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Factors of Carmichael numbers and an even weaker -tuples conjecture
No full textDOI:
10.1017/S000497271900013
Analysis of two-dimensional combustion waves arising in the presence of a competitive endothermic reaction
Get PDFWe consider a system of reaction-diffusion equations describing combustion dynamics. The reaction is assumed to undergo two competitive reactions, one which is exothermic and one which is endothermic. The one-dimensional model has been shown to exhibit complex behaviour, from propagating combustion waves with a constant speed to period doubling cascades and the possibility of chaotic wave speeds. In this study, we extend the combustion model from one to two dimensions by exploring a model of an insulated strip with no heat loss and axially symmetric spread. In particular, we compare and contrast the behaviour of the systems in one and two dimensions.
References J. D. Buckmaster and G. S. S. Ludford. Theory of Laminar Flames. Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University Press, 1982. doi:10.1017/CBO9780511569531. V. Gubernov, A. Kolobov, A. Polezhaev, H. Sidhu, and G. Mercer. Period doubling and chaotic transient in a model of chain-branching combustion wave propagation. P. Roy. Soc. A Math. Phy., 466:2747–2769, 2010. doi:10.1098/rspa.2009.0668. A. Hmaidi, A. C. McIntosh, and J. Brindley. A mathematical model of hotspot condensed phase ignition in the presence of a competitive endothermic reaction. Combust. Theor. Model., 14:893–920, 2010. doi:10.1080/13647830.2010.519050. G. N. Mercer and R. O. Weber. Combustion waves in two dimensions and their one-dimensional approximation. Combust. Theor. Model., 1:157–165, 1997. doi:10.1088/1364-7830/1/2/002. J. J. Sharples, H. S. Sidhu, A. C. McIntosh, J. Brindley, and V. V. Gubernov. Analysis of combustion waves arising in the presence of a competitive endothermic reaction. IMA J. Appl. Math., 77:18–31, 2012. doi:10.1093/imamat/hxr072. S. D. Watt, R. O. Weber, H. S. Sidhu, and G. N. Mercer. A weight-function approach for determining watershed initial conditions for combustion waves. IMA J. Appl. Math., 62:195–206, 1999. doi:10.1093/imamat/62.2.195
Best proximity points and fixed points with -functions in the framework of -distances
No full textDOI:
10.1017/S000497271800119
Estimates of the second derivative of bounded analytic functions
No full textDOI:
10.1017/S000497271900059
Derandomised lattice rules for high dimensional integration
Get PDFWe seek shifted lattice rules that are good for high dimensional integration over the unit cube in the setting of an unanchored weighted Sobolev space of functions with square-integrable mixed first derivatives. Many existing studies rely on random shifting of the lattice, whereas here we work with lattice rules with a deterministic shift. Specifically, we consider 'half-shifted' rules in which each component of the shift is an odd multiple of where is the number of points in the lattice. By applying the principle that there is always at least one choice as good as the average, we show that for a given generating vector there exists a half-shifted rule whose squared worst-case error differs from the shift-averaged squared worst-case error by a term of only order . We carry out numerical experiments where the generating vector is chosen component-by-component (CBC), as for randomly shifted lattices, and where the shift is chosen by a new `CBC for shift' algorithm. The numerical results are encouraging.
References J. Dick, F. Y. Kuo, and I. H. Sloan. High-dimensional integration: The quasi-Monte Carlo way. Acta Numer., 22:133–288, 2013. doi:10.1017/S0962492913000044. J. Dick, D. Nuyens, and F. Pillichshammer. Lattice rules for nonperiodic smooth integrands. Numer. Math., 126(2):259–291, 2014. doi:10.1007/s00211-013-0566-0. T. Goda, K. Suzuki, and T. Yoshiki. Lattice rules in non-periodic subspaces of sobolev spaces. Numer. Math., 141(2):399–427, 2019. doi:10.1007/s00211-018-1003-1. F. Y. Kuo. Lattice rule generating vectors. URL http://web.maths.unsw.edu.au/ fkuo/lattice/index.html. D. Nuyens and R. Cools. Fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces. Math. Comput., 75:903–920, 2006. doi:10.1090/S0025-5718-06-01785-6. I. H. Sloan and S. Joe. Lattice methods for multiple integration. Oxford Science Publications. Clarendon Press and Oxford University Press, 1994. URL https://global.oup.com/academic/product/lattice-methods-for-multiple-integration-9780198534723. I. H. Sloan and H. Wozniakowski. When are quasi-Monte Carlo algorithms efficient for high dimensional integrals? J. Complex., 14(1):1–33, 1998. doi:10.1006/jcom.1997.0463. I. H. Sloan, F. Y. Kuo, and S. Joe. On the step-by-step construction of quasi-Monte Carlo integration rules that achieve strong tractability error bounds in weighted Sobolev spaces. Math. Comput., 71:1609–1641, 2002. doi:10.1090/S0025-5718-02-01420-5
