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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DNA codes from skew dihedral group ring<p style='textindent:20px;'>In this work, we present a matrix construction for reversible codes derived from skew dihedral group rings. By employing this matrix construction, the ring <inlineformula><texmath id="M1">\begin{document}$ \mathcal{F}_{j, k} $\end{document}</texmath></inlineformula> and its associated Gray maps, we show how one can construct reversible codes of length <inlineformula><texmath id="M2">\begin{document}$ n2^{j+k} $\end{document}</texmath></inlineformula> over the finite field <inlineformula><texmath id="M3">\begin{document}$ \mathbb{F}_4. $\end{document}</texmath></inlineformula> As an application, we construct a number of DNA codes that satisfy the Hamming distance, the reverse, the reversecomplement, and the GCcontent constraints with better parameters than some good DNA codes in the literature.</p>

Miyamoto groups of code algebrasA code algebra A_C is a nonassociative commutative algebra defined via a binary linear code C. In a previous paper, we classified when code algebras are Z_2graded axial (decomposition) algebras generated by small idempotents. In this paper, for each algebra in our classification, we obtain the Miyamoto group associated to the grading. We also show that the code algebra structure can be recovered from the axial decomposition algebra structure.

Split spin factor algebrasMotivated by Yabe's classification of symmetric $2$generated axial algebras of Monster type \cite{yabe}, we introduce a large class of algebras of Monster type $(\alpha, \frac{1}{2})$, generalising Yabe's $\mathrm{III}(\alpha,\frac{1}{2}, \delta)$ family. Our algebras bear a striking similarity with Jordan spin factor algebras with the difference being that we asymmetrically split the identity as a sum of two idempotents. We investigate the properties of these algebras, including the existence of a Frobenius form and ideals. In the $2$generated case, where our algebra is isomorphic to one of Yabe's examples, we use our new viewpoint to identify the axet, that is, the closure of the two generating axes.

Enumerating 3generated axial algebras of Monster typeAn axial algebra is a commutative nonassociative algebra generated by axes, that is, primitive, semisimple idempotents whose eigenvectors multiply according to a certain fusion law. The Griess algebra, whose automorphism group is the Monster, is an example of an axial algebra. We say an axial algebra is of Monster type if it has the same fusion law as the Griess algebra. The 2generated axial algebras of Monster type, called NortonSakuma algebras, have been fully classified and are one of nine isomorphism types. In this paper, we enumerate a subclass of 3generated axial algebras of Monster type in terms of their groups and shapes. It turns out that the vast majority of the possible shapes for such algebras collapse; that is they do not lead to nontrivial examples. This is in sharp contrast to previous thinking. Accordingly, we develop a method of minimal forbidden configurations, to allow us to efficiently recognise and eliminate collapsing shapes.

A new perspective on the numerical and analytical treatment of a certain singular Volterra integral equationIn this thesis, the focus of our attention is on a certain linear Volterra integral equation with singular kernel. The equation is of great interest due to the fact that, under certain conditions, it possesses an in finite family of solutions, out of which only one has C1continuity. Numerous previous studies have been conducted and a variety of solution methods proposed. However, the emphasis has invariably been on determining just the differentiable solution. Thus, a significant gap in the research relating to this equation was identified and, therefore, our main objective here was to develop an effective solution method that allows us to approximate any chosen solution out of the infinite solution set. To this end, we converted the original integral equation into a singular differential form. Then, by applying a combination of analytical results from functional and real analysis, measure theory and the theory of Lebesgue integration, we reduced the problem to that of solving a regular initial value problem. Numerical methods were then applied and our experimental results proved that our method was highly effective, producing very accurate approximations to the true solution in a comparative study. Therefore, we feel our work here makes a significant contribution in this field of study, both from a theoretical viewpoint, as during the course of our research we established a direct relationship between the nonsmooth solutions of the integral equation and the weak solutions of our differential scheme, and in practice. Integral equations of this form arise in the study of heat conduction, diffusion and in thermodynamics. Therefore, another of our aims was to construct a method that could readily be applied in 'real world' modelling. Thus, as traditional models most often present as differential equations and, furthermore, as our method significantly simplifies the process of computing the solutions, we believe we have achieved this objective. Hence, in the final chapter, we highlight some of the ways in which our method could be adopted in order to help solve some of today's most challenging problems.

