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Cantor space, the set of infinite words over a finite alphabet, is a type of metric space with a 'self-similar' structure. This thesis explores three areas concerning Cantor space with regard to fractal geometry, group theory, and topology. We find first results on the dimension of intersections of fractal sets within the Cantor space. More specifically, we examine the intersection of a subset E of the n-ary Cantor space, C[sub]n with the image of another subset Funder a random isometry. We obtain almost sure upper bounds for the Hausdorff and upper box-counting dimensions of the intersection, and a lower bound for the essential supremum of the Hausdorff dimension. We then consider a class of groups, denoted by V[sub]n(G), of homeomorphisms of the Cantor space built from transducers. These groups can be seen as homeomorphisms that respect the self-similar and symmetric structure of C[sub]n, and are supergroups of the Higman-Thompson groups V[sub]n. We explore their isomorphism classes with our primary result being that V[sub]n(G) is isomorphic to (and conjugate to) V[sub]n if and only if G is a semiregular subgroup of the symmetric group on n points. Lastly, we explore invariant relations on Cantor space, which have quotients homeomorphic to fractals in many different classes. We generalize a method of describing these quotients by invariant relations as an inverse limit, before characterizing a specific class of fractals known as Sierpiński relatives as invariant factors. We then compare relations arising through edge replacement systems to invariant relations, detailing the conditions under which they are the same.

An inertial manifold (IM) is one of the key objects in the modern theory of dissipative systems generated by partial differential equations (PDEs) since it allows us to describe the limit dynamics of the considered system by the reduced finite-dimensional system of ordinary differential equations (ODEs). It is well known that the existence of an IM is guaranteed when the so called spectral gap conditions are satisfied, whereas their violation leads to the possibility of an infinite-dimensional limit dynamics, at least on the level of an abstract parabolic equation. However, these conditions restrict greatly the class of possible applications and are usually satisfied in the case of one spatial dimension only. Despite many efforts in this direction, the IMs in the case when the spectral gap conditions are violated remain a mystery especially in the case of parabolic PDEs. On the one hand, there is a number of interesting classes of such equations where the existence of IMs is established without the validity of the spectral gap conditions and, on the other hand there were no examples of dissipative parabolic PDEs where the non-existence of an IM is rigorously proved. The main aim of this thesis is to bring some light on this mystery by the comprehensive study of three model examples of parabolic PDEs where the spectral gap conditions are not satisfied, namely, 1D reaction-diffusion-advection (RDA) systems (see Chapter 3), the 3D Cahn-Hilliard equation on a torus (see Chapter 4) and the modified 3D Navier-Stokes equations (see Chapter 5). For all these examples the existence or non-existence of IM was an open problem. As shown in Chapter 3, the existence or non-existence of an IM for RDA systems strongly depends on the boundary conditions. In the case of Dirichlet or Neumann boundary conditions, we have proved the existence of an IM using a specially designed non-local in space diffeomorphism which transforms the equations to the new ones for which the spectral gap conditions are satisfied. In contrast to this, in the case of periodic boundary conditions, we construct a natural example of a RDA system which does not possess an IM. In Chapters 4 and 5 we develop an extension of the so-called spatial averaging principle (SAP) (which has been suggested by Sell and Mallet-Paret in order to treat scalar reaction-diffusion equation on a 3D torus) to the case of 4th order equations where the nonlinearity loses smoothness (the Cahn-Hilliard equation) as well as for systems of equations (modified Navier-Stokes equations).

