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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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.

New binary selfdual codes of lengths 56, 58, 64, 80 and 92 from a modification of the four circulant construction.In this work, we give a new technique for constructing selfdual codes over commutative Frobenius rings using $\lambda$circulant matrices. The new construction was derived as a modification of the wellknown four circulant construction of selfdual codes. Applying this technique together with the buildingup construction, we construct singlyeven binary selfdual codes of lengths 56, 58, 64, 80 and 92 that were not known in the literature before. Singlyeven selfdual codes of length 80 with $\beta \in \{2,4,5,6,8\}$ in their weight enumerators are constructed for the first time in the literature.

Composite Matrices from Group Rings, Composite GCodes and Constructions of SelfDual CodesIn this work, we define composite matrices which are derived from group rings. We extend the idea of Gcodes to composite Gcodes. We show that these codes are ideals in a group ring, where the ring is a finite commutative Frobenius ring and G is an arbitrary finite group. We prove that the dual of a composite Gcode is also a composite Gcode. We also define quasicomposite Gcodes. Additionally, we study generator matrices, which consist of the identity matrices and the composite matrices. Together with the generator matrices, the well known extension method, the neighbour method and its generalization, we find extremal binary selfdual codes of length 68 with new weight enumerators for the rare parameters $\gamma$ = 7; 8 and 9: In particular, we find 49 new such codes. Moreover, we show that the codes we find are inaccessible from other constructions.

High order algorithms for numerical solution of fractional differential equationsIn this paper, two novel high order numerical algorithms are proposed for solving fractional differential equations where the fractional derivative is considered in the Caputo sense. The total domain is discretized into a set of small subdomains and then the unknown functions are approximated using the piecewise Lagrange interpolation polynomial of degree three and degree four. The detailed error analysis is presented, and it is analytically proven that the proposed algorithms are of orders 4 and 5. The stability of the algorithms is rigorously established and the stability region is also achieved. Numerical examples are provided to check the theoretical results and illustrate the efficiency and applicability of the novel algorithms.

GCodes, selfdual GCodes and reversible GCodes over the Ring Bj,kIn this work, we study a new family of rings, Bj,k, whose base field is the finite field Fpr . We study the structure of this family of rings and show that each member of the family is a commutative Frobenius ring. We define a Gray map for the new family of rings, study Gcodes, selfdual Gcodes, and reversible Gcodes over this family. In particular, we show that the projection of a Gcode over Bj,k to a code over Bl,m is also a Gcode and the image under the Gray map of a selfdual Gcode is also a selfdual Gcode when the characteristic of the base field is 2. Moreover, we show that the image of a reversible Gcode under the Gray map is also a reversible G2j+kcode. The Gray images of these codes are shown to have a rich automorphism group which arises from the algebraic structure of the rings and the groups. Finally, we show that quasiG codes, which are the images of Gcodes under the Gray map, are also Gscodes for some s.

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.

Numerical approximation of the Stochastic CahnHilliard Equation near the Sharp Interface LimitAbstract. We consider the stochastic CahnHilliard equation with additive noise term that scales with the interfacial width parameter ε. We verify strong error estimates for a gradient flow structureinheriting timeimplicit discretization, where ε only enters polynomially; the proof is based on highermoment estimates for iterates, and a (discrete) spectral estimate for its deterministic counterpart. For γ sufficiently large, convergence in probability of iterates towards the deterministic HeleShaw/MullinsSekerka problem in the sharpinterface limit ε → 0 is shown. These convergence results are partly generalized to a fully discrete finite element based discretization. We complement the theoretical results by computational studies to provide practical evidence concerning the effect of noise (depending on its ’strength’ γ) on the geometric evolution in the sharpinterface limit. For this purpose we compare the simulations with those from a fully discrete finite element numerical scheme for the (stochastic) MullinsSekerka problem. The computational results indicate that the limit for γ ≥ 1 is the deterministic problem, and for γ = 0 we obtain agreement with a (new) stochastic version of the MullinsSekerka problem.

Entropydriven cell decisionmaking predicts "fluidtosolid" transition in multicellular systemsCellular decision making allows cells to assume functionally different phenotypes in response to microenvironmental cues, with or without genetic change. It is an open question, how individual cell decisions influence the dynamics at the tissue level. Here, we study spatiotemporal pattern formation in a population of cells exhibiting phenotypic plasticity, which is a paradigm of cell decision making. We focus on the migration/resting and the migration/proliferation plasticity which underly the epithelialmesenchymal transition (EMT) and the go or grow dichotomy. We assume that cells change their phenotype in order to minimize their microenvironmental entropy following the LEUP (Least microEnvironmental Uncertainty Principle) hypothesis. In turn, we study the impact of the LEUPdriven migration/resting and migration/proliferation plasticity on the corresponding multicellular spatiotemporal dynamics with a stochastic cellbased mathematical model for the spatiotemporal dynamics of the cell phenotypes. In the case of the go or rest plasticity, a corresponding meanfield approximation allows to identify a bistable switching mechanism between a diffusive (fluid) and an epithelial (solid) tissue phase which depends on the sensitivity of the phenotypes to the environment. For the go or grow plasticity, we show the possibility of Turing pattern formation for the "solid" tissue phase and its relation with the parameters of the LEUPdriven cell decisions.

