• 2^n Bordered Constructions of Self-Dual codes from Group Rings

      Dougherty, Steven; Gildea, Joe; Kaya, Abidin; University of Scranton; University of Chester; Sampoerna Academy (Elsevier, 2020-08-04)
      Self-dual codes, which are codes that are equal to their orthogonal, are a widely studied family of codes. Various techniques involving circulant matrices and matrices from group rings have been used to construct such codes. Moreover, families of rings have been used, together with a Gray map, to construct binary self-dual codes. In this paper, we introduce a new bordered construction over group rings for self-dual codes by combining many of the previously used techniques. The purpose of this is to construct self-dual codes that were missed using classical construction techniques by constructing self-dual codes with different automorphism groups. We apply the technique to codes over finite commutative Frobenius rings of characteristic 2 and several group rings and use these to construct interesting binary self-dual codes. In particular, we construct some extremal self-dual codes length 64 and 68, constructing 30 new extremal self-dual codes of length 68.
    • An algorithm to detect small solutions in linear delay differential equations

      Ford, Neville J.; Lumb, Patricia M. (Elsevier, 2006-08-15)
      This preprint discusses an algorithm that provides a simple reliable mechanism for the detection of small solutions in linear delay differential equations.
    • An Altered Four Circulant Construction for Self-Dual Codes from Group Rings and New Extremal Binary Self-dual Codes I

      Gildea, Joe; Kaya, Abidin; Yildiz, Bahattin; University of Chester; Sampoerna University; Northern Arizona University (Elsevier, 2019-08-07)
      We introduce an altered version of the four circulant construction over group rings for self-dual codes. We consider this construction over the binary field, the rings F2 + uF2 and F4 + uF4; using groups of order 4 and 8. Through these constructions and their extensions, we find binary self-dual codes of lengths 16, 32, 48, 64 and 68, many of which are extremal. In particular, we find forty new extremal binary self-dual codes of length 68, including twelve new codes with \gamma=5 in W68,2, which is the first instance of such a value in the literature.
    • An analysis of the L1 scheme for stochastic subdiffusion problem driven by integrated space-time white noise

      Yan, Yubin; Yan, Yuyuan; Wu, Xiaolei; University of Chester, Lvliang University, Jimei University (Elsevier, 2020-06-02)
      We consider the strong convergence of the numerical methods for solving stochastic subdiffusion problem driven by an integrated space-time white noise. The time fractional derivative is approximated by using the L1 scheme and the time fractional integral is approximated with the Lubich's first order convolution quadrature formula. We use the Euler method to approximate the noise in time and use the truncated series to approximate the noise in space. The spatial variable is discretized by using the linear finite element method. Applying the idea in Gunzburger \et (Math. Comp. 88(2019), pp. 1715-1741), we express the approximate solutions of the fully discrete scheme by the convolution of the piecewise constant function and the inverse Laplace transform of the resolvent related function. Based on such convolution expressions of the approximate solutions, we obtain the optimal convergence orders of the fully discrete scheme in spatial multi-dimensional cases by using the Laplace transform method and the corresponding resolvent estimates.
    • An analytic approach to the normalized Ricci flow-like equation: Revisited

      Kavallaris, Nikos I.; Suzuki, Takashi; University of Chester ; Osaka University (Elsevier, 2015-01-07)
      In this paper we revisit Hamilton’s normalized Ricci flow, which was thoroughly studied via a PDE approach in Kavallaris and Suzuki (2010). Here we provide an improved convergence result compared to the one presented Kavallaris and Suzuki (2010) for the critical case λ=8πλ=8π. We actually prove that the convergence towards the stationary normalized Ricci flow is realized through any time sequence.
    • Analytical and numerical investigation of mixed-type functional differential equations

      Lima, Pedro M.; Teodoro, M. Filomena; Ford, Neville J.; Lumb, Patricia M.; Instituto Superior Tecnico UTL, Lisbon : Instituto Politecnico de Setubal, Lisbon : University of Chester : University of Chester (Elsevier, 2009-11-09)
      This journal article is concerned with the approximate solution of a linear non-autonomous functional differential equation, with both advanced and delayed arguments.
    • Analytical and numerical treatment of oscillatory mixed differential equations with differentiable delays and advances

