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Identification of the initial function for discretized delay differential equationsIn the present work, we analyze a discrete analogue for the problem of the identification of the initial function for a delay differential equation (DDE) discussed by Baker and Parmuzin in 2004. The basic problem consists of finding an initial function that gives rise to a solution of a discretized DDE, which is a close fit to observed data.

Identification of the initial function for nonlinear delay differential equationsWe consider a 'data assimilation problem' for nonlinear delay differential equations. Our problem is to find an initial function that gives rise to a solution of a given nonlinear delay differential equation, which is a close fit to observed data. A role for adjoint equations and fundamental solutions in the nonlinear case is established. A 'pseudoNewton' method is presented. Our results extend those given by the authors in [(C. T. H. Baker and E. I. Parmuzin, Identification of the initial function for delay differential equation: Part I: The continuous problem & an integral equation analysis. NA Report No. 431, MCCM, Manchester, England, 2004.), (C. T. H. Baker and E. I. Parmuzin, Analysis via integral equations of an identification problem for delay differential equations. J. Int. Equations Appl. (2004) 16, 111–135.)] for the case of linear delay differential equations.

An implicit finite difference approximation for the solution of the diffusion equation with distributed order in timeIn this paper we are concerned with the numerical solution of a diffusion equation in which the time order derivative is distributed over the interval [0,1]. An implicit numerical method is presented and its unconditional stability and convergence are proved. A numerical example is provided to illustrate the obtained theoretical results.

Introducing delay dynamics to Bertalanffy's spherical tumour growth modelWe introduce delay dynamics to an ordinary differential equation model of tumour growth based upon von Bertalanffy's growth model, a model which has received little attention in comparison to other models, such as Gompterz, Greenspan and logistic models. Using existing, previously published data sets we show that our delay model can perform better than delay models based on a Gompertz, Greenspan or logistic formulation. We look for replication of the oscillatory behaviour in the data, as well as a low error value (via a LeastSquares approach) when comparing. We provide the necessary analysis to show that a unique, continuous, solution exists for our model equation and consider the qualitative behaviour of a solution near a point of equilibrium.

Layer Dynamics for the one dimensional $\eps$dependent CahnHilliard / AllenCahn EquationWe study the dynamics of the onedimensional εdependent CahnHilliard / AllenCahn equation within a neighborhood of an equilibrium of N transition layers, that in general does not conserve mass. Two different settings are considered which differ in that, for the second, we impose a massconservation constraint in place of one of the zeromass flux boundary conditions at x = 1. Motivated by the study of Carr and Pego on the layered metastable patterns of AllenCahn in [10], and by this of Bates and Xun in [5] for the CahnHilliard equation, we implement an Ndimensional, and a massconservative N−1dimensional manifold respectively; therein, a metastable state with N transition layers is approximated. We then determine, for both cases, the essential dynamics of the layers (ode systems with the equations of motion), expressed in terms of local coordinates relative to the manifold used. In particular, we estimate the spectrum of the linearized CahnHilliard / AllenCahn operator, and specify wide families of εdependent weights δ(ε), µ(ε), acting at each part of the operator, for which the dynamics are stable and rest exponentially small in ε. Our analysis enlightens the role of mass conservation in the classification of the general mixed problem into two main categories where the solution has a profile close to AllenCahn, or, when the mass is conserved, close to the CahnHilliard solution.

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.

Mathematical modelling and numerical simulations in nerve conductionIn the present work we analyse a functionaldifferential equation, sometimes known as the discrete FitzHughNagumo equation, arising in nerve conduction theory.

Mathematical models of DNA methylation dynamics: Implications for health and ageingDNA methylation status is a key epigenetic process which has been intimately associated with gene regulation. In recent years growing evidence has associated DNA methylation status with a variety of diseases including cancer, Alzheimers disease and cardiovascular disease. Moreover, changes to DNA methylation have also recently been implicated in the ageing process. The factors which underpin DNA methylation are complex, and remain to be fully elucidated. Over the years mathematical modelling has helped to shed light on the dynamics of this important molecular system. Although the existing models have contributed significantly to our overall understanding of DNA methylation, they fallshort of fully capturing the dynamics of this process. In this paper we develop a linear and nonlinear model which captures more fully the dynamics of the key intracellular events which characterise DNA methylation. In particular the outcomes of our linear model result in gene promoter specific methylation levels which are more biologically plausible than those revealed by previous mathematical models. In addition, our nonlinear model predicts DNA methylation promoter bistability which is commonly observed experimentally. The findings from our models have implications for our current understanding of how changes to the dynamics which underpin DNA methylation affect ageing and health.

