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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.

Quadruple Bordered Constructions of SelfDual Codes from Group RingsIn this paper, we introduce a new bordered construction for selfdual 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 selfdual codes and nonextremal selfdual codes of length 16, 32, 48, 64 and 68. In particular, we construct 33 new extremal selfdual codes of length 68.

Characteristic functions of differential equations with deviating argumentsThe 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 realvalued 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 RiemannStieltjes 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 nonoscillatory 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.

A Posteriori Analysis for SpaceTime, discontinuous in time Galerkin approximations for parabolic equations in a variable domainThis paper presents an a posteriori error analysis for the discontinuous in time spacetime scheme proposed by Jamet for the heat equation in multidimensional, noncylindrical domains [25]. Using a Cl ementtype interpolant, we prove abstract a posteriori error bounds for the numerical error. Furthermore, in the case of twodimensional spatial domains we transform the problem into an equivalent one, of parabolic type, with spacetime dependent coe cients but posed on a cylindrical domain. We formulate a discontinuous in time space{time scheme and prove a posteriori error bounds of optimal order. The a priori estimates of [19] for general parabolic initial and boundary value problems are used in the derivation of the upper bound. Our lower bound coincides with that of Picasso [36], proposed for adaptive, RungeKutta finite element methods for linear parabolic problems. Our theoretical results are verified by numerical experiments.

SpaceTime Discontinuous Galerkin Methods for the '\eps'dependent Stochastic AllenCahn Equation with mild noiseWe consider the $\eps$dependent stochastic AllenCahn equation with mild space time noise posed on a bounded domain of R^2. The positive parameter $\eps$ is a measure for the inner layers width that are generated during evolution. This equation, when the noise depends only on time, has been proposed by Funaki in [15]. The noise although smooth becomes white on the sharp interface limit as $\eps$ tends to zero. We construct a nonlinear dG scheme with spacetime finite elements of general type which are discontinuous in time. Existence of a unique discrete solution is proven by application of Brouwer's Theorem. We first derive abstract error estimates and then for the case of piecewise polynomial finite elements we prove an error in expectation of optimal order. All the appearing constants are estimated in terms of the parameter $\eps$. Finally, we present a linear approximation of the nonlinear scheme for which we prove existence of solution and optimal error in expectation in piecewise linear finite element spaces. The novelty of this work is based on the use of a finite element formulation in space and in time in 2+1dimensional subdomains for a nonlinear parabolic problem. In addition, this problem involves noise. These type of schemes avoid any RungeKutta type discretization for the evolutionary variable and seem to be very effective when applied to equations of such a difficulty.

Analysis of transient RivlinEricksen fluid and irreversibility of exothermic reactive hydromagnetic variable viscosityThe study analysed unsteady RivlinEricksen fluid and irreversibility of exponentially temperature dependent variable viscosity of hydromagnetic twostep exothermic chemical reactive flow along the channel axis with walls convective cooling. The nonNewtonian HeleShaw flow of RivlinErickson fluid is driven by bimolecular chemical kinetic and unvarying pressure gradient. The reactive fluid is induced by periodic changes in magnetic field and time. The Newtons law of cooling is satisfied by the constant heat coolant convection exchange at the wall surfaces with the neighboring regime. The dimensionless nonNewtonian reactive fluid equations are numerically solved using a convergent and consistence semiimplicit finite difference technique which are confirmed stable. The response of the reactive fluid flow to variational increase in the values of some entrenched fluid parameters in the momentum and energy balance equations are obtained. A satisfying equations for the ratio of irreversibility, entropy generation and Bejan number are solved with the results presented graphically and discussed quantitatively. From the study, it was obtained that the thermal criticality conditions with the right combination of thermofluid parameters, the thermal runaway can be prevented. Also, the entropy generation can minimize by at low dissipation rate and viscosity.

