Mathematics
We are an active university Mathematics Department with a strong teaching and research reputation. We offer students the chance to study at undergraduate or postgraduate level on degree programmes leading to: BSc in Mathematics, BSc/BA joint courses in Mathematics or Applied Statistics and a wide range of other subjects. We have an active research group focusing on Computational Applied Mathematics, with research students studying for the degrees of MPhil and PhD, postdoctoral workers and associated collaborators from across the world.
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Mathematical Modelling of Problems with Delay and AfterEffectThis paper provides a tutorial review of the use of delay differential equations in mathematical models of real problems. We use the COVID19 pandemic as an example to help explain our conclusions. We present the fundamental delay differential equation as a prototype for modelling problems where there is a delay or aftereffect, and we reveal (via the characteristic values) the infinite dimensional nature of the equation and the presence of oscillatory solutions not seen in corresponding equations without delay. We discuss how models were constructed for the COVID19 pandemic, particularly in view of the relative lack of understanding of the disease and the paucity of available data in the early stages, and we identify both strengths and weaknesses in the modelling predictions and how they were communicated and applied. We consider the question of whether equations with delay could have been or should have been utilised at various stages in order to make more accurate or more useful predictions.

Analysis of a ReactionDiffusion Model Towards Description and Prediction of Breast Cancer ProgressionThis thesis conducts a thorough investigation by combining mathematical analysis, empirical clinical data, and meticulous clinical observations to investigate the tumour dynamics which are controlled by two crucial biomarkers: cancerous cells (CK7) and active Tcells (CD4). Introducing a reactiondiffusion model facilitates the spatial and temporal distribution of the biomarkers CK7 and CD4. Subsequently, explore the stability, bifurcation and sensitivity analyses to show the effects of critical biological parameters on tumour dynamics. An exploration of the travelling wave solutions of the model, through numerical methods, aids in the visualisation of the dynamic behaviour of the system, particularly the invasion of the CK7 population towards the CD4 population. Employing a numerical approach, specifically utilising the finite volume method, provides a clear representation of the dynamic interplay within the tumour environment. Our analysis contributes not only to a heightened theoretical understanding but also holds potential implications for therapeutic decisions in tumour dynamics. Hence, our investigation shows efficacy not only in predicting tumour dynamics but also in providing a robust framework to present specific clinical observations.

Extremal binary selfdual codes from a bordered four circulant constructionIn this paper, we present a new bordered construction for selfdual codes which employs λcirculant matrices. We give the necessary conditions for our construction to produce selfdual codes over a finite commutative Frobenius ring of characteristic 2. Moreover, using our bordered construction together with the wellknown buildingup and neighbour methods, we construct many binary selfdual codes of lengths 56, 62, 78, 92 and 94 with parameters in their weight enumerators that were not known in the literature before.

New binary selfdual codes of lengths 80, 84 and 96 from composite matricesIn this work, we apply the idea of composite matrices arising from group rings to derive a number of different techniques for constructing selfdual codes over finite commutative Frobenius rings. By applying these techniques over different alphabets, we construct best known singlyeven binary selfdual codes of lengths 80, 84 and 96 as well as doublyeven binary selfdual codes of length 96 that were not known in the literature before.

Selfdual codes from a block matrix construction characterised by group ringsWe give a new technique for constructing selfdual codes based on a block matrix whose blocks arise from group rings and orthogonal matrices. The technique can be used to construct selfdual codes over finite commutative Frobenius rings of characteristic 2. We give and prove the necessary conditions needed for the technique to produce selfdual codes. We also establish the connection between selfdual codes generated by the new technique and units in group rings. Using the construction together with the buildingup construction, we obtain new extremal binary selfdual codes of lengths 64, 66 and 68 and new best known binary selfdual codes of length 80.

Quaternary Hermitian selfdual codes of lengths 26, 32, 36, 38 and 40 from modifications of wellknown circulant constructionsIn this work, we give three new techniques for constructing Hermitian selfdual codes over commutative Frobenius rings with a nontrivial involutory automorphism using λcirculant matrices. The new constructions are derived as modifications of various wellknown circulant constructions of selfdual codes. Applying these constructions together with the buildingup construction, we construct many new best known quaternary Hermitian selfdual codes of lengths 26, 32, 36, 38 and 40.

Tempered nonlocal integrodifferential equations with nonsmooth solution dataThis work on the time discretization for the solution of the tempered nonlocal integrodifferential equation with smooth and nonsmooth initial data are extended. We find the difficulty arising from theoretical analysis can be overcome by Laplace transform technique. A mount of proof skills have been provided based on Laplace transform technique for this kind of equations. The Laplace transform method is employed effectively to show that the proposed interpolating quadrature scheme is convergence for smooth and nonsmooth initial data in both homogeneous and inhomogeneous cases.