Binary selfdual and LCD codes from generator matrices constructed from two group ring elements by a heuristic search scheme<p style='textindent:20px;'>We present a generator matrix of the form <inlineformula><texmath id="M1">\begin{document}$ [ \sigma(v_1) \  \ \sigma(v_2)] $\end{document}</texmath></inlineformula>, where <inlineformula><texmath id="M2">\begin{document}$ v_1 \in RG $\end{document}</texmath></inlineformula> and <inlineformula><texmath id="M3">\begin{document}$ v_2\in RH $\end{document}</texmath></inlineformula>, for finite groups <inlineformula><texmath id="M4">\begin{document}$ G $\end{document}</texmath></inlineformula> and <inlineformula><texmath id="M5">\begin{document}$ H $\end{document}</texmath></inlineformula> of order <inlineformula><texmath id="M6">\begin{document}$ n $\end{document}</texmath></inlineformula> for constructing selfdual codes and linear complementary dual codes over the finite Frobenius ring <inlineformula><texmath id="M7">\begin{document}$ R $\end{document}</texmath></inlineformula>. In general, many of the constructions to produce selfdual codes forces the code to be an ideal in a group ring which implies that the code has a rich automorphism group. Unlike the traditional cases, codes constructed from the generator matrix presented here are not ideals in a group ring, which enables us to find selfdual and linear complementary dual codes that are not found using more traditional techniques. In addition to that, by using this construction, we improve <inlineformula><texmath id="M8">\begin{document}$ 10 $\end{document}</texmath></inlineformula> of the previously known lower bounds on the largest minimum weights of binary linear complementary dual codes for some lengths and dimensions. We also obtain <inlineformula><texmath id="M9">\begin{document}$ 82 $\end{document}</texmath></inlineformula> new binary linear complementary dual codes, <inlineformula><texmath id="M10">\begin{document}$ 50 $\end{document}</texmath></inlineformula> of which are either optimal or near optimal of lengths <inlineformula><texmath id="M11">\begin{document}$ 41 \leq n \leq 61 $\end{document}</texmath></inlineformula> which are new to the literature.</p>

Weak convergence of the L1 scheme for a stochastic subdiffusion problem driven by fractionally integrated additive noiseThe weak convergence of a fully discrete scheme for approximating a stochastic subdiffusion problem driven by fractionally integrated additive noise is studied. The Caputo fractional derivative is approximated by the L1 scheme and the RiemannLiouville fractional integral is approximated with the first order convolution quadrature formula. The noise is discretized by using the Euler method and the spatial derivative is approximated with the linear finite element method. Based on the nonsmooth data error estimates of the corresponding deterministic problem, the weak convergence orders of the fully discrete schemes for approximating the stochastic subdiffusion problem driven by fractionally integrated additive noise are proved by using the Kolmogorov equation approach. Numerical experiments are given to show that the numerical results are consistent with the theoretical results.

Numerical methods for CaputoHadamard fractional differential equations with graded and nonuniform meshesWe consider the predictorcorrector numerical methods for solving CaputoHadamard fractional differential equation with the graded meshes $\log t_{j} = \log a + \big ( \log \frac{t_{N}}{a} \big ) \big ( \frac{j}{N} \big )^{r}, \, j=0, 1, 2, \dots, N$ with $a \geq 1$ and $ r \geq 1$, where $\log a = \log t_{0} < \log t_{1} < \dots < \log t_{N}= \log T$ is a partition of $[\log t_{0}, \log T]$. We also consider the rectangular and trapezoidal methods for solving CaputoHadamard fractional differential equation with the nonuniform meshes $\log t_{j} = \log a + \big ( \log \frac{t_{N}}{a} \big ) \frac{j (j+1)}{N(N+1)}, \, j=0, 1, 2, \dots, N$. Under the weak smoothness assumptions of the CaputoHadamard fractional derivative, e.g., $\prescript{}{CH}D^\alpha_{a,t}y(t) \notin C^{1}[a, T]$ with $ \alpha \in (0, 2)$, the optimal convergence orders of the proposed numerical methods are obtained by choosing the suitable graded mesh ratio $r \geq 1$. The numerical examples are given to show that the numerical results are consistent with the theoretical findings.

