• New binary self-dual codes via a generalization of the four circulant construction

      Gildea, Joe; Kaya, Abidin; Yildiz, Bahattin; University of Chester ; Sampoerna University ; Northern Arizona University
      In this work, we generalize the four circulant construction for self-dual codes. By applying the constructions over the alphabets $\mathbb{F}_2$, $\mathbb{F}_2+u\mathbb{F}_2$, $\mathbb{F}_4+u\mathbb{F}_4$, we were able to obtain extremal binary self-dual codes of lengths 40, 64 including new extremal binary self-dual codes of length 68. More precisely, 43 new extremal binary self-dual codes of length 68, with rare new parameters have been constructed.
    • 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
      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.
    • Finite-time blow-up of a non-local stochastic parabolic problem

      Kavallaris, Nikos; Yan, Yubin; University of Chester
      The main aim of the current work is the study of the conditions under which (finite-time) blow-up of a non-local stochastic parabolic problem occurs. We first establish the existence and uniqueness of the local-in-time weak solution for such problem. The first part of the manuscript deals with the investigation of the conditions which guarantee the occurrence of noise-induced blow-up. In the second part we first prove the $C^{1}$-spatial regularity of the solution. Then, based on this regularity result, and using a strong positivity result we derive, for first in the literature of SPDEs, a Hopf's type boundary value point lemma. The preceding results together with Kaplan's eigenfunction method are then employed to provide a (non-local) drift term induced blow-up result. In the last part of the paper, we present a method which provides an upper bound of the probability of (non-local) drift term induced blow-up.
    • New Extremal Self-Dual Binary Codes of Length 68 via Composite Construction, F2 + uF2 Lifts, Extensions and Neighbors

      Dougherty, Steven; Gildea, Joe; Korban, Adrian; Kaya, Abidin; University of Scranton; University of Chester; University of Chester; Sampoerna Academy;
      We describe a composite construction from group rings where the groups have orders 16 and 8. This construction is then applied to find the extremal binary self-dual codes with parameters [32, 16, 8] or [32, 16, 6]. We also extend this composite construction by expanding the search field which enables us to find more extremal binary self-dual codes with the above parameters and with different orders of automorphism groups. These codes are then lifted to F2 + uF2, to obtain extremal binary images of codes of length 64. Finally, we use the extension method and neighbor construction to obtain new extremal binary self-dual codes of length 68. As a result, we obtain 28 new codes of length 68 which were not known in the literature before.
    • 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
      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.
    • High‐order ADI orthogonal spline collocation method for a new 2D fractional integro‐differential problem

      Yan, Yubin; Qiao, Leijie; Xu, Da; University of Chester, UK; Guangdong University of Technology, PR. China; Hunan Normal University, P. R. China
      We use the generalized L1 approximation for the Caputo fractional deriva-tive, the second-order fractional quadrature rule approximation for the inte-gral term, and a classical Crank-Nicolson alternating direction implicit (ADI)scheme for the time discretization of a new two-dimensional (2D) fractionalintegro-differential equation, in combination with a space discretization by anarbitrary-order orthogonal spline collocation (OSC) method. The stability of aCrank-Nicolson ADI OSC scheme is rigourously established, and error estimateis also derived. Finally, some numerical tests are given
    • Modified Quadratic Residue Constructions and New Exermal Binary Self-Dual Codes of Lengths 64, 66 and 68

      Gildea, Joe; Hamilton, Holly; Kaya, Abidin; Yildiz, Bahattin; University of Chester; University of Chester; Sampoerna University; Northern Arizona University (Elsevier, 2020-02-10)
      In this work we consider modified versions of quadratic double circulant and quadratic bordered double circulant constructions over the binary field and the rings F2 +uF2 and F4 +uF4 for different prime values of p. Using these constructions with extensions and neighbors we are able to construct a number of extremal binary self-dual codes of different lengths with new parameters in their weight enumerators. In particular we construct 2 new codes of length 64, 4 new codes of length 66 and 14 new codes of length 68. The binary generator matrices of the new codes are available online at [8].
    • Constructing Self-Dual Codes from Group Rings and Reverse Circulant Matrices

      Gildea, Joe; Kaya, Abidin; Korban, Adrian; Yildiz, Bahattin; University of Chester; Sampoerna Academy; Northern Arizona University
      In this work, we describe a construction for self-dual codes in which we employ group rings and reverse circulant matrices. By applying the construction directly over different alphabets, and by employing the well known extension and neighbor methods we were able to obtain extremal binary self-dual codes of different lengths of which some have parameters that were not known in the literature before. In particular, we constructed three new codes of length 64, twenty-two new codes of length 68, twelve new codes of length 80 and four new codes of length 92.
    • Developing A High-performance Liquid Chromatography Method for Simultaneous Determination of Loratadine and its Metabolite Desloratadine in Human Plasma.

