Knots and planar Skyrmions

In this thesis the research presented relates to topological solitons in (2+1) and (3+1)-dimensional Skyrme theories. Solutions in these theories have topologically invariant quantities which results in stable solutions which are topologically distinct from a vacuum. In Chapter 2 we discuss the broken baby Skyrme model, a theory which breaks symmetry to the dihedral group D_N. It has been shown that the unit soliton solution of the theory is formed of N distinct peaks, called partons. The multi-soliton solutions have already been numerically simulated for N = 3 and were found to be related to polyiamonds. We extend this for higher values of N and demonstrate that a polyform structure continues. We discuss our numerical simulations studying the dynamics of this model and show that the time dependent behaviour of solutions in the model can be understood by considering the interactions of individual pairs of partons. Results of these dynamics are then compared with those of the standard baby Skyrme model. Recently it has been demonstrated that Skyrmions of a fixed size are able to exist in theories without a Skyrme term so long as the Skyrmion is located on a domain wall. In Chapter 3 we present a (2+1)-dimensional O(3) sigma model, with a potential term of a particular form, in which such Skyrmions exist. We numerically compute domain wall Skyrmions of this type. We also investigate Skyrmion dynamics so that we can study Skyrmion stability and the scattering of multi-Skyrmions. We consider scattering events in which Skyrmions remain on the same domain wall and find they are effectively one-dimensional. At low speeds these scatterings are well-approximated by kinks in the integrable sine-Gordon model. We also present more exotic fully two-dimensional scatterings in which Skyrmions initially on different domain walls emerge on the same domain wall. The Skyrme-Faddeev model is a (3+1)-dimensional non-linear field theory that has topological soliton solutions, called hopfions. Solutions of this theory are unusual in that that they are string-like and take the form of knots and links. Solutions found to date take the form of torus knots and links of these. In Chapter 4 we show results which address the question of whether any non-torus knot hopfions exist. We present a construction of fields which are knotted in the form of cable knots to which an energy minimisation scheme can be applied. We find static hopfions of the theory which do not have the form of torus knots, but instead take the form of cable and hyperbolic knots. In Chapter 5 we consider an approximation to the Skyrme-Faddeev model in which the soliton is modelled by elastic rods. We use this as a mechanism to study examples of particular knots to attempt to gain an understanding of why such knots have not been found in the Skyrme-Faddeev model. The aim of this study is to focus the search for appropriate rational maps which can then be applied in the Skyrme-Faddeev model. The material presented in this thesis relates to two published papers and corresponding to Chapters 2 and 3 respectively, which were done as part of a collaboration. In this thesis my own results are presented. Chapter 4 concerns material which relates to the preprint which is all my own work. Chapter 5 discusses my own ongoing work

Geometric Properties and a Combinatorial Analysis of Convex Polygons Constructed of Tridrafters

The aim of this thesis is to show how the use of parity in tandem with the triangular grid as well as a newly introduced and similar method are insufficient to provide proof for why convex regions composed using the full set of shapes known as proper tridrafters have a portion shifted in a fashion known as against-the-grain. These two methods are applied in a combinatorial fashion

