The error of a distributed algorithm for big data classification with a support vector machine (SVM) is analysed in this paper. First, the given big data sets are divided into small subsets, on which the classical SVM with Gaussian kernels is used. Then, the classification error of the SVM for each subset is analysed based on the Tsybakov exponent, geometric noise, and width of the Gaussian kernels. Finally, the whole error of the distributed algorithm is estimated in terms of the error of each subset. doi:10.1017/S1446181116000390

As a new business form, the buy-online and pick-up-in-store (BOPS) mode allows consumers to pay for goods online and pick them up in a physical store. In this paper, an equilibrium model is constructed to formulate an optimal decision-making problem for online and offline retailers under the BOPS mode, where the online retailer determines the retail price of the goods and the consignment quantity in a physical store, while the offline retailer chooses the revenue share of profit by a consignment contract. Different to the existing models, the cost of overstocking and loss of understocking are incorporated into the profit function of the online retailer due to the randomness of demand. For the objective function of the offline retailer, the cross-sale quantity generated by the BOPS mode is taken into account. Then the game between the online and offline retailers is expressed as a stochastic Nash equilibrium model. Based on the analytic properties of the model, necessary conditions for the equilibrium solution are obtained. A case study and sensitivity analysis are employed to reveal the managerial implications of the model, which can provide a number of valuable suggestions on optimizing the strategies for the online and offline retailers under the BOPS mode. doi:10.1017/S1446181116000201

CTAC2016 was the 18th biennial Computational Techniques and Applications Conference, and took place at Monash University from 27--30 November, 2016. The ANZIAM Special Interest Group in Computational Techniques and Applications is responsible for the CTAC meetings, the first of which was held in 1981. The conference attracted 57 contributed talks. All articles appearing in the CTAC2016 proceedings have been peer reviewed by at least two referees, and in most cases undergone revision prior to acceptance.

A discounted Steiner tree is a weighted Steiner tree in which the costs of constructing the edges and values at the nodes are discounted over time. Discounted Steiner points can be located to maximise the sum of the discounted cash flows, known as the net present value, and an algorithm for doing this for a single Steiner point, known as the discounted Steiner point algorithm, was previously established. An application of this problem is underground mine planning. This article proposes an algorithm to optimally locate two junction points, given a surface portal and three ore resource points, for maximum net present value, which includes the value of the ore bodies and the construction costs. The discounted Steiner point algorithm is extended to locate two junction points where time delays may occur at a discounted Steiner point before constructing the adjacent edges. The optimal locations of the junction points are obtained for a range of discount rates. Numerical trials show that this algorithm works well. A generalisation of the algorithm to locate more discounted Steiner points is also discussed. References E. N. Gilbert and H. O. Pollak. Steiner minimal trees. SIAM J. Appl. Math., 16(1):1–29, 1968. doi:10.1137/0116001 . F. K. Hwang, D. S. Richards, and P. Winter. The Steiner Tree Problem. Elsevier, 1992. https://www.elsevier.com/books/the-steiner-tree-problem/hwang/978-0-444-89098-6 . K. G. Sirinanda, M. Brazil, P. A. Grossman, J. H. Rubinstein, and D. A. Thomas. Optimally locating a junction point for an underground mine to maximise the net present value. ANZIAM J., 55:C315–C328, 2014. doi:10.21914/anziamj.v55i0.7791 . K. G. Sirinanda, M. Brazil, P. A. Grossman, J. H. Rubinstein, and D. A. Thomas. Maximizing the net present value of a Steiner tree. J. Global Optim., 62(2):391–407, 2015. doi:10.1007/s10898-014-0246-3 .

In oscillatory shear experiments, the values of the storage and loss moduli, \(G′(𝜔)\) and \(G′′(𝜔)\), respectively, are only measured and recorded for a number of values of the frequency \(𝜔\) in some well-defined finite range \([𝜔_{\rm min},𝜔_{\rm max}]\). In many practical situations, when the range \([𝜔_{\rm min},𝜔_{\rm max}]\) is sufficiently large, information about the associated polymer dynamics can be assessed by simply comparing the interrelationship between the frequency dependence of \(G′(𝜔)\) and \(G′′(𝜔)\). For other situations, the required rheological insight can only be obtained once explicit knowledge about the structure of the relaxation time spectrum \(H(𝜏)\) has been determined through the inversion of the measured storage and loss moduli \(G′(𝜔)\) and \(G′′(𝜔)\). For the recovery of an approximation to \(H(𝜏)\), in order to cope with the limited range \([𝜔_{\rm min},𝜔_{\rm max}]\) of the measurements, some form of localization algorithm is required. A popular strategy for achieving this is to assume that \(H(𝜏)\) has a separated discrete point mass (Dirac delta function) structure. However, this expedient overlooks the potential information contained in the structure of a possibly continuous \(H(𝜏)\). In this paper, simple localization algorithms and, in particular, a joint inversion least squares procedure, are proposed for the rapid recovery of accurate approximations to continuous \(H(𝜏)\) from limited measurements of \(G′(𝜔)\) and \(G′′(𝜔)\}. doi: 10.1017/S1446181116000122