Insights into the Analysis of Fractional Delay Differential EquationsThis thesis is concerned with determining the analytic solution, using the method of steps, of the following fractional delay differential equation initial interval problem (FDDE IIP), c Dαy(s) = −y(t − τ ) for t > 0, τ > 0, 0 < α < 1, and y ∈ A1(0, T ] 0 t y(t) = ϕ(t) for t ∈ (−τ, 0] The properties of the analytic solution obtained are a surprise but they do sit comfortably when compared with those of the analytic solutions of an ordinary differential equation initial value problem (ODE IVP), a delay differential equation initial interval problem (DDE IIP) and an fractional ordinary differential equation initial value problem (FODE IVP). Further the analytic solution formula obtained is closely related to that of the analytic solution formula of the DDE IIP. However, these insights into the analytic solution of the FDDE IIP we have not seen before, and differ from those published elsewhere.

Isotopic signatures of methane emissions from tropical fires, agriculture and wetlands: the MOYA and ZWAMPS flightsWe report methane isotopologue data from aircraft and ground measurements in Africa and South America. Aircraft campaigns sampled strong methane fluxes over tropical papyrus wetlands in the Nile, Congo and Zambezi basins, herbaceous wetlands in Bolivian southern Amazonia, and over fires in African woodland, cropland and savannah grassland. Measured methane δ13CCH4 isotopic signatures were in the range −55 to −49‰ for emissions from equatorial Nile wetlands and agricultural areas, but widely −60 ± 1‰ from Upper Congo and Zambezi wetlands. Very similar δ13CCH4 signatures were measured over the Amazonian wetlands of NE Bolivia (around −59‰) and the overall δ13CCH4 signature from outer tropical wetlands in the southern Upper Congo and Upper Amazon drainage plotted together was −59 ± 2‰. These results were more negative than expected. For African cattle, δ13CCH4 values were around −60 to −50‰. Isotopic ratios in methane emitted by tropical fires depended on the C3 : C4 ratio of the biomass fuel. In smoke from tropical C3 dry forest fires in Senegal, δ13CCH4 values were around −28‰. By contrast, African C4 tropical grass fire δ13CCH4 values were −16 to −12‰. Methane from urban landfills in Zambia and Zimbabwe, which have frequent waste fires, had δ13CCH4 around −37 to −36‰. These new isotopic values help improve isotopic constraints on global methane budget models because atmospheric δ13CCH4 values predicted by global atmospheric models are highly sensitive to the δ13CCH4 isotopic signatures applied to tropical wetland emissions. Field and aircraft campaigns also observed widespread regional smoke pollution over Africa, in both the wet and dry seasons, and large urban pollution plumes. The work highlights the need to understand tropical greenhouse gas emissions in order to meet the goals of the UNFCCC Paris Agreement, and to help reduce air pollution over wide regions of Africa. This article is part of a discussion meeting issue 'Rising methane: is warming feeding warming? (part 2)'.

A Novel Averaging Principle Provides Insights in the Impact of Intratumoral Heterogeneity on Tumor ProgressionTypically stochastic differential equations (SDEs) involve an additive or multiplicative noise term. Here, we are interested in stochastic differential equations for which the white noise is nonlinearly integrated into the corresponding evolution term, typically termed as random ordinary differential equations (RODEs). The classical averaging methods fail to treat such RODEs. Therefore, we introduce a novel averaging method appropriate to be applied to a specific class of RODEs. To exemplify the importance of our method, we apply it to an important biomedical problem, in particular, we implement the method to the assessment of intratumoral heterogeneity impact on tumor dynamics. Precisely, we model gliomas according to a wellknown Go or Grow (GoG) model, and tumor heterogeneity is modeled as a stochastic process. It has been shown that the corresponding deterministic GoG model exhibits an emerging Allee effect (bistability). In contrast, we analytically and computationally show that the introduction of white noise, as a model of intratumoral heterogeneity, leads to monostable tumor growth. This monostability behavior is also derived even when spatial cell diffusion is taken into account.

