Mathematics
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L1 scheme for solving an inverse problem subject to a fractional diffusion equationThis paper considers the temporal discretization of an inverse problem subject to a time fractional diffusion equation. Firstly, the convergence of the L1 scheme is established with an arbitrary sectorial operator of spectral angle < π / 2 , that is the resolvent set of this operator contains { z ∈ C ∖ { 0 } :  Arg z  < θ } for some π / 2 < θ < π . The relationship between the time fractional order α ∈ ( 0 , 1 ) and the constants in the error estimates is precisely characterized, revealing that the L1 scheme is robust as α approaches 1. Then an inverse problem of a fractional diffusion equation is analyzed, and the convergence analysis of a temporal discretization of this inverse problem is given. Finally, numerical results are provided to confirm the theoretical results.

Higher moments for the Stochastic Cahn  Hilliard Equation with multiplicative Fourier noiseWe consider in dimensions $d=1,2,3$ the $\eps$dependent stochastic CahnHilliard equation with a multiplicative and sufficiently regular in space infinite dimensional Fourier noise with strength of order $\mathcal{O}(\eps^\gamma)$, $\gamma>0$. The initial condition is nonlayered and independent from $\eps$. Under general assumptions on the noise diffusion $\sigma$, we prove moment estimates in $H^1$ (and in $L^\infty$ when $d=1$). Higher $H^2$ regularity $p$moment estimates are derived when $\sigma$ is bounded, yielding as well space H\"older and $L^\infty$ bounds for $d=2,3$, and path a.s. continuity in space. All appearing constants are expressed in terms of the small positive parameter $\eps$. As in the deterministic case, in $H^1$, $H^2$, the bounds admit a negative polynomial order in $\eps$. Finally, assuming layered initial data of initial energy uniformly bounded in $\eps$, as proposed by X.F. Chen in \cite{chenjdg}, we use our $H^1$ $2$dmoment estimate and prove the stochastic solution's convergence to $\pm 1$ as $\eps\rightarrow 0$ a.s., when the noise diffusion has a linear growth.

Finite difference method for timefractional KleinGordon equation on an unbounded domain using artificial boundary conditionsA finite difference method for timefractional KleinGordon equation with the fractional order $\alpha \in (1, 2]$ on an unbounded domain is studied. The artificial boundary conditions involving the generalized Caputo derivative are derived using the Laplace transform technique. Stability and error estimates of the proposed finite difference scheme are proved in detail by using the discrete energy method. Numerical examples show that the artificial boundary method is a robust and efficient method for solving the timefractional KleinGordon equation on an unbounded domain.

Correction of HighOrder Lk Approximation for SubdiffusionThe subdiffusion equations with a Caputo fractional derivative of order $\alpha \in (0, 1)$ arise in a wide variety of practical problems, which describe the transport processes, in the forcefree limit, slower than Brownian diffusion. In this work, we derive the correction schemes of the Lagrange interpolation with degree $k ( \leq 6)$ convolution quadrature, called $L_{k}$ approximation, for the subdiffusion. The key step of designing correction algorithm is to calculate the explicit form of the coefficients of $L_{k}$ approximation by the polylogarithm function or BoseEinstein integral. To construct a $\tau_{8}$ approximation of BoseEinstein integral, the desired $(k+1\alpha)$thorder convergence rate can be proved for the correction $L_{k}$ scheme with nonsmooth data, which is higher than kthorder BDFk method in [Jin, Li, and Zhou, SIAM J. Sci. Comput., 39 (2017), A3129–A3152; Shi and Chen, J. Sci. Comput., (2020) 85:28]. The numerical experiments with spectral method are given to illustrate theoretical results.

