ChesterRep Collection:
http://hdl.handle.net/10034/6981
Thu, 09 Nov 2017 05:13:06 GMT2017-11-09T05:13:06ZGroup Rings, G-Codes and Constructions of Self-Dual and Formally Self-Dual Codes
http://hdl.handle.net/10034/620712
Title: Group Rings, G-Codes and Constructions of Self-Dual and Formally Self-Dual Codes
Authors: Dougherty, Steven; Gildea, Joe; Taylor, Rhian; Tylyschak, Alexander
Abstract: We describe G-codes, which are codes that are ideals in a group ring, where the ring
is a finite commutative Frobenius ring and G is an arbitrary finite group. We prove that
the dual of a G-code is also a G-code. We give constructions of self-dual and formally
self-dual codes in this setting and we improve the existing construction given in [13] by
showing that one of the conditions given in the theorem is unnecessary and, moreover,
it restricts the number of self-dual codes obtained by the construction. We show that
several of the standard constructions of self-dual codes are found within our general
framework. We prove that our constructed codes must have an automorphism group
that contains G as a subgroup. We also prove that a common construction technique
for producing self-dual codes cannot produce the putative [72, 36, 16] Type II code.
Additionally, we show precisely which groups can be used to construct the extremal
Type II codes over length 24 and 48. We define quasi-G codes and give a construction
of these codes.Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10034/6207122017-01-01T00:00:00ZA multi-species chemotaxis system: Lyapunov functionals, duality, critical mass
http://hdl.handle.net/10034/620705
Title: A multi-species chemotaxis system: Lyapunov functionals, duality, critical mass
Authors: Kavallaris, Nikos I.; Ricciardi, Tonia; Zecca, Gabriela
Abstract: We introduce a multi-species chemotaxis type system admitting an arbitrarily large number of population species, all of which are attracted vs.\ repelled by a single chemical substance. The production vs.\ destruction rates of the chemotactic substance by the species is described by a probability measure. For such a model we investigate the variational structures, in particular we prove the existence of Lyapunov functionals, we establish duality properties as well as a logarithmic Hardy-Littlewood-Sobolev type inequality for the associated free energy.
The latter inequality provides the optimal critical value for the conserved total population mass.
Description: This article has been accepted for publication and will appear in a revised form, subsequent to peer review and/or editorial input by Cambridge University Press, in European Journal of Applied Mathematics published by Cambridge University Press. Copyright Cambridge University Press 2017.Mon, 09 Oct 2017 00:00:00 GMThttp://hdl.handle.net/10034/6207052017-10-09T00:00:00ZSome time stepping methods for fractional diffusion problems with nonsmooth data
http://hdl.handle.net/10034/620655
Title: Some time stepping methods for fractional diffusion problems with nonsmooth data
Authors: Yang, Yan; Yan, Yubin; Ford, Neville J.
Abstract: We consider error estimates for some time stepping methods for solving fractional diffusion problems with nonsmooth data in both homogeneous and inhomogeneous cases. McLean and Mustapha \cite{mclmus} (Time-stepping error bounds for fractional diffusion problems with non-smooth initial data, Journal of Computational Physics, 293(2015), 201-217) established an $O(k)$ convergence rate for the piecewise constant discontinuous Galerkin method with nonsmooth initial data for the homogeneous problem when the linear operator $A$ is assumed to be self-adjoint, positive semidefinite and densely defined in a suitable Hilbert space, where $k$ denotes the time step size. In this paper, we approximate the Riemann-Liouville fractional derivative by Diethelm's method (or $L1$ scheme) and obtain the same time discretisation scheme as in McLean and Mustapha \cite{mclmus}. We first prove that this scheme has also convergence rate $O(k)$ with nonsmooth initial data for the homogeneous problem when $A$ is a closed, densely defined linear operator satisfying some certain resolvent estimates. We then introduce a new time discretization scheme for the homogeneous problem based on the convolution quadrature and prove that the convergence rate of this new scheme is $O(k^{1+ \alpha}), 0<\alpha <1 $ with the nonsmooth initial data. Using this new time discretization scheme for the homogeneous problem, we define a time stepping method for the inhomogeneous problem and prove that the convergence rate of this method is $O(k^{1+ \alpha}), 0<\alpha <1 $ with the nonsmooth data. Numerical examples are given to show that the numerical results are consistent with the theoretical results.Sat, 02 Sep 2017 00:00:00 GMThttp://hdl.handle.net/10034/6206552017-09-02T00:00:00ZAn analysis of the modified L1 scheme for time-fractional partial differential equations with nonsmooth data
http://hdl.handle.net/10034/620639
Title: An analysis of the modified L1 scheme for time-fractional partial differential equations with nonsmooth data
Authors: Yan, Yubin; Khan, Monzorul; Ford, Neville J.
Abstract: We introduce a modified L1 scheme for solving time fractional partial differential equations and obtain error estimates for smooth and nonsmooth initial data in both homogeneous and inhomogeneous cases. Jin \et (2016, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA J. of Numer. Anal., 36, 197-221) established an $O(k)$ convergence rate for the L1 scheme for smooth and nonsmooth initial data for the homogeneous problem, where $k$ denotes the time step size. We show that the modified L1 scheme has convergence rate $O(k^{2-\alpha}), 0< \alpha <1$ for smooth and nonsmooth initial data in both homogeneous and inhomogeneous cases. Numerical examples are given to show that the numerical results are consistent with the theoretical results.Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10034/6206392017-01-01T00:00:00Z