm of Stoer and Bulirsch, using the Clifford algebra foundations for vector rational interpolants of Roberts.

This letter presents a noninvasive microwave application method for breast tumor detection. Forward equations are developed using the signals' scattering effect at the presence of an object inside the breast model. It is shown how the complex backscattered signals which characterize the internal material properties are analyzed using cylindrical coordinates to obtain required information. The algorithm developed to solve the inverse problem can compute the unknown tumor size and its location inside the breast model

A planetary microlensing signal is generally characterized by a short-term perturbation to the standard single lensing light curve. A subset of binary-source events can produce perturbations that mimic planetary signals, thereby introducing an ambiguity between the planetary and binary-source interpretations. In this paper, we present analysis of the microlensing event MOA-2012-BLG-486, for which the light curve exhibits a short-lived perturbation. Routine modeling not considering data taken in different passbands yields a best-fit planetary model that is slightly preferred over the best-fit binary-source model. However, when allowed for a change in the color during the perturbation, we find that the binary-source model yields a significantly better fit and thus the degeneracy is clearly resolved. This event not only signifies the importance of considering various interpretations of short-term anomalies, but also demonstrates the importance of multi-band data for checking the possibility of false-positive planetary signals., 5 pages, 4 figures and ApJ submitted

We analyze MOA-2010-BLG-311, a high magnification (A_max>600) microlensing event with complete data coverage over the peak, making it very sensitive to planetary signals. We fit this event with both a point lens and a 2-body lens model and find that the 2-body lens model is a better fit but with only Delta chi^2~80. The preferred mass ratio between the lens star and its companion is $q=10^(-3.7+/-0.1), placing the candidate companion in the planetary regime. Despite the formal significance of the planet, we show that because of systematics in the data the evidence for a planetary companion to the lens is too tenuous to claim a secure detection. When combined with analyses of other high-magnification events, this event helps empirically define the threshold for reliable planet detection in high-magnification events, which remains an open question., 29 pages, 6 Figures, 3 Tables. For a brief video presentation on this paper, please see http://www.youtube.com/user/OSUAstronomy 10/25/2012 - Updated author list. Replaced 10/10/13 to reflect the version published in ApJ

We report analysis of high microlensing event MOA-2008-BLG-379, which has a strong microlensing anomaly at its peak, due to a massive planet with a mass ratio of q = 6.9 x 10^{-3}. Because the faint source star crosses the large resonant caustic, the planetary signal dominates the light curve. This is unusual for planetary microlensing events, and as a result, the planetary nature of this light curve was not immediately noticed. The planetary nature of the event was found when the MOA Collaboration conducted a systematic study of binary microlensing events previously identified by the MOA alert system. We have conducted a Bayesian analysis based on a standard Galactic model to estimate the physical parameters of the lens system. This yields a host star mass of M_L = 0.66_{-0.33}^{+0.29} M_Sun orbited by a planet of mass m_P = 4.8_{-2.4}^{+2.1} M_Jup at an orbital separation of a = 4.1_{-1.5}^{+1.9} AU at a distance of D_L = 3.6 +/- 1.3 kpc. The faint source magnitude of I_S = 21.30 and relatively high lens-source relative proper motion of mu_rel = 7.6 +/- 1.6 mas/yr implies that high angular resolution adaptive optics or Hubble Space Telescope observations are likely to be able to detect the source star, which would determine the masses and distance of the planet and its host star., Comment: 25 pages, 5 figures, accepted for publication in ApJ

Observations of accretion disks around young brown dwarfs have led to the speculation that they may form planetary systems similar to normal stars. While there have been several detections of planetary-mass objects around brown dwarfs (2MASS 1207-3932 and 2MASS 0441-2301), these companions have relatively large mass ratios and projected separations, suggesting that they formed in a manner analogous to stellar binaries. We present the discovery of a planetary-mass object orbiting a field brown dwarf via gravitational microlensing, OGLE-2012-BLG-0358Lb. The system is a low secondary/primary mass ratio (0.080 +- 0.001), relatively tightly-separated (~0.87 AU) binary composed of a planetary-mass object with 1.9 +- 0.2 Jupiter masses orbiting a brown dwarf with a mass 0.022 M_Sun. The relatively small mass ratio and separation suggest that the companion may have formed in a protoplanetary disk around the brown dwarf host, in a manner analogous to planets., Comment: 6 pages, 3 figures, 3 tables, Submitted in ApJ