Detailed Error Analysis for a Fractional Adams Method on CaputoHadamard Fractional Differential EquationsWe consider a predictorcorrector numerical method for solving CaputoHadamard fractional differential equation over the uniform mesh $\log t_{j} = \log a + \big ( \log \frac{t_{N}}{a} \big ) \big ( \frac{j}{N} \big ), \, j=0, 1, 2, \dots, N$~with $a \geq 1$, where $\log a = \log t_{0} < \log t_{1} < \dots < \log t_{N}= \log T$ is a partition of $[\log a, \log T]$. The error estimates under the different smoothness properties of the solution $y$ and the nonlinear function $f$ are studied. Numerical examples are given to verify that the numerical results are consistent with the theoretical results.

DNA codes from skew dihedral group ringIn this work, we present a matrix construction for reversible codes derived from skew dihedral group rings. By employing this matrix construction, the ring {F}_{j, k} and its associated Gray maps, we show how one can construct reversible codes of length n2^{j+k} over the finite field {F}_4. 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.

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.

Group LCD and group reversible LCD codesIn this paper, we give a new method for constructing LCD codes. We employ group rings and a well known map that sends group ring elements to a subring of the n × n matrices to obtain LCD codes. Our construction method guarantees that our LCD codes are also group codes, namely, the codes are ideals in a group ring. We show that with a certain condition on the group ring element v, one can construct nontrivial group LCD codes. Moreover, we also show that by adding more constraints on the group ring element v, one can construct group LCD codes that are reversible. We present many examples of binary group LCD codes of which some are optimal and group reversible LCD codes with different parameters.

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.

A novel genetic search scheme based on natureinspired evolutionary algorithms for binary selfdual codesIn this paper, a genetic algorithm, one of the evolutionary algorithm optimization methods, is used for the first time for the problem of computing extremal binary selfdual codes. We present a comparison of the computational times between the genetic algorithm and a linear search for different size search spaces and show that the genetic algorithm is capable of computing binary selfdual codes significantly faster than the linear search. Moreover, by employing a known matrix construction together with the genetic algorithm, we are able to obtain new binary selfdual codes of lengths 68 and 72 in a significantly short time. In particular, we obtain 11 new binary selfdual codes of length 68 and 17 new binary selfdual codes of length 72.

New type I binary $[72, 36, 12]$ selfdual codes from $M_6(\mathbb{F}_2)G$  Group matrix rings by a hybrid search technique based on a neighbourhoodvirus optimisation algorithmIn this paper, a new search technique based on a virus optimisation algorithm is proposed for calculating the neighbours of binary selfdual codes. The aim of this new technique is to calculate neighbours of selfdual codes without reducing the search field in the search process (this technique is known in the literature due to the computational time constraint) but still obtaining results in a reasonable time (significantly faster when compared to the standard linear computational search). We employ this new search algorithm to the wellknown neighbour method and its extension, the $ k^{th} $ range neighbours, and search for binary $ [72, 36, 12] $ selfdual codes. In particular, we present six generator matrices of the form $ [I_{36} \  \ \tau_6(v)], $ where $ I_{36} $ is the $ 36 \times 36 $ identity matrix, $ v $ is an element in the group matrix ring $ M_6(\mathbb{F}_2)G $ and $ G $ is a finite group of order 6, to which we employ the proposed algorithm and search for binary $ [72, 36, 12] $ selfdual codes directly over the finite field $\mathbb{F}_2 $. We construct 1471 new Type I binary $ [72, 36, 12] $ selfdual codes with the rare parameters $ \gamma = 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 32 $\ in their weight enumerators.

Binary selfdual codes of various lengths with new weight enumerators from a modified bordered construction and neighbours<p style='textindent:20px;'>In this work, we define a modification of a bordered construction for selfdual codes which utilises <inlineformula><texmath id="M1">\begin{document}$ \lambda $\end{document}</texmath></inlineformula>circulant matrices. We provide the necessary conditions for the construction to produce selfdual codes over finite commutative Frobenius rings of characteristic 2. Using the modified construction together with the neighbour construction, we construct many binary selfdual codes of lengths 54, 68, 82 and 94 with weight enumerators that have previously not been known to exist.</p>

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)'.