Automorphism groups of axial algebrasAxial algebras are a class of commutative nonassociative algebras which have a natural group of automorphisms, called the Miyamoto group. The motivating example is the Griess algebra which has the Monster sporadic simple group as its Miyamoto group. Previously, using an expansion algorithm, about 200 examples of axial algebras in the same class as the Griess algebra have been constructed in dimensions up to about 300. In this list, we see many reoccurring dimensions which suggests that there may be some unexpected isomorphisms. Such isomorphisms can be found when the full automorphism groups of the algebras are known. Hence, in this paper, we develop methods for computing the full automorphism groups of axial algebras and apply them to a number of examples of dimensions up to 151.

Axial algebras of Jordan and Monster typeAxial algebras are a class of nonassociative commutative algebras whose properties are defined in terms of a fusion law. When this fusion law is graded, the algebra has a naturally associated group of automorphisms and thus axial algebras are inherently related to group theory. Examples include most Jordan algebras and the Griess algebra for the Monster sporadic simple group. In this survey, we introduce axial algebras, discuss their structural properties and then concentrate on two specific classes: algebras of Jordan and Monster type, which are rich in examples related to simple groups.

Existence, uniqueness and regularity for a semilinear stochastic subdiffusion with integrated multiplicative noiseWe investigate a semilinear stochastic timespace fractional subdiffusion equation driven by fractionally integrated multiplicative noise. The equation involves the ψCaputo derivative of order α∈(0, 1) and the spectral fractional Laplacian of order β∈(12, 1]. The existence and uniqueness of the mild solution are proved in a suitable Banach space by using the Banach contraction mapping principle. The spatial and temporal regularities of the mild solution are established in terms of the smoothing properties of the solution operators.

Codes over a ring of order 32 with two Gray mapsWe describe a ring of order 32 and prove that it is a local Frobenius ring. We study codes over this ring and we give two distinct nonequivalent linear orthogonalitypreserving Gray maps to the binary space. Selfdual codes are studied over this ring as well as the binary selfdual codes that are the Gray images of those codes. Specifically, we show that the image of a selfdual code over this ring is a binary selfdual code with an automorphism consisting of 2n transpositions for the first map and n transpositions for the second map. We relate the shadows of binary codes to additive codes over the ring. As Gray images of codes over the ring, binary selfdual [ 70 , 35 , 12 ] codes with 91 distinct weight enumerators are constructed for the first time in the literature.

Numerical Approximation for a Stochastic Fractional Differential Equation Driven by Integrated Multiplicative NoiseWe consider a numerical approximation for stochastic fractional differential equations driven by integrated multiplicative noise. The fractional derivative is in the Caputo sense with the fractional order α∈(0,1), and the nonlinear terms satisfy the global Lipschitz conditions. We first approximate the noise with the piecewise constant function to obtain the regularized stochastic fractional differential equation. By applying Minkowski’s inequality for double integrals, we establish that the error between the exact solution and the solution of the regularized problem has an order of O(Δtα) in the mean square norm, where Δt denotes the step size. To validate our theoretical conclusions, numerical examples are presented, demonstrating the consistency of the numerical results with the established theory.

BDF2 ADI orthogonal spline collocation method for the fractional integrodifferential equations of parabolic type in three dimensionsIn this paper, we are concerned with constructing a fast and an efficient alternating direction implicit (ADI) scheme for the fractional parabolic integrodifferential equations (FPIDE) with a weakly singular kernel in three dimensions (3D). Our constructed scheme is based on a secondorder backward differentiation formula (BDF2) for temporal discretization, orthogonal spline collocation (OSC) method for spatial discretization and a secondorder fractional quadrature rule proposed by Lubich for the RiemannLiouville fractional integral. The stability and convergence of the constructed numerical scheme are derived. Finally, some numerical examples are given to illustrate the accuracy and validity of the BDF2 ADI OSC method. Based on the obtained results, the numerical results are in line with the theoretical ones.

Spatial discretization for stochastic semilinear superdiffusion driven by fractionally integrated multiplicative spacetime white noiseWe investigate the spatial discretization of a stochastic semilinear superdiffusion problem driven by fractionally integrated multiplicative spacetime white noise. The white noise is characterized by its properties of being white in both space and time and the time fractional derivative is considered in the Caputo sense with an order $\alpha \in (1, 2)$. A spatial discretization scheme is introduced by approximating the spacetime white noise with the Euler method in the spatial direction and approximating the secondorder space derivative with the central difference scheme. By using the Green functions, we obtain both exact and approximate solutions for the proposed problem. The regularities of both the exact and approximate solutions are studied and the optimal error estimates that depend on the smoothness of the initial values are established. This paper builds upon the research presented in Mathematics. 2021. 9, 1917, where we originally focused on error estimates in the context of subdiffusion with $\alpha \in (0, 1)$. We extend our investigation to the spatial approximation of stochastic superdiffusion with $\alpha \in (1, 2)$ and place particular emphasis on refining our understanding of the superdiffusion phenomenon by analyzing the error estimates associated with the time derivative at the initial point.