Oscillatory and stability of a mixed type difference equation with variable coefficientsThe goal of this paper is to study the oscillatory and stability of the mixed type difference equation with variable coefficients \[ \Delta x(n)=\sum_{i=1}^{\ell}p_{i}(n)x(\tau_{i}(n))+\sum_{j=1}^{m}q_{j}(n)x(\sigma_{i}(n)),\quad n\ge n_{0}, \] where $\tau_{i}(n)$ is the delay term and $\sigma_{j}(n)$ is the advance term and they are positive real sequences for $i=1,\cdots,l$ and $j=1,\cdots,m$, respectively, and $p_{i}(n)$ and $q_{j}(n)$ are real functions. This paper generalise some known results and the examples illustrate the results.

Spatial Discretization for Stochastic SemiLinear Subdiffusion Equations Driven by Fractionally Integrated Multiplicative SpaceTime White NoiseSpatial discretization of the stochastic semilinear subdiffusion driven by integrated multiplicative spacetime white noise is considered. The spatial discretization scheme discussed in Gy\"ongy \cite{gyo_space} and Anton et al. \cite{antcohque} for stochastic quasilinear parabolic partial differential equations driven by multiplicative spacetime noise is extended to the stochastic subdiffusion. The nonlinear terms $f$ and $\sigma$ satisfy the global Lipschitz conditions and the linear growth conditions. The space derivative and the integrated multiplicative spacetime white noise are discretized by using finite difference methods. Based on the approximations of the Green functions which are expressed with the MittagLeffler functions, the optimal spatial convergence rates of the proposed numerical method are proved uniformly in space under the suitable smoothness assumptions of the initial values.

Error estimates of a continuous Galerkin time stepping method for subdiffusion problemA continuous Galerkin time stepping method is introduced and analyzed for subdiffusion problem in an abstract setting. The approximate solution will be sought as a continuous piecewise linear function in time $t$ and the test space is based on the discontinuous piecewise constant functions. We prove that the proposed time stepping method has the convergence order $O(\tau^{1+ \alpha}), \, \alpha \in (0, 1)$ for general sectorial elliptic operators for nonsmooth data by using the Laplace transform method, where $\tau$ is the time step size. This convergence order is higher than the convergence orders of the popular convolution quadrature methods (e.g., Lubich's convolution methods) and Ltype methods (e.g., L1 method), which have only $O(\tau)$ convergence for the nonsmooth data. Numerical examples are given to verify the robustness of the time discretization schemes with respect to data regularity.

A Comprehensive Review of the Composition, Nutritional Value, and Functional Properties of Camel Milk FatRecently, camel milk (CM) has been considered as a healthpromoting icon due to its medicinal and nutritional benefits. CM fat globule membrane has numerous healthpromoting properties, such as antiadhesion and antibacterial properties, which are suitable for people who are allergic to cow’s milk. CM contains milk fat globules with a small size, which accounts for their rapid digestion. Moreover, it also comprises lower amounts of cholesterol and saturated fatty acids concurrent with higher levels of essential fatty acids than cow milk, with an improved lipid profile manifested by reducing cholesterol levels in the blood. In addition, it is rich in phospholipids, especially plasmalogens and sphingomyelin, suggesting that CM fat may meet the daily nutritional requirements of adults and infants. Thus, CM and its dairy products have become more attractive for consumers. In view of this, we performed a comprehensive review of CM fat’s composition and nutritional properties. The overall goal is to increase knowledge related to CM fat characteristics and modify its unfavorable perception. Future studies are expected to be directed toward a better understanding of CM fat, which appears to be promising in the design and formulation of new products with significant healthpromoting benefits.