We present an analysis of the anomalous microlensing event, MOA-2010-BLG-073, announced by the Microlensing Observations in Astrophysics survey on 2010-03-18. This event was remarkable because the source was previously known to be photometrically variable. Analyzing the pre-event source lightcurve, we demonstrate that it is an irregular variable over time scales >200d. Its dereddened color, $(V-I)_{S,0}$, is 1.221$\pm$0.051mag and from our lens model we derive a source radius of 14.7$\pm$1.3 $R_{\odot}$, suggesting that it is a red giant star. We initially explored a number of purely microlensing models for the event but found a residual gradient in the data taken prior to and after the event. This is likely to be due to the variability of the source rather than part of the lensing event, so we incorporated a slope parameter in our model in order to derive the true parameters of the lensing system. We find that the lensing system has a mass ratio of q=0.0654$\pm$0.0006. The Einstein crossing time of the event, $T_{\rm{E}}=44.3$\pm$0.1d, was sufficiently long that the lightcurve exhibited parallax effects. In addition, the source trajectory relative to the large caustic structure allowed the orbital motion of the lens system to be detected. Combining the parallax with the Einstein radius, we were able to derive the distance to the lens, $D_L$=2.8$\pm$0.4kpc, and the masses of the lensing objects. The primary of the lens is an M-dwarf with $M_{L,p}$=0.16$\pm0.03M_{\odot}$ while the companion has $M_{L,s}$=11.0$\pm2.0M_{\rm{J}}$ putting it in the boundary zone between planets and brown dwarfs., 24 pages, 8 figures, best viewed in colour, accepted by ApJ

We study the doses generated by instantaneously released dense gas clouds. The dose is of practical importance since it is widely used as a measure of the harmful effect of an exposure. We take the doses from wind-tunnel experimental data. We extend from Davies' (1992) work to show that for later times after release the dose across repeat experiments may be fitted with a lognormal distribution. A simple box model is constructed from the dataset. It produces a reasonable model under some of the experimental conditions. We show how to use the repeats of experiments to produce a representative model.

This paper describes the investigations conducted in a Mathematics-in-Industry Study Group project from the Australian meeting at Queensland University of Technology in 2015. This concerned the management of stock levels of raw materials used to construct aortic stents. The approaches used included network visualisation, classification and regression trees, and time series modelling. This work will be of general interest to those who are managing stock levels in a highly volatile context. The methods applied show that there is potential value in taking a statistical approach to understand and make decisions within such volatility. The work provides a basis for developing more advanced statistical approaches for specific inventory problems. References Breiman, L. (1996) Bagging Predictors. Machine Learning, 24 , 2, 123–140. doi:10.1023/A:1018054314350 Breiman, L. (2001) Statistical Modeling: The Two Cultures Statistical Science, 16 , 3, 199–231. https://projecteuclid.org/download/pdf_1/euclid.ss/1009213726 Breiman, L., Friedman, J.H., Olshen, R.A., Stone, C.J. (1984) Classification and Regression Trees Wadsworth, Belmont, Ca. ISBN-13: 978-0412048418 ISBN-10: 0412048418 Csardi, G., Nepusz, T. (2006) The igraph software package for complex network research. InterJournal: Complex Systems, 1695 , 5, 1–9. http://www.interjournal.org/manuscript_abstract.php?361100992 Fruchterman, T. M. J., Reingold, E. M. (1991) Graph drawing by force-directed placement. Software: Practice and Experience, 21 , 11, 1129–1164. doi:10.1002/spe.4380211102 Hastie, T., Tibshirani, R., Friedman, J. H. (2009) The elements of statistical learning : Data mining, inference, and prediction, 2nd edition New York: Springer Verlag. ISBN 978-0-387-84858-7 (eBook), ISBN 978-0-387-84857-0 (Hardcover) doi:10.1007/978-0-387-84858-7 Kim, J. H., Wong, K., Athanasopoulos, G., Liu, S. (2011) Beyond point forecasting: Evaluation of alternative prediction intervals for tourist arrivals. International Journal of Forecasting, 27 , 887–901. doi:10.1016/j.ijforecast.2010.02.014 Liu, S., Maharaj, E. A. (2013). A hypothesis test using bias-adjusted AR estimators for classifying time series in small samples. Computational Statistics and Data Analysis, 60, 32–49. doi:10.1016/j.csda.2012.11.014 Liu, S., Maharaj, E. A., Inder, B. (2014) Polarization of forecast densities: A new approach to time series classification. Computational Statistics and Data Analysis, 70, 345–361. doi:10.1016/j.csda.2013.10.008 Liu, S., McGree, J., Ge, Z., Xie, Y. (2015) Computational and Statistical Methods for Analysing Big Data with Applications. Elsevier, London. ISBN: 978-0-12-803732-4 Reingold, E. M., Tilford, J. S. (1981) Tidier drawings of trees. IEEE Transactions on Software Engineering, 7 , 2, 223–228. doi:10.1109/TSE.1981.234519