Extending an Established Isomorphism between Group Rings and a Subring of the n × n MatricesIn this work, we extend an established isomorphism between group rings and a subring of the n × n matrices. This extension allows us to construct more complex matrices over the ring R. We present many interesting examples of complex matrices constructed directly from our extension. We also show that some of the matrices used in the literature before can be obtained by a direct application of our extended isomorphism.

Two highorder time discretization schemes for subdiffusion problems with nonsmooth dataTwo new highorder time discretization schemes for solving subdiffusion problems with nonsmooth data are developed based on the corrections of the existing time discretization schemes in literature. Without the corrections, the schemes have only a first order of accuracy for both smooth and nonsmooth data. After correcting some starting steps and some weights of the schemes, the optimal convergence orders $O(k^{3 \alpha})$ and $O(k^{4 \alpha})$ with $0< \alpha <1$ can be restored for any fixed time $t$ for both smooth and nonsmooth data, respectively. The error estimates for these two new highorder schemes are proved by using Laplace transform method for both homogeneous and inhomogeneous problem. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

Dynamics of shadow system of a singular GiererMeinhardt system on an evolving domainThe main purpose of the current paper is to contribute towards the comprehension of the dynamics of the shadow system of a singular GiererMeinhardt model on an isotropically evolving domain. In the case where the inhibitor's response to the activator's growth is rather weak, then the shadow system of the GiererMeinhardt model is reduced to a single though nonlocal equation whose dynamics is thoroughly investigated throughout the manuscript. The main focus is on the derivation of blowup results for this nonlocal equation, which can be interpreted as instability patterns of the shadow system. In particular, a diffusiondriven instability (DDI), or Turing instability, in the neighbourhood of a constant stationary solution, which then is destabilised via diffusiondriven blowup, is observed. The latter indicates the formation of some unstable patterns, whilst some stability results of globalintime solutions towards nonconstant steady states guarantee the occurrence of some stable patterns. Most of the theoretical results are verified numerically, whilst the numerical approach is also used to exhibit the dynamics of the shadow system when analytical methods fail.

DOMestic Energy Systems and Technologies InCubator (DOMESTIC) and indoor air quality of the built environmentOral presentation at RMetS Students and Early Career Scientists Conference 2020 on research project DOMESTIC (DOMestic Energy Systems and Technologies InCubator), which aims to build a facility for the demonstration of domestic technologies and design methodologies (i.e. air quality, energy efficiency).

New SelfDual Codes of Length 68 from a 2 × 2 Block Matrix Construction and Group RingsMany generator matrices for constructing extremal binary selfdual codes of different lengths have the form G = (In  A); where In is the n x n identity matrix and A is the n x n matrix fully determined by the first row. In this work, we define a generator matrix in which A is a block matrix, where the blocks come from group rings and also, A is not fully determined by the elements appearing in the first row. By applying our construction over F2 +uF2 and by employing the extension method for codes, we were able to construct new extremal binary selfdual codes of length 68. Additionally, by employing a generalised neighbour method to the codes obtained, we were able to con struct many new binary selfdual [68,34,12]codes with the rare parameters $\gamma = 7$; $8$ and $9$ in $W_{68,2}$: In particular, we find 92 new binary selfdual [68,34,12]codes.

SelfDual Codes using Bisymmetric Matrices and Group RingsIn this work, we describe a construction in which we combine together the idea of a bisymmetric matrix and group rings. Applying this construction over the ring F4 + uF4 together with the well known extension and neighbour methods, we construct new selfdual codes of length 68: In particular, we find 41 new codes of length 68 that were not known in the literature before.

New Extremal binary selfdual codes of length 68 from generalized neighborsIn this work, we use the concept of distance between selfdual codes, which generalizes the concept of a neighbor for selfdual codes. Using the $k$neighbors, we are able to construct extremal binary selfdual codes of length 68 with new weight enumerators. We construct 143 extremal binary selfdual codes of length 68 with new weight enumerators including 42 codes with $\gamma=8$ in their $W_{68,2}$ and 40 with $\gamma=9$ in their $W_{68,2}$. These examples are the first in the literature for these $\gamma$ values. This completes the theoretical list of possible values for $\gamma$ in $W_{68,2}$.

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.