      Ferreira, José M.; Ford, Neville J.; Malique, Md A.; Pinelas, Sandra; Yan, Yubin; Instituto Superior Técnico, Lisbon : University of Chester : University of Chester : Universidade dos Açores : University of Chester (Elsevier, 2011-04-12)
      This article discusses the oscillatory behaviour of the differential equation of mixed type.
    • Blending low-order stabilised finite element methods: a positivity preserving local projection method for the convection-diffusion equation

      Barrenechea, Gabriel; Burman, Erik; Karakatsani, Fotini; University of Strathclyde; UCL; University of Chester (Elsevier, 2017-01-20)
      In this work we propose a nonlinear blending of two low-order stabilisation mechanisms for the convection–diffusion equation. The motivation for this approach is to preserve monotonicity without sacrificing accuracy for smooth solutions. The approach is to blend a first-order artificial diffusion method, which will be active only in the vicinity of layers and extrema, with an optimal order local projection stabilisation method that will be active on the smooth regions of the solution. We prove existence of discrete solutions, as well as convergence, under appropriate assumptions on the nonlinear terms, and on the exact solution. Numerical examples show that the discrete solution produced by this method remains within the bounds given by the continuous maximum principle, while the layers are not smeared significantly.
    • Bordered Constructions of Self-Dual Codes from Group Rings and New Extremal Binary Self-Dual Codes

      Dougherty, Steven; Gildea, Joe; Kaya, Abidin; Korban, Adrian; Tylyshchak, Alexander; Yildiz, Bahattin; University of Scranton; University of Chester; Sampoerna Academy; Uzhgorod State University; Northern Arizona University (Elsevier, 2019-02-22)
      We introduce a bordered construction over group rings for self-dual codes. We apply the constructions over the binary field and the ring $\F_2+u\F_2$, using groups of orders 9, 15, 21, 25, 27, 33 and 35 to find extremal binary self-dual codes of lengths 20, 32, 40, 44, 52, 56, 64, 68, 88 and best known binary self-dual codes of length 72. In particular we obtain 41 new binary extremal self-dual codes of length 68 from groups of orders 15 and 33 using neighboring and extensions. All the numerical results are tabulated throughout the paper.
    • Characteristic functions of differential equations with deviating arguments

      Baker, Christopher T. H.; Ford, Neville J.; University of Manchester; University of Chester (Elsevier, 2019-04-24)
      The material here is motivated by the discussion of solutions of linear homogeneous and autonomous differential equations with deviating arguments. If $a, b, c$ and $\{\check{\tau}_\ell\}$ are real and ${\gamma}_\natural$ is real-valued and continuous, an example with these parameters is \begin{equation} u'(t) = \big\{a u(t) + b u(t+\check{\tau}_1) + c u(t+\check{\tau}_2) \big\} { \red +} \int_{\check{\tau}_3}^{\check{\tau}_4} {{\gamma}_\natural}(s) u(t+s) ds \tag{\hbox{$\rd{\star}$}} . \end{equation} A wide class of equations ($\rd{\star}$), or of similar type, can be written in the {\lq\lq}canonical{\rq\rq} form \begin{equation} u'(t) =\DSS \int_{\tau_{\rd \min}}^{\tau_{\rd \max}} u(t+s) d\sigma(s) \quad (t \in \Rset), \hbox{ for a suitable choice of } {\tau_{\rd \min}}, {\tau_{\rd \max}} \tag{\hbox{${\rd \star\star}$}} \end{equation} where $\sigma$ is of bounded variation and the integral is a Riemann-Stieltjes integral. For equations written in the form (${\rd{\star\star}}$), there is a corresponding characteristic function \begin{equation} \chi(\zeta) ):= \zeta - \DSS \int_{\tau_{\rd \min}}^{\tau_{\rd \max}} \exp(\zeta s) d\sigma(s) \quad (\zeta \in \Cset), \tag{\hbox{${\rd{\star\star\star}}$}} \end{equation} %%($ \chi(\zeta) \equiv \chi_\sigma (\zeta)$) whose zeros (if one considers appropriate subsets of equations (${\rd \star\star}$) -- the literature provides additional information on the subsets to which we refer) play a r\^ole in the study of oscillatory or non-oscillatory solutions, or of bounded or unbounded solutions. We show that the related discussion of the zeros of $\chi$ is facilitated by observing and exploiting some simple and fundamental properties of characteristic functions.
    • Computational approaches to parameter estimation and model selection in immunology