Highorder schemes based on extrapolation for semilinear fractional differential equationBy rewriting the Riemann–Liouville fractional derivative as Hadamard finitepart integral and with the help of piecewise quadratic interpolation polynomial approximations, a numerical scheme is developed for approximating the Riemann–Liouville fractional derivative of order α∈(1, 2). The error has the asymptotic expansion (d3τ3α+d4τ4α+d5τ5α+⋯)+(d2∗τ4+d3∗τ6+d4∗τ8+⋯) at any fixed time tN=T, N∈Z+, where di, i=3, 4, … and di∗, i=2, 3, … denote some suitable constants and τ=T/N denotes the step size. Based on this discretization, a new scheme for approximating the linear fractional differential equation of order α∈(1, 2) is derived and its error is shown to have a similar asymptotic expansion. As a consequence, a highorder scheme for approximating the linear fractional differential equation is obtained by extrapolation. Further, a highorder scheme for approximating a semilinear fractional differential equation is introduced and analyzed. Several numerical experiments are conducted to show that the numerical results are consistent with our theoretical findings.

Unconditionally stable and convergent difference scheme for superdiffusion with extrapolationApproximating the Hadamard finitepart integral by the quadratic interpolation polynomials, we obtain a scheme for approximating the RiemannLiouville fractional derivative of order α∈(1, 2) and the error is shown to have the asymptotic expansion (d3τ3α+d4τ4α+d5τ5α+⋯)+(d2∗τ4+d3∗τ6+d4∗τ8+⋯) at any fixed time, where τ denotes the step size and dl, l=3, 4, ⋯ and dl∗, l=2, 3, ⋯ are some suitable constants. Applying the proposed scheme in temporal direction and the central difference scheme in spatial direction, a new finite difference method is developed for approximating the time fractional wave equation. The proposed method is unconditionally stable, convergent with order O(τ3α), α∈(1, 2) and the error has the asymptotic expansion. Richardson extrapolation is applied to improve the accuracy of the numerical method. The convergence orders are O(τ4α) and O(τ2(3α)), α∈(1, 2), respectively, after first two extrapolations. Numerical examples are presented to show that the numerical results are consistent with the theoretical findings.

Quotients of the Highwater algebra and its coverPrimitive axial algebras of Monster type are a class of nonassociative algebras with a strong link to finite (especially simple) groups. The motivating example is the Griess algebra, with the Monster as its automorphism group. A crucial step towards the understanding of such algebras is the explicit description of the 2generated symmetric objects. Recent work of Yabe, and Franchi and Mainardis shows that any such algebra is either explicitly known, or is a quotient of the infinitedimensional Highwater algebra H, or its characteristic 5 cover Ĥ. In this paper, we complete the classification of symmetric axial algebras of Monster type by determining the quotients of H and Ĥ. We proceed in a unified way, by defining a cover of H in all characteristics. This cover has a previously unseen fusion law and provides an insight into why the Highwater algebra has a cover which is of Monster type only in characteristic 5.

The weight enumerators of singlyeven selfdual [88,44,14] codes and new binary selfdual [68,34,12] and [88,44,14] codesIn this work, we focus on constructing binary selfdual [68, 34, 12] and [88, 44, 14] codes with new parameters in their weight enumerators. For this purpose, we present a new bordered matrix construction for selfdual codes which is derived as a modification of two known bordered matrix constructions. We provide the necessary conditions for the new construction to produce selfdual codes over finite commutative Frobenius rings of characteristic 2. We also construct the possible weight enumerators for singlyeven selfdual [88, 44, 14] codes and their shadows as this has not been done in the literature yet. We employ the modified bordered matrix together with the wellknown neighbour method to construct binary selfdual codes that could not be obtained from the other, known bordered matrix constructions. Many of the codes turn out to have parameters in their weight enumerators that were not known in the literature before.

Numerical Methods for Stochastic AllenCahn Equation and Stochastic Subdiffusion and SuperdiffusionIn this Thesis, we consider the numerical solution of stochastic partial differential equations with particular interest on the Ɛdependent AllenCahn equation, and the stochastic time fractional partial differential equations in both subdiffusion and superdiffusion cases.

Strong approximation of stochastic semilinear subdiffusion and superdiffusion driven by fractionally integrated additive noiseRecently, Kovács et al. considered a Mittag‐Leffler Euler integrator for a stochastic semilinear Volterra integral‐differential equation with additive noise and proved the strong convergence error estimates [see SIAM J. Numer. Anal. 58(1) 2020, pp. 66‐85]. In this article, we shall consider the Mittag‐Leffler integrators for more general models: stochastic semilinear subdiffusion and superdiffusion driven by fractionally integrated additive noise. The mild solutions of our models involve four different Mittag‐Leffler functions. We first consider the existence, uniqueness and the regularities of the solutions. We then introduce the full discretization schemes for solving the problems. The temporal discretization is based on the Mittag‐Leffler integrators and the spatial discretization is based on the spectral method. The optimal strong convergence error estimates are proved under the reasonable assumptions for the semilinear term and for the regularity of the noise. Numerical examples are given to show that the numerical results are consistent with the theoretical results.