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System dynamics (SD) simulation models are differential equation models that often contain a complex network of relationships between variables. These models are widely used, but have a number of limitations. SD models cannot represent individual entities, or model the stochastic behaviour of these individuals. In addition, model parameters are often not observable and so values of these are based on expert opinion, rather than being derived directly from historical data. This thesis aims to address these limitations and hence enhance system dynamics modelling. This research is undertaken in the context of SD models from a major telecommunications provider. In the first part of the thesis we investigate the advantages of adding a discreteevent simulation model to an existing SD model, to form a hybrid model. There are few examples of previous attempts to build models of this type and we therefore provide an account of the approach used and its potential for larger models. Results demonstrate the advantages of the hybrid’s ability to track individuals and represent stochastic variation. In the second part of the thesis we investigate data-driven methods to validate model assumptions and estimate model parameters from historical data. This commences with use of regression based methods to assess core structural assumptions of the organisation’s SD model. This is a complex, highly nonlinear model used by the organisation for service delivery. We then attempt to estimate the parameters of this model, using a modified version of an existing approach based on state-space modelling and Kalman filtering, known as FIMLOF. One such modification, is the use of the unscented Kalman filter for nonlinear systems. After successfully estimating parameters in simulation studies, we attempt to calibrate the model for 59 geographical regions. Results demonstrate the success of our estimated parameters compared to the organisation’s default parameters in replicating historical data.

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In this thesis we explore identities which can be proven on multiple zeta values using the derivation operators $ D_r $ from Brown's motivic MZV framework. We then explore identities which occur on multiple polylogarithms by way of the symbol map $ \mathcal{S} $, and the multiple polylogarithm coproduct $ \Delta $. On multiple zeta values, we consider Borwein, Bradley, Broadhurst, and Lisoněk's cyclic insertion conjecture about inserting blocks of $ \{2\}^{a_i} $ between the arguments of $ \zeta(\{1,3\}^n) $. We generalise this conjecture to a much broader setting, and give a proof of a symmetrisation of this generalised cyclic insertion conjecture. This proof is by way of the block-decomposition of iterated integrals introduced here, and Brown's motivic MZV framework. This symmetrisation allows us to prove (or to make progress towards) various conjectural identities, including the original cyclic insertion conjecture, and Hoffman's $ 2\zeta(3,3,\{2\}^n) - \zeta(3,\{2\}^n,1,2) $ identity. Moreover, we can then generate unlimited new conjectural identities, and give motivic proofs of their symmetrisations. We then consider the task of relating weight 5 multiple polylogarithms. Using the symbol map, we determine all of the symmetries and functional equations between depth 2 and between depth 3 iterated integrals with 'coupled-cross ratio' arguments $ [\mathrm{cr}(a,b,c,d_1), \ldots, \mathrm{cr}(a,b,c,d_k)] $. We lift the identity for $ I_{4,1}(x,y) + I_{4,1}(\frac{1}{x}, \frac{1}{y}) $ to an identity holding exactly on the level of the symbol and prove a generalisation of this for $ I_{a,b}(x,y) $. Moreover, we further lift the subfamily $ I_{n,1} $ to a candidate numerically testable identity using slices of the coproduct. We review Dan's reduction method for reducing the iterated integral $ I_{1,1,\ldots,1} $ to a sum in $ \leq n-2 $ variables. We provide proofs for Dan's claims, and run the method in the case $ I_{1,1,1,1} $ to correct Dan's original reduction of $ I_{1,1,1,1} $ to $ I_{3,1} $ and $ I_4 $. We can then compare this with another reduction to find $ I_{3,1} $ functional equations, and their nature. We then give a reduction of $ I_{1,1,1,1,1} $ to $ I_{3,1,1} $, $ I_{3,2} $ and $ I_{5} $, and indicate how one might be able to further reduce to $ I_{3,2} $ and $ I_5 $. Lastly, we use and generalise an idea suggested by Goncharov at weight 4 and weight 5. We find $ \mathrm{Li}_n $ terms when certain $ \mathrm{Li}_2 $, $ \mathrm{Li}_3 $ and $ \mathrm{Li}_4 $ functional equations are substituted into the arguments of symmetrisations of $ I_{m,1}(x,y) $. By expanding $ I_{m,1}(\text{$\mathrm{Li}_k$ equation}, \text{$\mathrm{Li}_\ell $ equation}) $ in two different ways we obtain functional equations for $ \mathrm{Li}_5 $ and $ \mathrm{Li}_6 $. We make some suggestions for how this might work at weight 7 and weight 8 giving a potential route to $ \mathrm{Li}_7 $ and $ \mathrm{Li}_8 $ functional equations.