      Baker, Christopher T. H.; Bocharov, Gennady; Ford, Judith M.; Lumb, Patricia M.; Norton, Stewart J.; Paul, C. A. H.; Junt, Tobias; Krebs, Philippe; Ludewig, Burkhard (Elsevier, 2005-12-01)
      This article seeks to illustrate the computational implementation of an information-theoretic approach (associated with a maximum likelihood treatment) to modelling in immunology.
    • Computational methods for a mathematical model of propagation of nerve impulses in myelinated axons

      Lima, Pedro M.; Ford, Neville J.; Lumb, Patricia M.; CEMAT, IST, Lisbon ; University of Chester ; University of Chester (Elsevier, 2014-06-26)
      This paper is concerned with the approximate solution of a nonlinear mixed type functional differential equation (MTFDE) arising from nerve conduction theory. The equation considered describes conduction in a myelinated nerve axon. We search for a monotone solution of the equation defined in the whole real axis, which tends to given values at ±∞. We introduce new numerical methods for the solution of the equation, analyse their performance, and present and discuss the results of the numerical simulations.
    • Computational modelling with functional differential equations: Identification, selection, and sensitivity

      Baker, Christopher T. H.; Bocharov, Gennady; Paul, C. A. H.; Rihan, F. A. R.; University College Chester ; Institute of Numerical Mathematics, Russian Academy of Sciences ; University of Salford (Elsevier, 2005-05)
      Mathematical models based upon certain types of differential equations, functional differential equations, or systems of such equations, are often employed to represent the dynamics of natural, in particular biological, phenomena. We present some of the principles underlying the choice of a methodology (based on observational data) for the computational identification of, and discrimination between, quantitatively consistent models, using scientifically meaningful parameters. We propose that a computational approach is essential for obtaining meaningful models. For example, it permits the choice of realistic models incorporating a time-lag which is entirely natural from the scientific perspective. The time-lag is a feature that can permit a close reconciliation between models incorporating computed parameter values and observations. Exploiting the link between information theory, maximum likelihood, and weighted least squares, and with distributional assumptions on the data errors, we may construct an appropriate objective function to be minimized computationally. The minimizer is sought over a set of parameters (which may include the time-lag) that define the model. Each evaluation of the objective function requires the computational solution of the parametrized equations defining the model. To select a parametrized model, from amongst a family or hierarchy of possible best-fit models, we are able to employ certain indicators based on information-theoretic criteria. We can evaluate confidence intervals for the parameters, and a sensitivity analysis provides an expression for an information matrix, and feedback on the covariances of the parameters in relation to the best fit. This gives a firm basis for any simplification of the model (e.g., by omitting a parameter).
    • Constructions for Self-Dual Codes Induced from Group Rings

      Gildea, Joe; Kaya, Abidin; Taylor, Rhian; Yildiz, Bahattin; University of Chester; Sampoerna Academy, University of Chester; Northern Arizona University (Elsevier, 2018-02-03)
      In this work, we establish a strong connection between group rings and self-dual codes. We prove that a group ring element corresponds to a self-dual code if and only if it is a unitary unit. We also show that the double-circulant and four-circulant constructions come from cyclic and dihedral groups, respectively. Using groups of order 8 and 16 we find many new construction methods, in addition to the well-known methods, for self-dual codes. We establish the relevance of these new constructions by finding many extremal binary self-dual codes using them, which we list in several tables. In particular, we construct 10 new extremal binary self-dual codes of length 68.
    • Data-driven selection and parameter estimation for DNA methylation mathematical models

      Larson, Karen; Zagkos, Loukas; Mc Auley, Mark T.; Roberts, Jason A.; Kavallaris, Nikos I.; Matzavinos, Anastasios; Brown University; University of Chester (Elsevier, 2019-01-10)
      Epigenetics is coming to the fore as a key process which underpins health. In particular emerging experimental evidence has associated alterations to DNA methylation status with healthspan and aging. Mammalian DNA methylation status is maintained by an intricate array of biochemical and molecular processes. It can be argued changes to these fundamental cellular processes ultimately drive the formation of aberrant DNA methylation patterns, which are a hallmark of diseases, such as cancer, Alzheimer's disease and cardiovascular disease. In recent years mathematical models have been used as e ective tools to help advance our understanding of the dynamics which underpin DNA methylation. In this paper we present linear and nonlinear models which encapsulate the dynamics of the molecular mechanisms which de ne DNA methylation. Applying a recently developed Bayesian algorithm for parameter estimation and model selection, we are able to estimate distributions of parameters which include nominal parameter values. Using limited noisy observations, the method also identifed which methylation model the observations originated from, signaling that our method has practical applications in identifying what models best match the biological data for DNA methylation.
    • The diffusion-driven instability and complexity for a single-handed discrete Fisher equation

