Mathematics
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Higher Order Time Stepping Methods for Subdiffusion Problems Based on Weighted and Shifted Grünwald–Letnikov Formulae with Nonsmooth DataTwo 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\"unwaldLetnikov formulae introduced in Tian et al. [Math. Comp. 84 (2015), pp. 27032727]. 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.

An analysis of the L1 scheme for stochastic subdiffusion problem driven by integrated spacetime white noiseWe consider the strong convergence of the numerical methods for solving stochastic subdiffusion problem driven by an integrated spacetime 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. 17151741), 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 multidimensional cases by using the Laplace transform method and the corresponding resolvent estimates.

New binary selfdual codes via a generalization of the four circulant constructionIn this work, we generalize the four circulant construction for selfdual 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 selfdual codes of lengths 40, 64 including new extremal binary selfdual codes of length 68. More precisely, 43 new extremal binary selfdual codes of length 68, with rare new parameters have been constructed.

2^n Bordered Constructions of SelfDual codes from Group RingsSelfdual 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 selfdual codes. In this paper, we introduce a new bordered construction over group rings for selfdual codes by combining many of the previously used techniques. The purpose of this is to construct selfdual codes that were missed using classical construction techniques by constructing selfdual 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 selfdual codes. In particular, we construct some extremal selfdual codes length 64 and 68, constructing 30 new extremal selfdual codes of length 68.

Finitetime blowup of a nonlocal stochastic parabolic problemThe main aim of the current work is the study of the conditions under which (finitetime) blowup of a nonlocal stochastic parabolic problem occurs. We first establish the existence and uniqueness of the localintime weak solution for such problem. The first part of the manuscript deals with the investigation of the conditions which guarantee the occurrence of noiseinduced blowup. 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 (nonlocal) drift term induced blowup result. In the last part of the paper, we present a method which provides an upper bound of the probability of (nonlocal) drift term induced blowup.

New Extremal SelfDual Binary Codes of Length 68 via Composite Construction, F2 + uF2 Lifts, Extensions and NeighborsWe 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 selfdual 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 selfdual 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 selfdual codes of length 68. As a result, we obtain 28 new codes of length 68 which were not known in the literature before.

The diffusiondriven instability and complexity for a singlehanded discrete Fisher equationFor 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 diffusiondriven instability/Turing instability for a singlehanded discrete Fisher equation with the Neumann boundary conditions may occur and a series of 2periodic patterns have been observed. Motivated by these pattern formations, the existence of 2periodic 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 problemWe use the generalized L1 approximation for the Caputo fractional derivative, the secondorder fractional quadrature rule approximation for the integral term, and a classical CrankNicolson alternating direction implicit (ADI)scheme for the time discretization of a new twodimensional (2D) fractionalintegrodifferential equation, in combination with a space discretization by anarbitraryorder orthogonal spline collocation (OSC) method. The stability of aCrankNicolson 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 SelfDual Codes of Lengths 64, 66 and 68In 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 selfdual 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 SelfDual Codes from Group Rings and Reverse Circulant MatricesIn this work, we describe a construction for selfdual 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 selfdual 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, twentytwo new codes of length 68, twelve new codes of length 80 and four new codes of length 92.

Developing A Highperformance Liquid Chromatography Method for Simultaneous Determination of Loratadine and its Metabolite Desloratadine in Human Plasma.Allergic diseases are considered among the major burdons of public health with increased prevalence globally. Histamine H1receptor 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 antihistaminic action of 2.5 to 4 times greater than loratadine. To develop and validate a novel isocratic reversedphase high performance liquid chromatography (RPHPLC) 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 SelfDual Codes from Group Rings over Frobenius RingsIn this work, we describe a double bordered construction of selfdual 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 selfdual 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 equationA 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.

Gcodes over Formal Power Series Rings and Finite Chain RingsIn 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 SelfDual Codes from Group RingsWe describe a bordered construction for selfdual codes coming from group rings. We apply the constructions coming from the cyclic and dihedral groups over several alphabets to obtain extremal binary selfdual codes of various lengths. In particular we find a new extremal binary selfdual code of length 78.

Composite Constructions of SelfDual Codes from Group Rings and New Extremal SelfDual Binary Codes of Length 68We describe eight composite constructions from group rings where the orders of the groups are 4 and 8, which are then applied to find selfdual 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 selfdual binary codes of length 64. Finally, we use a buildingup method over F2 + uF2 to obtain new extremal binary selfdual codes of length 68. We construct 11 new codes via the buildingup method and 2 new codes by considering possible neighbors.

Numerical methods for solving space fractional partial differential equations by using Hadamard finitepart integral approachWe introduce a novel numerical method for solving twosided space fractional partial differential equation in two dimensional case. The approximation of the space fractional RiemannLiouville derivative is based on the approximation of the Hadamard finitepart 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 RiemannLiouville 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\"unwaldLetnikov 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 finitepart 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 finitepart integral approach for solving space fractional partial differential equation with nonhomogeneous Dirichlet boundary conditions is indeed higher than the convergence order obtained by using the numerical methods constructed with the standard shifted Gr\"unwaldLetnikov formula or Lubich's higer order approximation schemes.

Numerical Approximation of Stochastic TimeFractional DiffusionWe develop and analyze a numerical method for stochastic timefractional 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 RiemannLiouville 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\"unwaldLetnikov 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 twodimensional numerical results are presented to support the theoretical findings.

A discrete mutualism model: analysis and exploration of a financial applicationWe 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 SelfDual Codes from Group Rings and New Extremal Binary Selfdual Codes IWe introduce an altered version of the four circulant construction over group rings for selfdual 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 selfdual codes of lengths 16, 32, 48, 64 and 68, many of which are extremal. In particular, we find forty new extremal binary selfdual 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.