• 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 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.
    • 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 (Croatian Mathematical Society, 2020-05-31)
      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.
    • Higher Order Time Stepping Methods for Subdiffusion Problems Based on Weighted and Shifted Grünwald–Letnikov Formulae with Nonsmooth Data

      Yan, yubin; Wang, Yanyong; Yan, Yuyuan; Pani, Amiya K.; University of Chester, Lvliang University, Jimei University, Indian Institute of Technology Bombay (Springer Link, 2020-05-19)
      Two higher order time stepping methods for solving subdiffusion problems are studied in this paper. The Caputo time fractional derivatives are approximated by using the weighted and shifted Gr\"unwald-Letnikov formulae introduced in Tian et al. [Math. Comp. 84 (2015), pp. 2703-2727]. After correcting a few starting steps, the proposed time stepping methods have the optimal convergence orders $O(k^2)$ and $ O(k^3)$, respectively for any fixed time $t$ for both smooth and nonsmooth data. The error estimates are proved by directly bounding the approximation errors of the kernel functions. Moreover, we also present briefly the applicabilities of our time stepping schemes to various other fractional evolution equations. Finally, some numerical examples are given to show that the numerical results are consistent with the proven theoretical results.
    • 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, 2020-05-07)
      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.
    • G-codes over Formal Power Series Rings and Finite Chain Rings

      Dougherty, Steven; Gildea, Joe; Korban, Adrian; University of Scranton; University of Chester (2020-02-29)
      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.
    • 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; (Inderscience, 2020-02-29)
      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.
    • 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].
    • 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 (John Wiley & Sons Ltd, 2020-02-05)
      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
    • 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 (American Institute of Mathematical Sciences, 2020-01-20)
      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.
    • 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.
    • 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.
    • 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.]
    • 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.
    • New Self-Dual and Formally Self-Dual Codes from Group Ring Constructions

      Dougherty, Steven; Gildea, Joe; Kaya, Abidin; Yildiz, Bahattin; University of Scranton; University of Chester; Sampoerna Academy; University of Chester; Northern Arizona University (American Institute of Mathematical Sciences, 2019-08-31)
      In this work, we study construction methods for self-dual and formally self-dual codes from group rings, arising from the cyclic group, the dihedral group, the dicyclic group and the semi-dihedral group. Using these constructions over the rings $_F2 +uF_2$ and $F_4 + uF_4$, we obtain 9 new extremal binary self-dual codes of length 68 and 25 even formally self-dual codes with parameters [72,36,14].
    • 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.
    • 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.
    • 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 (EDP Sciences, 2019-07-09)
      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.
    • 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.