Anaerobic digestion of municipal and animal wastes provides a potentially important means of producing energy sustainably. The reliable production of methane from anaerobic digestion requires that bacteria metabolise municipal and agricultural wastes to produce mainly carbon dioxide and methane. Our model uses Monod based kinetics and is complex since the biomass concentration growth is inhibited with the increase in substrate. The production of methane depends on the number of methanogenic bacteria; there is evidence from previous chemostats experiments that there is a time delay in population number of these cells to respond to changes in substrate concentrations. In this article we focus on determining the dynamical response of the methane production rate under different dynamics of feed conditions and temperatures. Since there is a delay in the methane bacteria mass response, we include the effects of the delay on the dynamics of the system based on modification of kinetics in a simple fermentation anaerobic model. We compare the dynamics of the system with and without delay to show the contribution of delay to the overall stability of the system. References O. Bernard, Z. Hadj-Sadok, D. Dochain, A. Genovesi, J. P. Steyer. Dynamical model development and parameter identification for an anaerobic wastewater treament process. Biotechnol. Bioeng. 75:424–438, 2001. doi:10.1002/bit.10036 T. F. Thingstad and T. I. Langeland. Dynamics of chemostat culture: The effect of a delay in cell response. J. Theor. Biol. , 48:149–159, 1974. doi:10.1016/0022-5193(74)90186-6 J. R. Beddington and R. M. May. Time delays are not nessarily destabilizing. Math. Biosci. 27:109–119, 1975. doi:10.1016/0025-5564(75)90028-0 K. Y. San and G. Stephanopoulos. The effect of growth rate delays in substrate-inhibited kinetics on the optimal profile of fed-batch reactors. J. Theor. Biol. 28:356–361, 1986. doi:10.1002/bit.260280308 G. Stephanopoulos and K. Y. San. Studies on-line bioreactor identification. I. Theory. Biotechnol. Bioeng. 26:1176–1188, 1984. doi:10.1002/bit.260261006 A. A. Khan, A. J. Stacey and J. J. Shepherd. Optimization of methane output for an anaerobic waste digester. ANZIAM J. 54:C523–C539, 2013. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/6322 J. F. Andrews. A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates. Biotechnol. Bioeng. 10:707–723, 1968. doi:10.1002/bit.260100602 D. J. Batstone, J. Keller, I. Angelidaki, S. V. Kalyuzzhni, S. G. Pavlostathis, A. Rozzi, W. T. M. Sanders, H. Siegrist, and V. A. Vavilin. Anaerobic digestion model No. 1 (ADM1). Water Sci. Technol. 45(10):65–73, 2002 http://wst.iwaponline.com/content/45/10/65 K. J. Beers. Numerical methods for chemical engineering: Applications in Matlab . Cambridge, New York, USA, 2007. http://www.cambridge.org/us/academic/subjects/engineering/chemical-engineering/numerical-methods-chemical-engineering-applications-matlab J. F. Andrews. The development of a dynamic model and control strategies for the anaerobic digestion process. In Mathematical Models in Water Pollution Control , A. James (ed). Wiley, 1978. A. W. Bush and A. E. Cook. The effect of time delay and growth rate inhibition in the bacterial treatment of wastewater J. Theor. Biol. 63:385–395, 1976. doi:10.1016/0022-5193(76)90041-2 L. F. Shampine, I. Gladwell and S. Thompson. Solving ODEs with Matlab . Cambridge, New York, USA, 2003. http://www.cambridge.org/us/academic/subjects/mathematics/numerical-analysis/solving-odes-matlab?format=PB