Mixedtype functional differential equations: A numerical approachThis preprint discusses mixedtype functional equations.

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.

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].

Motion of a droplet for the Stochastic mass conserving AllenCahn equationWe study the stochastic massconserving AllenCahn equation posed on a smoothly bounded domain of R2 with additive, spatially smooth, spacetime noise. This equation describes the stochastic motion of a small almost semicircular droplet attached to domain's boundary and moving towards a point of locally maximum curvature. We apply It^o calculus to derive the stochastic dynamics of the center of the droplet by utilizing the approximately invariant manifold introduced by Alikakos, Chen and Fusco [2] for the deterministic problem. In the stochastic case depending on the scaling, the motion is driven by the change in the curvature of the boundary and the stochastic forcing. Moreover, under the assumption of a su ciently small noise strength, we establish stochastic stability of a neighborhood of the manifold of boundary droplet states in the L2 and H1norms, which means that with overwhelming probability the solution stays close to the manifold for very long timescales.

The multidimensional Stochastic Stefan Financial Model for a portfolio of assetsThe financial model proposed in this work involves the liquidation process of a portfolio of n assets through sell or (and) buy orders placed, in a logarithmic scale, at a (vectorial) price with volatility. We present the rigorous mathematical formulation of this model in a financial setting resulting to an ndimensional outer parabolic Stefan problem with noise. The moving boundary encloses the areas of zero trading, the socalled solid phase. We will focus on a case of financial interest when one or more markets are considered. In particular, our aim is to estimate for a short time period the areas of zero trading, and their diameter which approximates the minimum of the n spreads of the portfolio assets for orders from the n limit order books of each asset respectively. In dimensions n = 3, and for zero volatility, this problem stands as a mean field model for Ostwald ripening, and has been proposed and analyzed by Niethammer in [25], and in [7] in a more general setting. There in, when the initial moving boundary consists of well separated spheres, a first order approximation system of odes had been rigorously derived for the dynamics of the interfaces and the asymptotic pro le of the solution. In our financial case, we propose a spherical moving boundaries approach where the zero trading area consists of a union of spherical domains centered at portfolios various prices, while each sphere may correspond to a different market; the relevant radii represent the half of the minimum spread. We apply It^o calculus and provide second order formal asymptotics for the stochastic version dynamics, written as a system of stochastic differential equations for the radii evolution in time. A second order approximation seems to disconnect the financial model from the large diffusion assumption for the trading density. Moreover, we solve the approximating systems numerically.

Multiorder fractional differential equations and their numerical solutionThis article considers the numerical solution of (possibly nonlinear) fractional differential equations of the form y(α)(t)=f(t,y(t),y(β1)(t),y(β2)(t),…,y(βn)(t)) with α>βn>βn−1>>β1 and α−βn1, βj−βj−11, 0

A multispecies chemotaxis system: Lyapunov functionals, duality, critical massWe introduce a multispecies chemotaxis type system admitting an arbitrarily large number of population species, all of which are attracted vs.\ repelled by a single chemical substance. The production vs.\ destruction rates of the chemotactic substance by the species is described by a probability measure. For such a model we investigate the variational structures, in particular we prove the existence of Lyapunov functionals, we establish duality properties as well as a logarithmic HardyLittlewoodSobolev type inequality for the associated free energy. The latter inequality provides the optimal critical value for the conserved total population mass.

Neutral delay differential equations in the modelling of cell growthIn this contribution, we indicate (and illustrate by example) roles that may be played by neutral delay differential equations in modelling of certain cell growth phenomena that display a time lag in reacting to events. We explore, in this connection, questions involving the sensitivity analysis of models and related mathematical theory; we provide some associated numerical results.