A high order numerical method for solving nonlinear fractional differential equation with nonuniform meshesWe introduce a highorder numerical method for solving nonlinear fractional differential equation with nonuniform meshes. We first transform the fractional nonlinear differential equation into the equivalent Volterra integral equation. Then we approximate the integral by using the quadratic interpolation polynomials. On the first subinterval $[t_{0}, t_{1}]$, we approximate the integral with the quadratic interpolation polynomials defined on the nodes $t_{0}, t_{1}, t_{2}$ and in the other subinterval $[t_{j}, t_{j+1}], j=1, 2, \dots N1$, we approximate the integral with the quadratic interpolation polynomials defined on the nodes $t_{j1}, t_{j}, t_{j+1}$. A highorder numerical method is obtained. Then we apply this numerical method with the nonuniform meshes with the step size $\tau_{j}= t_{j+1} t_{j}= (j+1) \mu$ where $\mu= \frac{2T}{N (N+1)}$. Numerical results show that this method with the nonuniform meshes has the higher convergence order than the standard numerical methods obtained by using the rectangle and the trapzoid rules with the same nonuniform meshes.

Datadriven selection and parameter estimation for DNA methylation mathematical modelsEpigenetics 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.

Bordered Constructions of SelfDual Codes from Group Rings and New Extremal Binary SelfDual CodesWe introduce a bordered construction over group rings for selfdual 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 selfdual codes of lengths 20, 32, 40, 44, 52, 56, 64, 68, 88 and best known binary selfdual codes of length 72. In particular we obtain 41 new binary extremal selfdual codes of length 68 from groups of orders 15 and 33 using neighboring and extensions. All the numerical results are tabulated throughout the paper.

On hereditary reducibility of 2monomial matrices over commutative ringsA 2monomial matrix over a commutative ring $R$ is by definition any matrix of the form $M(t,k,n)=\Phi\left(\begin{smallmatrix}I_k&0\\0&tI_{nk}\end{smallmatrix}\right)$, $0<k<n$, where $t$ is a noninvertible element of $R$, $\Phi$ the compa\nion matrix to $\lambda^n1$ and $I_k$ the identity $k\times k$matrix. In this paper we introduce the notion of hereditary reducibility (for these matrices) and indicate one general condition of the introduced reducibility.

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.

New SelfDual and Formally SelfDual Codes from Group Ring ConstructionsIn this work, we study construction methods for selfdual and formally selfdual codes from group rings, arising from the cyclic group, the dihedral group, the dicyclic group and the semidihedral group. Using these constructions over the rings $_F2 +uF_2$ and $F_4 + uF_4$, we obtain 9 new extremal binary selfdual codes of length 68 and 25 even formally selfdual codes with parameters [72,36,14].

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.

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.

Optimal convergence rates for semidiscrete finite element approximations of linear spacefractional partial differential equations under minimal regularity assumptionsWe consider the optimal convergence rates of the semidiscrete finite element approximations for solving linear spacefractional partial differential equations by using the regularity results for the fractional elliptic problems obtained recently by Jin et al. \cite{jinlazpasrun} and Ervin et al. \cite{ervheuroo}. The error estimates are proved by using two approaches. One approach is to apply the duality argument in Johnson \cite{joh} for the heat equation to consider the error estimates for the linear spacefractional partial differential equations. This argument allows us to obtain the optimal convergence rates under the minimal regularity assumptions for the solution. Another approach is to use the approximate solution operators of the corresponding fractional elliptic problems. This argument can be extended to consider more general linear spacefractional partial differential equations. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

A highorder scheme to approximate the Caputo fractional derivative and its application to solve the fractional diffusion wave equationA new highorder finite difference scheme to approximate the Caputo fractional derivative $\frac{1}{2} \big ( \, _{0}^{C}D^{\alpha}_{t}f(t_{k})+ \, _{0}^{C}D^{\alpha}_{t}f(t_{k1}) \big ), k=1, 2, \dots, N, $ with the convergence order $O(\Delta t^{4\alpha}), \, \alpha\in(1,2)$ is obtained when $f^{\prime \prime \prime} (t_{0})=0$, where $\Delta t$ denotes the time step size. Based on this scheme we introduce a finite difference method for solving fractional diffusion wave equation with the convergence order $O(\Delta t^{4\alpha} + h^2)$, where $h$ denotes the space step size. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