      Sebaiy, Mahmoud M; Ziedan, Noha I (2019-11-24)
      Allergic diseases are considered among the major burdons of public health with increased prevalence globally. Histamine H1-receptor antagonists are the foremost commonly used drugs in the treatment of allergic disorders. Our target drug is one of this class, loratadine and its biometabolite desloratadine which is also a non sedating H1 receptor antagonist with anti-histaminic action of 2.5 to 4 times greater than loratadine. To develop and validate a novel isocratic reversed-phase high performance liquid chromatography (RP-HPLC) method for rapid and simultaneous separation and determination of loratadine and its metabolite, desloratadine in human plasma. The drug extraction method from plasma was based on protein precipitation technique. The separation was carried out on a Thermo Scientific BDS Hypersil C18 column (5µm, 250 x 4.60 mm) using a mobile phase of MeOH : 0.025M KH2PO4 adjusted to pH 3.50 using orthophosphoric acid (85 : 15, v/v) at ambient temperature. The flow rate was maintained at 1 mL/min and maximum absorption was measured using PDA detector at 248 nm. The retention times of loratadine and desloratadine in plasma samples were recorded to be 4.10 and 5.08 minutes respectively, indicating a short analysis time. Limits of detection were found to be 1.80 and 1.97 ng/mL for loratadine and desloratadine, respectively, showing a high degree of method sensitivity. The method was then validated according to FDA guidelines for the determination of the two analytes in human plasma. The results obtained indicate that the proposed method is rapid, sensitive in the nanogram range, accurate, selective, robust and reproducible compared to other reported methods. [Abstract copyright: Copyright© Bentham Science Publishers; For any queries, please email at epub@benthamscience.net.]
    • Double Bordered Constructions of Self-Dual Codes from Group Rings over Frobenius Rings

      Gildea, Joe; Kaya, Abidin; Taylor, Rhian; Tylyshchak, Alexander; University of Chester; Sampoerna University; Uzhgorod State University
      In this work, we describe a double bordered construction of self-dual codes from group rings. We show that this construction is effective for groups of order 2p where p is odd, over the rings F2 + uF2 and F4 + uF4. We demonstrate the importance of this new construction by finding many new binary self-dual codes of lengths 64, 68 and 80; the new codes and their corresponding weight enumerators are listed in several tables
    • On the behavior of the solutions for linear autonomous mixed type difference equation

      Yan, Yubin; Yenicerioglu, Ali Fuat; Pinelas, Sandra; University of Chester; Kocaeli University, Turkey; RUDN University, Russia (Springer Link, 2019-07-30)
      A class of linear autonomous mixed type difference equations is considered, and some new results on the asymptotic behavior and the stability are given, via a positive root of the corresponding characteristic equation.
    • G-codes over Formal Power Series Rings and Finite Chain Rings

      Dougherty, Steven; Gildea, Joe; Korban, Adrian; University of Scranton; University of Chester
      In this work, we define $G$-codes over the infinite ring $R_\infty$ as ideals in the group ring $R_\infty G$. We show that the dual of a $G$-code is again a $G$-code in this setting. We study the projections and lifts of $G$-codes over the finite chain rings and over the formal power series rings respectively. We extend known results of constructing $\gamma$-adic codes over $R_\infty$ to $\gamma$-adic $G$-codes over the same ring. We also study $G$-codes over principal ideal rings.
    • A Modified Bordered Construction for Self-Dual Codes from Group Rings

      Kaya, Abidin; Tylyshchak, Alexander; Yildiz, Bahattin; Gildea, Joe; University of Chester; Sampoerna University; Uzhgorod State University; Northern Arizona University (Jacodesmath Institute, 2019)
      We describe a bordered construction for self-dual codes coming from group rings. We apply the constructions coming from the cyclic and dihedral groups over several alphabets to obtain extremal binary self-dual codes of various lengths. In particular we find a new extremal binary self-dual code of length 78.
    • Composite Constructions of Self-Dual Codes from Group Rings and New Extremal Self-Dual Binary Codes of Length 68