Group Codes, Composite Group Codes and Constructions of SelfDual CodesThe main research presented in this thesis is around constructing binary selfdual codes using group rings together with some wellknown code construction methods and the study of group codes and composite group codes over different alphabets. Both these families of codes are generated by the elements that come from group rings. A search for binary selfdual codes with new weight enumerators is an ongoing research area in algebraic coding theory. For this reason, we present a generator matrix in which we employ the idea of a bisymmetric matrix with its entries being the block matrices that come from group rings and give the necessary conditions for this generator matrix to produce a selfdual code over a fi nite commutative Frobenius ring. Together with our generator matrix and some wellknown code construction methods, we find many binary selfdual codes with parameters [68, 34, 12] that have weight enumerators that were not known in the literature before. There is an extensive literature on the study of different families of codes over different alphabets and speci fically finite fi elds and finite commutative rings. The study of codes over rings opens up a new direction for constructing new binary selfdual codes with a rich automorphism group via the algebraic structure of the rings through the Gray maps associated with them. In this thesis, we introduce a new family of rings, study its algebraic structure and show that each member of this family is a commutative Frobenius ring. Moreover, we study group codes over this new family of rings and show that one can obtain codes with a rich automorphism group via the associated Gray map. We extend a well established isomorphism between group rings and the subring of the n x n matrices and show its applications to algebraic coding theory. Our extension enables one to construct many complex n x n matrices over the ring R that are fully de ned by the elements appearing in the first row. This property allows one to build generator matrices with these complex matrices so that the search field is practical in terms of the computational times. We show how these complex matrices are constructed using group rings, study their properties and present many interesting examples of complex matrices over the ring R. Using our extended isomorphism, we de ne a new family of codes which we call the composite group codes or for simplicity, composite Gcodes. We show that these new codes are ideals in the group ring RG and prove that the dual of a composite Gcode is also a composite Gcode. Moreover, we study generator matrices of the form [In  Ω(v)]; where In is the n x n identity matrix and Ω(v) is the composite matrix that comes from the extended isomorphism mentioned earlier. In particular, we show when such generator matrices produce selfdual codes over finite commutative Frobenius rings. Additionally, together with some generator matrices of the type [In  Ω(v)] and the wellknown extension and neighbour methods, we fi nd many new binary selfdual codes with parameters [68, 34, 12]. Lastly in this work, we study composite Gcodes over formal power series rings and finite chain rings. We extend many known results on projections and lifts of codes over these alphabets. We also extend some known results on γadic codes over the infi nite ring R∞

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.

New Extremal Binary Selfdual Codes from block circulant matrices and block quadratic residue circulant matricesIn this paper, we construct selfdual codes from a construction that involves both block circulant matrices and block quadratic residue circulant matrices. We provide conditions when this construction can yield selfdual codes. We construct selfdual codes of various lengths over F2 and F2 + uF2. Using extensions, neighbours and sequences of neighbours, we construct many new selfdual codes. In particular, we construct one new selfdual code of length 66 and 51 new selfdual codes of length 68.

New Selfdual Codes from 2 x 2 block circulant matrices, Group Rings and Neighbours of NeighboursIn this paper, we construct new selfdual codes from a construction that involves a unique combination; $2 \times 2$ block circulant matrices, group rings and a reverse circulant matrix. There are certain conditions, specified in this paper, where this new construction yields selfdual codes. The theory is supported by the construction of selfdual codes over the rings $\FF_2$, $\FF_2+u\FF_2$ and $\FF_4+u\FF_4$. Using extensions and neighbours of codes, we construct $32$ new selfdual codes of length $68$. We construct 48 new best known singlyeven selfdual codes of length 96.

Galerkin finite element approximation of a stochastic semilinear fractional subdiffusion with fractionally integrated additive noiseA Galerkin finite element method is applied to approximate the solution of a semilinear stochastic space and time fractional subdiffusion problem with the Caputo fractional derivative of the order $ \alpha \in (0, 1)$, driven by fractionally integrated additive noise. After discussing the existence, uniqueness and regularity results, we approximate the noise with the piecewise constant function in time in order to obtain a regularized stochastic fractional subdiffusion problem. The regularized problem is then approximated by using the finite element method in spatial direction. The mean squared errors are proved based on the sharp estimates of the various MittagLeffler functions involved in the integrals. Numerical experiments are conducted to show that the numerical results are consistent with the theoretical findings.