      Yan, Yubin; Zhang, Guang; Zhang, Ruixuan; University of Chester; Tianjin University of Commerce (Elsevier, 2019-12-19)
      For a reaction diffusion system, it is well known that the diffusion coefficient of the inhibitor must be bigger than that of the activator when the Turing instability is considered. However, the diffusion-driven instability/Turing instability for a single-handed discrete Fisher equation with the Neumann boundary conditions may occur and a series of 2-periodic patterns have been observed. Motivated by these pattern formations, the existence of 2-periodic solutions is established. Naturally, the periodic double and the chaos phenomenon should be considered. To this end, a simplest two elements system will be further discussed, the flip bifurcation theorem will be obtained by computing the center manifold, and the bifurcation diagrams will be simulated by using the shooting method. It proves that the Turing instability and the complexity of dynamical behaviors can be completely driven by the diffusion term. Additionally, those effective methods of numerical simulations are valid for experiments of other patterns, thus, are also beneficial for some application scientists.
    • Discontinuous Galerkin time stepping method for solving linear space fractional partial differential equations

      Liu, Yanmei; Yan, Yubin; Khan, Monzorul; LuLiang University; University of Chester (Elsevier, 2017-01-23)
      In this paper, we consider the discontinuous Galerkin time stepping method for solving the linear space fractional partial differential equations. The space fractional derivatives are defined by using Riesz fractional derivative. The space variable is discretized by means of a Galerkin finite element method and the time variable is discretized by the discontinuous Galerkin method. The approximate solution will be sought as a piecewise polynomial function in $t$ of degree at most $q-1, q \geq 1$, which is not necessarily continuous at the nodes of the defining partition. The error estimates in the fully discrete case are obtained and the numerical examples are given.
    • A discrete mutualism model: analysis and exploration of a financial application

      Roberts, Jason A.; Kavallaris, Nikos I.; Rowntree, Andrew P.; University of Chester (Elsevier, 2019-09-16)
      We perform a stability analysis on a discrete analogue of a known, continuous model of mutualism. We illustrate how the introduction of delays affects the asymptotic stability of the system’s positive nontrivial equilibrium point. In the second part of the paper we explore the insights that the model can provide when it is used in relation to interacting financial markets. We also note the limitations of such an approach.
    • Distributed order equations as boundary value problems

      Ford, Neville J.; Morgado, Maria L.; University of Chester ; University of Tras-os-Montes e Alto Douro (Elsevier, 2012-01-20)
      This preprint discusses the existence and uniqueness of solutions and proposes a numerical method for their approximation in the case where the initial conditions are not known and, instead, some Caputo-type conditions are given away from the origin.
    • Enumerating 3-generated axial algebras of Monster type

      McInroy, Justin; Shpectorov, Sergey; Khasraw, Sanhan; Salahaddin University-Erbil; University of Bristol; Heilbronn Institute for Mathematical Research, Bristol; University of Birmingham; University of Chester (Elsevier, 2021-06-17)
      An axial algebra is a commutative non-associative algebra generated by axes, that is, primitive, semisimple idempotents whose eigenvectors multiply according to a certain fusion law. The Griess algebra, whose automorphism group is the Monster, is an example of an axial algebra. We say an axial algebra is of Monster type if it has the same fusion law as the Griess algebra. The 2-generated axial algebras of Monster type, called Norton-Sakuma algebras, have been fully classified and are one of nine isomorphism types. In this paper, we enumerate a subclass of 3-generated axial algebras of Monster type in terms of their groups and shapes. It turns out that the vast majority of the possible shapes for such algebras collapse; that is they do not lead to non-trivial examples. This is in sharp contrast to previous thinking. Accordingly, we develop a method of minimal forbidden configurations, to allow us to efficiently recognise and eliminate collapsing shapes.