Australian Mathematical Society (AustMS): E-Journals
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Implementation of high-order, discontinuous Galerkin time stepping for fractional diffusion problems
No full textThe discontinuous Galerkin (DG) method provides a robust and flexible technique for the time integration of fractional diffusion problems. However, a practical implementation uses coefficients defined by integrals that are not easily evaluated. We describe specialized quadrature techniques that efficiently maintain the overall accuracy of the DG method. In addition, we observe in numerical experiments that known superconvergence properties of DG time stepping for classical diffusion problems carry over in a modified form to the fractional-order setting.
doi: 10.1017/S144618112000015
Connectivity properties of McKay quivers
No full textWe present several results regarding the connectivity of McKay quivers of finite-dimensional complex representations of finite groups, with no restriction on the faithfulness or self-duality of the representations. We give examples of McKay quivers, as well as quivers that cannot arise as McKay quivers, and discuss a necessary and sufficient condition for two finite groups to share a connected McKay quiver
Weighted composition operators between Lorentz spaces
No full texthttp://dx.doi.org/10.1017/S000497271200033
Linear response in dynamical systems: optimisation and finite-time coherent sets
No full texthttp://dx.doi.org/10.1017/S000497271200033
Analysis of cell transmission model for traffic flow simulation with application to network traffic
No full textThe cell transmission model (CTM) is a macroscopic model that describes the dynamics of traffic flow over time and space. The effectiveness and accuracy of the CTM are discussed in this paper. First, the CTM formula is recognized as a finite-volume discretization of the kinematic traffic model with a trapezoidal flux function. To validate the constructed scheme, the simulation of shock waves and rarefaction waves as two important elements of traffic dynamics was performed. Adaptation of the CTM for intersecting and splitting cells is discussed. Its implementation on the road segment with traffic influx produces results that are consistent with the analytical solution of the kinematic model. Furthermore, a simulation on a simple road network shows the back and forth propagation of shock waves and rarefaction waves. Our numerical result agrees well with the existing result of Godunov’s finite-volume scheme. In addition, from this accurately proven scheme, we can extract information for the average travel time on a certain route, which is the most important information a traveller needs. It appears from simulations of different scenarios that, depending on the circumstances, a longer route may have a shorter travel time. Finally, there is a discussion on the possible application for traffic management in Indonesia during the Eid al-Fitr exodus.
doi:10.1017/S144618112100008
Spectrally accurate option pricing under the time fractional Black-Scholes model
No full textWe propose a Legendre–Laguerre spectral approximation to price the European and double barrier options in the time-fractional framework. By choosing an appropriate basis function, the spectral discretization is used for the approximation of the spatial derivatives of the time-fractional Black–Scholes equation. For the time discretization, we consider the popular finite difference approximation, which converges with order for functions which are twice continuously differentiable. However, when using the scheme for problems with nonsmooth initial data, only the first-order accuracy in time is achieved. This low-order accuracy is also observed when solving the time-fractional Black–Scholes European and barrier option pricing problems for which the payoffs are all nonsmooth. To increase the temporal convergence rate, we therefore consider a Richardson extrapolation method, which when combined with the spectral approximation in space, exhibits higher order convergence such that high accuracies over the whole discretization grid are obtained. Compared with the traditional finite difference scheme, numerical examples clearly indicate that the spectral approximation converges exponentially over a small number of grid points. Also, as demonstrated, such high accuracies can be achieved in much fewer time steps using the extrapolation approach.
doi:10.1017/S1446181121000286
Asymptotic analysis for the mean first passage time in finite or spatially periodic 2D domains with a cluster of small traps
No full textA hybrid asymptotic-numerical method is developed to approximate the mean first passage time (MFPT) and the splitting probability for a Brownian particle in a bounded two-dimensional (2D) domain that contains absorbing disks, referred to as "traps”, of asymptotically small radii. In contrast to previous studies that required traps to be spatially well separated, we show how to readily incorporate the effect of a cluster of closely spaced traps by adapting a recently formulated least-squares approach in order to numerically solve certain local problems for the Laplacian near the cluster. We also provide new asymptotic formulae for the MFPT in 2D spatially periodic domains where a trap cluster is centred at the lattice points of an oblique Bravais lattice. Over all such lattices with fixed area for the primitive cell, and for each specific trap set, the average MFPT is smallest for a hexagonal lattice of traps.
doi:10.1017/S144618112100001
A new algorithm for decomposing modular tensor products
No full textLet be a prime and let denote a full Jordan block matrix with eigenvalue over a field of characteristic . For positive integers and with , the Jordan canonical form of the matrix has the form where \lambda_1 \geq \lambda_2 \geq \dots \geq \lambda_{r}>0. This decomposition determines a partition of but the values of the parts depend on , , and . Shifting focus to the multiplicities of the parts of , write
where \mu_1>\mu_2>\dots>\mu_k>0. Then is a composition of , from which can be computed easily. We present a new bottom-up algorithm for computing directly from the base- expansions for and
Controlling the false discovery rate through multiple competition
No full text
As their name suggests, competition-based procedures use direct head-to-head competition between the originally observed score and a randomly generated null score for each hypothesis in order to control the FDR amongst the resulting list of discoveries. In this thesis we extend this competition framework and develop multiple testing procedures that utilize competition between the original scores and multiple, rather than a single, set of null scores.
We construct these methods in the frameworks of peptide identification through target-decoy competition and the classical linear regression problem with Barber and Cand`es' recent knockoff procedure. In both these cases we show through simulations and real data experiments that utilizing multiple competition properly can lead to significant power gains without losing FDR control
A computational model of the initial/pre-collecting lymphatics, and a study of lymphatic valvogenesis
No full texthttp://dx.doi.org/10.1017/S000497271200033