The Galactic bulge source MOA-2010-BLG-523S exhibited short-term deviations from a standard microlensing lightcurve near the peak of an Amax ~ 265 high-magnification microlensing event. The deviations originally seemed consistent with expectations for a planetary companion to the principal lens. We combine long-term photometric monitoring with a previously published high-resolution spectrum taken near peak to demonstrate that this is an RS CVn variable, so that planetary microlensing is not required to explain the lightcurve deviations. This is the first spectroscopically confirmed RS CVn star discovered in the Galactic bulge., 29 pp, 6 figs, submitted to ApJ

Context. Caustic crossing is the clearest signature of binary lenses in microlensing. In the present context, this signature is diluted by the large source star but a detailed analysis has allowed the companion signal to be extracted.Aims. MOA 2009-BLG-411 was detected on August 5, 2009 by the MOA-Collaboration. Alerted as a high-magnification event, it was sensitive to planets. Suspected anomalies in the light curve were not confirmed by a real-time model, but further analysis revealed small deviations from a single lens extended source fit.Methods. Thanks to observations by all the collaborations, this event was well monitored. We first decided to characterize the source star properties by using a more refined method than the classical one: we measure the interstellar absorption along the line of sight in five different passbands (VIJHK). Secondly, we model the lightcurve by using the standard technique: make (s,q,α) grids to look for local minima and refine the results by using a downhill method (Markov chain Monte Carlo). Finally, we use a Galactic model to estimate the physical properties of the lens components.Results. We find that the source star is a giant G star with radius 9 R⊙. The grid search gives two local minima, which correspond to the theoretical degeneracy s ≡ s-1. We find that the lens is composed of a brown dwarf secondary of mass MS = 0.05 M⊙ orbiting a primary M-star of mass MP = 0.18 M⊙. We also reveal a new mass-ratio degeneracy for the central caustics of close binaries.Conclusions. As far as we are aware, this is the first detection using the microlensing technique of a binary system in our Galaxy composed of an M-star and a brown dwarf. Context. Caustic crossing is the clearest signature of binary lenses in microlensing. In the present context, this signature is diluted by the large source star but a detailed analysis has allowed the companion signal to be extracted. Aims. MOA 2009-BLG-411 was detected on August 5, 2009 by the MOA-Collaboration. Alerted as a high-magnification event, it was sensitive to planets. Suspected anomalies in the light curve were not confirmed by a real-time model, but further analysis revealed small deviations from a single lens extended source fit. Methods. Thanks to observations by all the collaborations, this event was well monitored. We first decided to characterize the source star properties by using a more refined method than the classical one: we measure the interstellar absorption along the line of sight in five different passbands (VIJHK). Secondly, we model the lightcurve by using the standard technique: make (s,q,α) grids to look for local minima and refine the results by using a downhill method (Markov chain Monte Carlo). Finally, we use a Galactic model to estimate the physical properties of the lens components. Results. We find that the source star is a giant G star with radius 9 R . The grid search gives two local minima, which correspond to the theoretical degeneracy s s . We find that the lens is composed of a brown dwarf secondary of mass M = 0.05 M orbiting a primary M-star of mass M = 0.18 M . We also reveal a new mass-ratio degeneracy for the central caustics of close binaries. Conclusions. As far as we are aware, this is the first detection using the microlensing technique of a binary system in our Galaxy composed of an M-star and a brown dwarf.

Steel becomes brittle during the cold rolling process which is used to produce sheet metal. Heat treatment (annealing) is required to release stresses and reform the crystalline structure. The 2008 Mathematics-and Statistics-in- Industry Study Group in Wollongong (MISG08) modelled the approach used by New Zealand Steel for which steel coils are heated in a batch annealing furnace. Determining the temperature within each coil is complicated by height-dependent gaps within the coils. Deciding on suitable boundary conditions for the outside of the coils provides a further challenge. This is explored with two alternative models. Having made reasonable assumptions, a linear model is found to be sufficient for modelling the heating process and allows the cold point in the steel coil to be established.