After introducing the topics that will be covered in this work we review important concepts from the calculus of variations in elasticity theory. Subsequently the following three topics are discussed: The first originates from the work of Post and Sivaloganathan [\emph{Proceedings of the Royal Society of Edinburgh, Section: A Mathematics}, 127(03):595--614, 1997] in the form of two scenarios involving the twisting of the outer boundary of an annulus $A$ around the inner. It seeks minimisers of $∫_A \frac{1}{2}|∇u|^2 \d x$ among deformations $u$ with the constraint $\det ∇u ≥0$ a.e.~as well as of $∫_A \frac{1}{2}|∇u|^2 + h(\det ∇u) \d x$ in which $h$ penalises volume compression so that $\det ∇u > 0$ a.e.~is imposed on minimisers. In the former case we find infinitely many explicit solutions for which $\det ∇u = 0$ holds on a region around the inner boundary of $A$. In the latter we expand on known results by showing similar growth properties of the solutions compared to the previous case while contrasting that $\det ∇u>0$ holds everywhere. In the second we introduce a new semiconvexity called $n$-polyconvexity that unifies poly- and rank-one convexity in the sense that for $f:ℝ^{d×D}→\bar{ℝ}$ we have that $n$-polyconvexity is equivalent to polyconvexity for $n=\min\{d,D\}=:d∧D$ and equivalent to rank-one convexity for $n=1$. For $d,D≥3$ we gain previously unknown semiconvexities in hierarchical order ($2$-polyconvexity, \dots, $(d∧D-1)$-polyconvexity, weakest to strongest). We further define functions which are `$n$-polyaffine at $F$' and find that they are not necessarily polyaffine for $n<d∧D$ (unlike rank-one affine functions). As one of the main results we obtain that $1$-polyconvex (i.e.~rank-one convex) functions $f:ℝ^{d×D} →ℝ$ are the pointwise supremum of $1$-polyaffine functions at $F$ for every $F∈ℝ^{d×D}$. In addition and among other things, we discuss envelopes, generalised $T_k$ configurations and relations to quasiconvexity. The third involves a generalisation of the theory of abstract convexity which allows one to include cases like $1$-polyconvex functions as the pointwise supremum of $1$-polyconvex functions at $F$ for every $F$, while this is not possible within the classical theory. We review the most important results of the classical theory and present results on generalised hull operators, subgradients, conjugations and Legendre-Fenchel transforms for our new theory. In particular we obtain an operator that is reminiscent of the rank-one convexification process via lamination steps for a function. Moreover, we show that directional convexity is a special case of the generalised abstract convexity theory. Finally, we conclude each topic, pointing out possible directions of further research.

In this thesis we consider two types of ill-posed problem: (13 the numerical solution of Fredholm integral equations of the first kind of convolution type, with non-negativity constraints on the solution, and (2) the numerical inversion of some truncated integral transforms of nonnegative functions. In (2) the transforms are with respect to simple kernel functions of the form cos ax and sin(ax)/(ax), a constant. In solving both types of problem we extend ideas of Biraud for solving convolution type integral equations. This method is based on extrapolation and in Chapter 1a brief resume of extrapolation methods is given by way of introduction. In Chapter 2 we discuss Biraud's method and the choice of approximating function spaces. In this thesis we consider two distinct types of approximations; band-limited approximations, which-are not analytic in Fourier space, and also approximations which are analytic in Fourier space. In Chapter 3, which deals with numerical deconvolution, Wahba's idea of weighted cross-validation is used to determine an optimal rectangular filter, and the resulting cut-off point. is compared on several' test examples, with the best cut-off point from which to perform Biraud extrapolation. In many cases the two cut-off point coincide, or very nearly. In Chapter 4, trigonometric Biraud extrapolation is used to numerically invert several autocorrelation functions arising in laser technology. In Chapter 5, two of these problems are repeated using Hermite wave-functions (HWFs) as bases. 8oth. linear and Biraud extrapolation 'methods are compared. A new formula for the convolutions of HWFs is developed and used.