Theoretical and numerical analysis of unsteady fractional viscoelastic flows in simple geometries.In this work we discuss the connection between classical and fractional viscoelastic Maxwell models, presenting the basic theory supporting these constitutive equations, and establishing some background on the admissibility of the fractional Maxwell model. We then develop a numerical method for the solution of two coupled fractional differential equations (one for the velocity and the other for the stress), that appear in the pure tangential annular ow of fractional viscoelastic fluids. The numerical method is based on finite differences, with the approximation of fractional derivatives of the velocity and stress being inspired by the method proposed by Sun and Wu for the fractional diffusionwave equation [ Z.Z. Sun, X. Wu, A fully discrete difference scheme for a diffusionwave system, Applied Numerical Mathematics 56 (2006) 193209]. We prove solvability, study numerical convergence of the method, and also discuss the applicability of this method for simulating the rheological response of complex fluids in a real concentric cylinder rheometer. By imposing a torsional stepstrain, we observe the different rates of stress relaxation obtained with different values of \alpha and \beta (the fractional order exponents that regulate the viscoelastic response of the complex fluids).

Error estimates of highorder numerical methods for solving time fractional partial differential equationsError estimates of some highorder numerical methods for solving time fractional partial differential equations are studied in this paper. We first provide the detailed error estimate of a highorder numerical method proposed recently by Li et al. \cite{liwudin} for solving time fractional partial differential equation. We prove that this method has the convergence order $O(\tau^{3 \alpha})$ for all $\alpha \in (0, 1)$ when the first and second derivatives of the solution are vanish at $t=0$, where $\tau$ is the time step size and $\alpha$ is the fractional order in the Caputo sense. We then introduce a new time discretization method for solving time fractional partial differential equations, which has no requirements for the initial values as imposed in Li et al. \cite{liwudin}. We show that this new method also has the convergence order $O(\tau^{3 \alpha})$ for all $\alpha \in (0, 1)$. The proofs of the error estimates are based on the energy method developed recently by Lv and Xu \cite{lvxu}. We also consider the space discretization by using the finite element method. Error estimates with convergence order $O(\tau^{3 \alpha} + h^2)$ are proved in the fully discrete case, where $h$ is the space step size. Numerical examples in both one and twodimensional cases are given to show that the numerical results are consistent with the theoretical results.

Malliavin Calculus for the stochastic Cahn Hilliard/AllenCahn equation with unbounded noise diffusionThe stochastic partial di erential equation analyzed in this work, is motivated by a simplified mesoscopic physical model for phase separation. It describes pattern formation due to adsorption and desorption mechanisms involved in surface processes, in the presence of a stochastic driving force. This equation is a combination of CahnHilliard and AllenCahn type operators with a multiplicative, white, spacetime noise of unbounded di usion. We apply Malliavin calculus, in order to investigate the existence of a density for the stochastic solution u. In dimension one, according to the regularity result in [5], u admits continuous paths a.s. Using this property, and inspired by a method proposed in [8], we construct a modi ed approximating sequence for u, which properly treats the new second order AllenCahn operator. Under a localization argument, we prove that the Malliavin derivative of u exists locally, and that the law of u is absolutely continuous, establishing thus that a density exists.

A posteriori error estimates for fully discrete schemes for the time dependent Stokes problemThis work is devoted to a posteriori error analysis of fully discrete finite element approximations to the time dependent Stokes system. The space discretization is based on popular stable spaces, including Crouzeix–Raviart and Taylor–Hood finite element methods. Implicit Euler is applied for the time discretization. The finite element spaces are allowed to change with time steps and the projection steps include alternatives that is hoped to cope with possible numerical artifices and the loss of the discrete incompressibility of the schemes. The final estimates are of optimal order in L∞(L2) for the velocity error.