Interface theory and percolation

This thesis is mainly concerned with percolation on general infinite graphs, as well as the approximation of conformal maps by square tilings, which are defined using electrical networks. The first chapter is concerned with the smoothness of the percolation density on various graphs. In particular, we prove that for Bernoulli percolation on Z d , d ≥ 2, the percolation density is an analytic function of the parameter in the supercritical interval (pc(Z d ), 1]. This answers a question of Kesten [1981]. The analogous result is also proved for the Boolean model of continuum percolation in R 2 , answering a question of Last et al. [2017]. In order to prove these results, we introduce the notion of interfaces, which is studied extensively in the current thesis. For dimensions d ≥ 3, we use renormalisation tecnhiques. Furthermore, we prove that the susceptibility is analytic in the subcritical interval for all transitive short- or long-range models, and that pc < 1/2 for bond percolation on certain families of triangulations for which Benjamini & Schramm conjectured that pc ≤ 1/2 for site percolation. For the latter result, we use the well-known circle packing theorem of He and Schramm [1995], a discrete analogue of the Riemann mapping theorem. In Chapter 2, we continue the study of interfaces, and in particular, we consider the exponential growth rate br of the number of interfaces of a given size as a function of their surface-to-volume ratio r. We prove that the values of the percolation parameter p for which the interface size distribution has an exponential tail are uniquely determined by br by comparison with a dimension-independent function f(r) := (1+r) 1+r r r . We also point out a formula for translating any upper bound on the percolation threshold of a lattice G into a lower bound on the exponential growth rate of lattice animals a(G) and vice-versa. We exploit this in both directions. We obtain the rigorous lower bound pc(Z 3 ) > 0.2522 for 3-dimensional site percolation. We also improve on the best known asymptotic lower and upper bounds on a(Z d ) as d → ∞. We also prove that the rate of the exponential decay of the cluster size distribution, defined as c(p) := limn→∞ (Pp(|Co| = n))1/n, is a continuous function of p. The proof makes use of the Arzel`a-Ascoli theorem but otherwise boils down to elementary calculations. The analogous statement is also proved for the interface size distribution. For this we first establish that the rate of exponential decay is well-defined. In Chapter 3, we use interfaces to obtain upper bounds for the site percolation threshold of plane graphs with given minimum degree conditions. The results of this chapter are inspired by well-known conjectures of Benjamini and Schramm [1996b] for percolation on general graphs. We prove a conjecture by Benjamini and Schramm [1996b] stating that plane graphs of minimum degree at least 7 have site percolation threshold bounded away from 1/2. We also make progress on a conjecture of Angel et al. [2018] that the critical probability is at most 1/2 for plane triangulations of minimum degree 6. In the process, we prove tight new isoperimetric bounds for certain classes of hyperbolic graphs. This establishes the vertex isoperimetric constant for all triangular and square hyperbolic lattices, answering a question of [Lyons and Peres, 2016, Question 6.20]. Another topic of this thesis is the discrete approximation of conformal maps using another discrete analogue of the Riemann mapping theorem, namely the square tilings of Brooks et al. [1940]. This result is analogous to a well-known the orem of Rodin & Sullivan, previously conjectured by Thurston, which states that the circle packing of the intersection of a lattice with a simply connected planar domain Ω into the unit disc D converges to a Riemann map from Ω to D when the mesh size converges to 0. As a result, we obtain a new algorithm that allows us to numerically compute the Riemann map from any Jordan domain onto a square

Theoretical Optimization of Enzymatic Biomass Processes

This dissertation introduces a complete, stochastically-based algorithmic framework Cellulect to study, optimize and predict hydrolysis processes of the structured biomass cellulose. The framework combines a comprehensive geometric model for the cellulosic substrate with microstructured crystalline/amorphous regions distribution, distinctive monomers, polymer chain lengths distribution and free surface area tracking. An efficient tracking algorithm, formulated in a serial fashion, performs the updates of the system. The updates take place reaction-wise. The notion of real time is preserved. Advanced types of enzyme actions (random cuts, reduced/non-reduced end cuts, orientation, and the possibility of a fixed position of active centers) and their modular structure (carbohydrate-binding module with a flexible linker and a catalytic domain) are taken into account within the framework. The concept of state machines is adopted to model enzyme entities. This provides a reliable, powerful and maintainable approach for modelling already known enzyme features and can be extended with additional features not taken into account in the present work. The provided extensive probabilistic catalytic mechanism description further includes adsorption, desorption, competitive inhibition by soluble product polymers, and dynamical bond-breaking reactions with inclusive dependence on monomers and their polymers states within the substrate. All incorporated parameters refer to specific system properties, providing a one to one relationship between degrees of freedom and available features of the model. Finally, time propagation of the system is based on the modified stochastic Gillespie algorithm. It provides an exact stochastic time-reaction propagation algorithm, taking into account the random nature of reaction events as well as its random occurrences. The framework is ready for constrained input parameter estimation with empirical data sets of product concentration profiles by utilizing common optimization routines. Verification of the available data for the most common enzyme kinds (EG, β-G, CBH) in the literature has been accomplished. Sensitivity analysis of estimated model parameters were carried out. Dependency of various experimental input is shown. Optimization behavior in underdetermined conditions is inspected and visualized. Results and predictions for mixtures of optimized enzymes, as well as a practical way to implement and utilize the Cellulect framework are also provided. The obtained results were compared to experimental literature data demonstrate the high flexibility, efficiency and accuracy of the presented framework for the prediction of the cellulose hydrolysis process