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q-discrete Painleve equations: their hierarchies and properties
No full textThe main objective of this thesis is to derive hierarchies of q-discrete Painelevé equations. Some of the important properties of these hierarchies will also be given, namely Lax pairs, Bäcklund transformations, solutions of their asso- ciated linear problems for special values of parameters and their symmetry groups.
To construct these hierarchies, we apply a geometric reduction and a stair- case method on a multi-parameteric generalized lattice modified Korteweg-de Vries equation. In addition, the property of consistency around the cube is used in order to find Bäcklund transformations.
Starting with the base case of q-discrete second, third and fourth Painlevé equations on A5 initial-values surface, new hierarchies of q-discrete third and fourth Painlevé equations are discovered, and we also rediscover the hierarchy of q-discrete second Painlevé equation. In this thesis, we provide the Lax pairs for each member in these hierarchies. Using the consistency around the cube, we also provide the Bäcklund transformation for the entire hierarchy of q-discrete second and third Painlevé hierarchies. We generate a hierarchy of special solutions starting with seed solutions for q-discrete second and third Painlevé hierarchies.
An assumption made is that particular parameter values would enable the ability to diagonalize the Lax pair. As a consequence, we found that the as- sociated linear problem for the three hierarchies can be solved in terms of q-Gamma function. Furthermore, the hierarchy of q-discrete fourth Painlevé hierarchy can be reduced to one equation that can be linearlized to become Riccati equation which has hypergeometric special solutions.
Finally, we investigated the affine Weyl group structure of the symmetry group for each hierarchy. In this thesis, we construct the explicit representation of the symmetry group for the first and second member of these hierarchies.
The collection of new hierarchies, their Lax pairs, Bäcklund transforma- tions, the resultant symmetry groups and special solutions comprise the new results of this thesis. This thesis contains material published in [10] in collabo- ration with N. Joshi and D. Tran and myself. The material of this paper is pre- sented in Chapter 3, and is related to qPII and qPIII hierarchies, their Lax pair and examples. In Chapter 4, Bäcklund transformation of qPII and qPIII hierarchies, includes material from the above-mentioned paper. Similarly, Chapter 5 reports on results about solutions of the linear problem from the above paper. However, we emphasize that all the results about qPIV through out the thesis are completely new and unpublished. Chapter 6 includes unpublished material even for qPII and qPIII hierarchies
Evolutionary dynamics in discrete time for the perturbed positive definite replicator equation
No full textThe population dynamics for the replicator equation has been well studied in continuous time, but there is less work that explicitly considers the evolution in discrete time. The discrete-time dynamics can often be justified indirectly by establishing the relevant evolutionary dynamics for the corresponding continuous-time system, and then appealing to an appropriate approximation property. In this paper we study the discrete-time system directly, and establish basic stability results for the evolution of a population defined by a positive definite system matrix, where the population is disrupted by random perturbations to the genotype distribution either through migration or mutation, in each successive generation.
doi: 10.1017/S144618112000014
Large values of -functions on the -line
No full textIn this paper, we study lower bounds of a general family of - functions on the -line. More precisely, we show that for any in this family, there exists arbitrary large such that , where is the order of the pole of at . This is a generalization of the same result of Aistleitner, Munsch and the second author for the Riemann zeta-function. As a consequence, we get lower bounds for large values of Dedekind zeta-functions and Rankin-Selberg - functions of the type on the -line
Proceedings of the 2020 Computational Techniques and Applications Conference
No full textCTAC-2020
Sydney, Australia
30 August -- 2 September 2020
The 20th biennial Computational Techniques and Applications Conference was originally planned to take place at the University of New South Wales from 30 August until 2 September, 2020. As the Covid-19 pandemic worsened, it became clear that travel restrictions and other measures meant that the conference could not be held in its usual form, and the organising committee decided to move the conference online.
The ANZIAM Special Interest Group in Computational Mathematics is responsible for this series of conferences, the first of which was held in 1981. Participants of the conference are able to submit a short article based on their presentation for publication in this special section of the ANZIAM Journal (Electronic Supplement). The editors, William McLean and Shev MacNamara, thank the referees whose efforts have helped improve the quality of these conference proceedings.
This Special Section of the ANZIAM Journal (Electronic Supplement) contains the refereed conference papers.
The eight keynote presentations were as follows.
Konstantin Brenner, University of Nice Sophia-Antipolis Numerical modeling of two-phase flow in fractured porous media
Michael Feischl, TU ViennaNumerical analysis and machine learning.
David Harvey, UNSWFast Fourier transforms of prime length.
Trevor McDougall, UNSWSome mathematical aspects of physical oceanography.
Kate Smith-Miles, University of MelbourneIn search of algorithmic trust \dots show us the stress-testing!
Catherine Powell, University of ManchesterAdaptive and multilevel stochastic Galerkin methods for PDEs with uncertain inputs.
Aretha Teckentrup, University of EdinburghConvergence of Gaussian process emulators with estimated hyper-parameters.
Alex Townsend, Cornell UniversityThe ultraspherical spectral method.
The presentation by Trevor McDougall was a public lecture. The conference attracted 131 registered participants, and featured a total of 81 contributed talks.
CTAC2020 Organising Committee
Josef Dick, UNSW
Bishnu Lamichhane, University of Newcastle
Ngan Le, Monash University
Quoc Thong Le Gia (Chair), UNSW
Shev MacNamara, UTS
William McLean, UNSW
Vera Roschchina, UNSW
Thanh Tran, UNSW
CTAC2020 Scientific Committee
Steve Armfield, University of Sydney
Jerome Droniou, Monash University
Frances Kuo, UNSW
Markus Hegland, ANU
Stephen Roberts, ANU
Ian H. Sloan (Chair), UNSW
Ian Turner, QUT
Acknowledgements
We gratefully acknowledge support from the following sponsors:
School of Mathematics and Statistics, UNSW.
The New South Wales Government.
The Modelling and Simulation Society of Australia and New Zealand.
The Mathematics of Computation and Optimization (MoCaO) special interest group of the Australian Mathematical Society.
Proceedings editors
William McLean
Shev Macnamara
Judith Bunde
Equivalence of semi-norms related to super weakly compact operators
No full texthttp://dx.doi.org/10.1017/S000497271200033
Draping woven sheets
No full textMotivated by the manufacture of carbon fibre components, this paper considers the smooth draping of loosely woven fabric over rigid obstacles, both smooth and nonsmooth. The draped fabric is modelled as the continuum limit of a Chebyshev net of two families of short rigid rods that are freely pivoted at their joints. This approach results in a system of nonlinear hyperbolic partial differential equations whose characteristics are the fibres in the fabric. The analysis of this system gives useful information about the drapability of obstacles of many shapes and also poses interesting theoretical questions concerning well-posedness, smoothness and computability of the solutions.
doi:10.1017/S144618112000019
Solutions of fourth-order evolution equations in material science
No full texthttp://dx.doi.org/10.1017/S000497271200033
On unramified solvable extensions of small number fields
No full texthttp://dx.doi.org/10.1017/S000497271200033