The mathematical modelling of deformable, permeable screen doors was explored to assess their behaviour under wind loading. A load-response model was proposed whereby the wind load was modelled using a simplified approach that allowed it to be approximated as a uniformly distributed pressure load with empirical modification factors applied to relate it to the real case of a door on a building. Several approaches were adopted to model the mechanical behaviour of the door system in response to load, including discrete models based on mass-spring systems, continuum models based on the membrane equations (including tension modulation in some cases), and computational models using finite element packages. The primary aim of the work was to determine the distribution of wind load to the door's supporting `tabs' and estimate a failure wind speed. The mass-spring model and the membrane models without tension modulation both generated unrealistic deflection magnitudes in response to wind load, but could be calibrated in future work, and then used to obtain an estimate of the total force on the tabs. A tension-modulated version of the membrane model performed better with regards to deflected shape and magnitude, but time constraints meant that the load on the tabs was not calculated. Preliminary validation experiments were undertaken and deflected shape and magnitude were successfully measured in response to given wind speeds. References Standards Australia. AS/NZS1170.2–-Structural design actions part 2: Wind actions , 2011. F. Avanzini and R. Maronga. A modular physically based approach to the sound synthesis of membrane percussion instruments. IEEE Transactions on Audio, Speech, and Language Processing , 18(4):891–902, 2010. doi:10.1109/TASL.2009.2036903 B. Bank. Energy-based synthesis of tension modulation in strings. In Proceedings of the 12th International Conference on Digital Audio Effects (DAFx-09) , 2009. http://dafx09.como.polimi.it/proceedings/papers/paper_76.pdf Bert Blocken. 50 years of computational wind engineering: Past, present and future. Journal of Wind Engineering and Industrial Aerodynamics , 129(0):69–102, 2014. doi:10.1016/j.jweia.2014.03.008 Demetres Briassoulis, Antonis Mistriotis, and Anastasios Giannoulis. Wind forces on porous elevated panels. Journal of Wind Engineering and Industrial Aerodynamics , 98(12):919–928, 2010. doi:10.1016/j.jweia.2010.09.006 John Cheung and William Melbourne. Wind loading on porous cladding. In 9th Australasian fluid mechanics conference , 1986. http://people.eng.unimelb.edu.au/imarusic/proceedings/9/CheungMelbourne.pdf John Holmes. Wind Loading of Structures . Spon Press, 2001. P. D. Howell, G. Kozyreff, and J. R. Ockendon. Applied Solid Mechanics . Cambridge University Press, 2009. C.W Letchford. Wind loads on rectangular signboards and hoardings. Journal of Wind Engineering and Industrial Aerodynamics , 89(2):135–151, 2001. doi:10.1016/S0167-6105(00)00068-4 Wojciech Okrasinski and \T1\L ukasz Plociniczak. A nonlinear mathematical model of the corneal shape. Nonlinear Analysis: Real World Applications , 13(3):1498–1505, 2012. doi:10.1016/j.nonrwa.2011.11.014 D. W. Oplinger. Frequency response of a nonlienar stretched string. Journal of the Acoustical Society of America , 32(12):1529–1538, 1960. doi:10.1121/1.1907948 Xavier Provot. Deformation constraints in a mass-spring model to describe rigid cloth behaviour. In Graphics interface , pages 147–147. Canadian Information Processing Society, 1995. http://kucg.korea.ac.kr/education/2005/CSCE352/paper/provot95.pdf P.J Richards and M Robinson. Wind loads on porous structures. Journal of Wind Engineering and Industrial Aerodynamics , 83(13):455–465, 1999. doi:10.1016/S0167-6105(99)00093-8 A.P. Robertson, Ph. Roux, J. Gratraud, G. Scarascia, S. Castellano, M. Dufresne de Virel, and P. Palier. Wind pressures on permeably and impermeably-clad structures. Journal of Wind Engineering and Industrial Aerodynamics , 90(45):461–474, 2002. Bluff Body Aerodynamics and Applications. doi:10.1016/S0167-6105(01)00210-0 T. Tolonen, V. Valimaki, and M. Karjalainen. Modeling of tension modulation nonlinearity in plucked strings. IEEE Transactions on Speech and Audio Processing , 8(3):300–310, 2000. doi:10.1109/89.841212

This special Section of the ANZIAM Journal (Electronic Supplement) contains the refereed papers from the 2014 Mathematics and Statistics in Industry Study Group (MISG 2014) held at QUT from 28 January to 1 February 2014. The MISG is a special interest meeting of ANZIAM, the Australia and New Zealand Industrial and Applied Mathematics (ANZIAM) division of the Australian Mathematics Society. The MISG meetings take place annually and provide a forum where projects proposed by industry can be worked on intensively, by high profile scientists in the fields of Applied Mathematics, Statistics and Operations Research, from Australia, New Zealand and the world beyond, along with representatives from the industries proposing the projects.