      Dougherty, Steven; Gildea, Joe; Kaya, Abidin; Korban, Adrian; University of Scranton; University of Chester; Sampoerna University ; University of Chester (American Institute of Mathematical Sciences, 2019-11-30)
      We describe eight composite constructions from group rings where the orders of the groups are 4 and 8, which are then applied to find self-dual codes of length 16 over F4. These codes have binary images with parameters [32, 16, 8] or [32, 16, 6]. These are lifted to codes over F4 + uF4, to obtain codes with Gray images extremal self-dual binary codes of length 64. Finally, we use a building-up method over F2 + uF2 to obtain new extremal binary self-dual codes of length 68. We construct 11 new codes via the building-up method and 2 new codes by considering possible neighbors.
    • Numerical methods for solving space fractional partial differential equations by using Hadamard finite-part integral approach

      Yan, Yubin; Wang, Yanyong; Hu, Ye; University of Chester; Lvliang University (Springer, 2019-07-26)
      We introduce a novel numerical method for solving two-sided space fractional partial differential equation in two dimensional case. The approximation of the space fractional Riemann-Liouville derivative is based on the approximation of the Hadamard finite-part integral which has the convergence order $O(h^{3- \alpha})$, where $h$ is the space step size and $\alpha\in (1, 2)$ is the order of Riemann-Liouville fractional derivative. Based on this scheme, we introduce a shifted finite difference method for solving space fractional partial differential equation. We obtained the error estimates with the convergence orders $O(\tau +h^{3-\alpha}+ h^{\beta})$, where $\tau$ is the time step size and $\beta >0$ is a parameter which measures the smoothness of the fractional derivatives of the solution of the equation. Unlike the numerical methods for solving space fractional partial differential equation constructed by using the standard shifted Gr\"unwald-Letnikov formula or higher order Lubich'e methods which require the solution of the equation satisfies the homogeneous Dirichlet boundary condition in order to get the first order convergence, the numerical method for solving space fractional partial differential equation constructed by using Hadamard finite-part integral approach does not require the solution of the equation satisfies the Dirichlet homogeneous boundary condition. Numerical results show that the experimentally determined convergence order obtained by using the Hadamard finite-part integral approach for solving space fractional partial differential equation with non-homogeneous Dirichlet boundary conditions is indeed higher than the convergence order obtained by using the numerical methods constructed with the standard shifted Gr\"unwald-Letnikov formula or Lubich's higer order approximation schemes.
    • Numerical Approximation of Stochastic Time-Fractional Diffusion

      Yan, Yubin; Jin, Bangti; Zhou, Zhi; University of Chester; University College London; The Hong Kong Polytechnic University
      We develop and analyze a numerical method for stochastic time-fractional diffusion driven by additive fractionally integrated Gaussian noise. The model involves two nonlocal terms in time, i.e., a Caputo fractional derivative of order $\alpha\in(0,1)$, and fractionally integrated Gaussian noise (with a Riemann-Liouville fractional integral of order $\gamma \in[0,1]$ in the front). The numerical scheme approximates the model in space by the standard Galerkin method with continuous piecewise linear finite elements and in time by the classical Gr\"unwald-Letnikov method, and the noise by the $L^2$-projection. Sharp strong and weak convergence rates are established, using suitable nonsmooth data error estimates for the deterministic counterpart. One- and two-dimensional numerical results are presented to support the theoretical findings.
    • 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.
    • 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.
    • Quadruple Bordered Constructions of Self-Dual Codes from Group Rings

      Dougherty, Steven; Gildea, Joe; Kaya, Abidin; University of Scranton; University of Chester; Sampoerna University (Springer Verlag, 2019-07-05)
      In this paper, we introduce a new bordered construction for self-dual codes using group rings. We consider constructions over the binary field, the family of rings Rk and the ring F4 + uF4. We use groups of order 4, 12 and 20. We construct some extremal self-dual codes and non-extremal self-dual codes of length 16, 32, 48, 64 and 68. In particular, we construct 33 new extremal self-dual codes of length 68.
    • 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.