The ice core time series from Vostok Station in Antarctica and the North Greenland Ice Core Project have seasonal variation corresponding to the Milankovitch cycles. After removing these cycles, and interpolating to equal time intervals, stationary time series models are fitted. The series show clear directionality and this feature is modelled by either non-Gaussian errors or non-linear time series models. Threshold autoregressive models are fitted by penalized least squares and compared with non-threshold autoregressive models. Since both ice core time series are reasonably modelled as first order autoregressive series with parameters close to one, directionality will arise from non-symmetric error distributions. However, two regime threshold autoregressive models, of order one and two for Greenland and Vostok, respectively, give an improved match to the observed directionality and a reduced sum of squared residuals. Realizations from the threshold autoregressive models are noticeably different from the non-threshold models. Since the non-threshold models are a restricted case of the threshold models, and the threshold models are a better fit to the observed time series, threshold models should provide more realistic realizations. References C. Chatfield. The Analysis of Time Series: An Introduction . CRC Press, 2004. https://www.crcpress.com/The-Analysis-of-Time-Series-An-Introduction-Sixth-Edition/Chatfield/p/book/9781584883173 A. J. Lawrance. Directionality and reversibility in time series. Int. Stat. Rev. 59(1):67–79, 1991. doi:10.2307/1403575 M. M. Mansor, M. E. Glonek, D. A. Green and A. V. Metcalfe. Threshold autoregressive models for directional time series. In I. Rojas and H. Pomares (Eds.), Time Series Analysis and Forecasting Selected Contributions from the ITISE Conference (ITISE 2015) . pp. 13–25, 2016. doi:10.1007/978-3-319-28725-6 M. M. Mansor, M. E. Glonek, D. A. Green and A. V. Metcalfe. Modelling directionality in stationary geophysical time series. International work-conference on Time Series (ITISE 2015) . http://www.researchgate.net/publication/281835075 M. M. Mansor, D. A. Green and A. V. Metcalfe. Modelling and simulation of directional financial time series. Proceedings of the 21st International Congress on Modelling and Simulation (MODSIM 2015) , pp. 1022–1028, 2015. http://www.mssanz.org.au/modsim2015/E4/mansor.pdf M. M. Mansor, D. A. Green and A. V. Metcalfe. Directionality and volatility in electroencephalogram time series. Proceedings of the 2nd International Conference on Mathematical Sciences and Statistics (ICMSS 2016), AIP Conf. Proc. 1739:020080, 2016. doi:10.1063/1.4952560 North Greenland Ice Core Project members. High-resolution record of Northern Hemisphere climate extending into the last interglacial period. Nature , 431:147–151, 2004. doi:10.1038/nature02805 J. R. Petit, J. Jouzel, D. Raynaud, N. I. Barkov, J.-M. Barnola, I. Basile, M. Bender, J. Chappellaz, M. Davis, G. Delaygue, M. Delmotte, V. M. Kotlyakov, M. Legrand, V. Y. Lipenkov, C. Lorius, L. Pepin, C. Ritz, E. Saltzman and M. Stievenard. Climate and atmospheric history of the past 420,000 years from the Vostok ice core, Antarctica. Nature , 399:429–436, 1999. doi:10.1038/20859 S. Soubeyrand, C. E. Morris and E. K. Bigg. Analysis of fragmented time directionality in time series to elucidate feedbacks in climate data. Environ. Modell. Softw. 61:78–86, 2014. doi:10.1016/j.envsoft.2014.07.003

We consider travelling front solutions of a one-dimensional reaction-diffusion system corresponding to two-stage competitively exothermic reactions. We suppose all reactions occurring during the combustion may be lumped together as two different paths. Both exothermic reactions compete for the same reactant. Properties of travelling wave fronts, particularly flame speed, are determined numerically by solving the governing partial differential equations. The flame speed is analysed for different values of the heat loss parameter. It is demonstrated that, as the heat loss coefficient increases, the flame speed decays gradually until the front ceases to exist due to insufficient energy being available to sustain the flame front. Earlier studies for the adiabatic case showed the existence of bi-stability (fast and slow waves co-exist for the same parameter values). We study how heat loss affects the size of the bi-stable region. Furthermore, we investigate how the extinction limit depends on the heat loss parameter as well as the parameter representing the ratio of the activation energy to the heat release of the second reaction. Numerical solutions show that there is no travelling front when these parameters are above threshold values. The dependence of flame speed on the temperature profile is also investigated. The bi-stability phenomenon is demonstrated by perturbing the temperature profile. References W. Choi, S. Hong, J. T. Abrahamson, J.-H. Han, C. Song, N. Nair, S. Baik and M. S. Strano. Chemically driven carbon-nanotube-guided thermopower waves. Nat. Mater. 9:423–429, 2010. doi:10.1038/nmat2714 A. G. Merzhanov. Combustion and explosion processes in physical chemistry and technology of inorganic materials. Russ. Chem. Rev. 72:289–310, 2003. doi:10.1070/rc2003v072n04abeh000766 J. K. Bechtold and C. K. Law. The structure of premixed methane-air flames with large activation energy. Combust. Flame 97:317–338, 1994. doi:10.1016/0010-2180(94)90024-8 K. Seshadri, N. Peters and F. A. Williams. Asymptotic analysis of stoichiometric and lean hydrogen-air flames. Combust. Flame 96:407–427, 1994. doi:10.1016/0010-2180(94)90108-2 A. L. Sanchez, A. Lepinette, M. Bollig, A. Linan and B. Lazaro. The reduced kinetic description of lean premixed combustion. Combust. Flame 123:436–464, 2000. doi:10.1016/S0010-2180(00)00177-2 A. L. Sanchez, G. Balakrishnan, A. Linan and F. A. Williams. Relationships between bifurcation and numerical analyses for ignition of hydrogen-air diffusion flames. Combust. Flame 105:569–590, 1996. doi:10.1016/0010-2180(95)00241-3 V. V. Gubernov, A. V. Kolobov, A. A. Polezhaev and H. S. Sidhu. Analysing the stability of premixed rich hydrogen-air flame with the use of two-step models. Combust. Flame 160:1060–1069, 2013. doi:10.1016/j.combustflame.2013.01.021 R. Ball, A. McIntosh and J. Brindley. Thermokinetic models for simultaneous reactions: a comparative study. Combust. Theor. Model. 3:447–468, 1999. doi:10.1088/1364-7830/3/3/302 N. A. Martirosyan, S. K. Dolukhanyan and A. G. Merzhanov. Nonuniqueness of stationary states in combustion of mixtures of zirconium and soot powders in hydrogen. Combust. Explo. Shock 19:569–571, 1983. doi:10.1007/BF00750423 H. S. Sidhu, I. N. Towers, V. V. Gubernov, A. V. Kolobov and A. V. Polezhaev. Investigation of flame propagation in a model with competing exothermic reactions. Chemeca 2013 , Brisbane, Australia, 29 September–2 October, 2013. http://www.conference.net.au/chemeca2013/papers/29350.pdf I. N. Towers, V. V. Gubernov, A. V. Kolobov, A. V. Polezhaev and H. S. Sidhu. Bistability of flame propagation in a model with competing exothermic reactions. Proc. R. Soc. Lond. A 469:20130315, 2013. doi:10.1098/rspa.2013.0315 L. K. Forbes and W. Derrick. A combustion wave of permanent form in a compressible gas. Anziam J. 43:35–58, 2001. doi:10.1017/S144618110001141X Z. Huang, H. S. Sidhu, I. N. Towers, Z. Jovanoski and V. V. Gubernov. Investigating flame fronts in competitive exothermic reactions. MODSIM2015 , 21st International congress on Modelling and simulation, Gold coast, Australia, 29 November–4 December, 2015. http://www.mssanz.org.au/modsim2015/A1/huang.pdf F. A. Williams. Combustion theory . Westview Press, 1985. https://westviewpress.com/?s=Combustion+theory FlexPDE\(^TM\), PDE Solutions Inc. http://www.pdesolutions.com W. E. Schiesser. The numerical method of lines: integration of partial differential equations . Academic Press, San Diego, 1991. http://store.elsevier.com/The-Numerical-Method-of-Lines/isbn-9780128015513/ A. C. McIntosh, R. O. Weber and G. N. Mercer. Non-adiabatic combustion waves for general Lewis numbers: wave speed and extinction conditions. Anziam J. 46:1–16, 2004. doi:10.1017/S1446181100013638 A. R. Hall and H. G. Wolfhard. Multiple reaction zones in low pressure flames with ethyl and methyl nitrate, methyl nitrite and nitromethane. Proc. Combust. Inst , 6:190–199, 1957. doi:10.1016/S0082-0784(57)80029-0