In this thesis we derive mathematical models for the glass sheet redraw process for the production of very thin glass sheets. In the redraw process, a prefabricated glass block is fed into a furnace, where it is heated and stretched by the application of draw rollers to reduce its thickness. Redrawn sheets may be used in various applications including smartphone and battery technology. Our aims are to investigate the factors determining the final thickness profile of a glass sheet produced by this process, as well as the growth of out-of-plane deformations in the sheet during redraw. Our method is to model the glass sheet using Navier–Stokes equations and free-surface conditions, and exploit small aspect ratios in the sheet to simplify and solve these equations using asymptotic expansions. We first consider a simple two-dimensional sheet to determine which physical effects should be taken into account in modelling the redraw process. Next, we derive a mathematical model for redraw of a thin threedimensional sheet. We consider the limits in which the heater zone is either short or long compared with the sheet half-width. The resulting reduced models predict the thickness profile of the redrawn sheet and the initial shape required to redraw a product of uniform thickness. We then derive mathematical models for buckling of thin viscous sheets during redraw. For buckling of a two-dimensional glass sheet due to gravity-induced compression, we predict the evolution of the centreline and investigate the early- and late-time behaviour of the system. For a three-dimensional glass sheet undergoing redraw, we use numerical solutions to investigate the behaviour of the sheet mid-surface.

<p>The main topic of this doctoral thesis is asymptotic properties of zeros in polynomial families arising as eigenfunctions to exactly-solvable differential operators. The study was initially inspired by a number of striking results from computer experiments performed by G. Masson and B. Shapiro for a more restrictive class of operators. Our research is also motivated by a classical question going back to S. Bochner on a general classification of differential operators possessing an infinite sequence of orthogonal eigenpolynomials. In general however, the sequence of eigenpolynomials of an exactly-solvable operator is not an orthogonal system and it can therefore not be studied by means of the extensive theory known for such systems. Our study can thus be considered as the first steps to a natural generalization of the asymptotic behaviour of the roots of classical orthogonal polynomials. Exactly-solvable operators split into two major classes: non-degenerate and degenerate. We prove that in the former case, as the degree tends to infinity, the zeros of the eigenpolynomial are distributed according to a certain probability measure which is compactly supported on a tree and which depends only on the leading term of the operator. Computer experiments indicate the existence of a limiting root measure in the degenerate case too, but that it is compactly supported (conjecturally on a tree) only after an appropriate scaling which is conjectured (and partially proved) in this thesis. One of the main technical tools in this thesis is the Cauchy transform of a probability measure, which in the considered situation satisfies an algebraic equation. Due to the connection between the asymptotic root measure and its Cauchy transform it is therefore possible to obtain detailed information on the limiting zero distribution.</p>

The main topic of this doctoral thesis is asymptotic properties of zeros in polynomial families arising as eigenfunctions to exactly-solvable differential operators. The study was initially inspired by a number of striking results from computer experiments performed by G. Masson and B. Shapiro for a more restrictive class of operators. Our research is also motivated by a classical question going back to S. Bochner on a general classification of differential operators possessing an infinite sequence of orthogonal eigenpolynomials. In general however, the sequence of eigenpolynomials of an exactly-solvable operator is not an orthogonal system and it can therefore not be studied by means of the extensive theory known for such systems. Our study can thus be considered as the first steps to a natural generalization of the asymptotic behaviour of the roots of classical orthogonal polynomials. Exactly-solvable operators split into two major classes: non-degenerate and degenerate. We prove that in the former case, as the degree tends to infinity, the zeros of the eigenpolynomial are distributed according to a certain probability measure which is compactly supported on a tree and which depends only on the leading term of the operator. Computer experiments indicate the existence of a limiting root measure in the degenerate case too, but that it is compactly supported (conjecturally on a tree) only after an appropriate scaling which is conjectured (and partially proved) in this thesis. One of the main technical tools in this thesis is the Cauchy transform of a probability measure, which in the considered situation satisfies an algebraic equation. Due to the connection between the asymptotic root measure and its Cauchy transform it is therefore possible to obtain detailed information on the limiting zero distribution.

In the first part of this thesis, we will explore an iterative procedure to determine the detailed asymptotic behaviour of solutions of a certain class of nonlinear vector differential equations which approach a nonlinear sink as time tends to infinity. This procedure is indifferent to resonance in the eigenvalues. Some attention will be given to finding approximations to solutions which are themselves flows. Moreover, we will address the writing of one component in terms of another in the case of a planar system. In the second part of this thesis, we will explore the Michaelis-Menten mechanism of a single enzyme-substrate reaction. The focus is an analysis of the planar reduction in phase space or, equivalently, solutions of the scalar reduction. In particular, we will prove the existence and uniqueness of a slow manifold between the horizontal and vertical isoclines. Also, we will determine the concavity of all solutions in the first quadrant. Moreover, we will establish the asymptotic behaviour of all solutions near the origin, which generally is not given by a Taylor series. Finally, we will determine the asymptotic behaviour of the slow manifold at infinity. Additionally, we will study the planar reduction. In particular, we will find non-trivial bounds on the length of the pre-steady-state period, determine the asymptotic behaviour of solutions as time tends to infinity, and determine bounds on the solutions valid for all time. In the third part of this thesis, we explore the (nonlinear) Lindemann mechanism of unimolecular decay. The analysis will be similar to that for the Michaelis-Menten mechanism with an emphasis on the differences. In the fourth and final part of this thesis, we will present some open problems.

In the first part of this thesis, we will explore an iterative procedure to determine the detailed asymptotic behaviour of solutions of a certain class of nonlinear vector differential equations which approach a nonlinear sink as time tends to infinity. This procedure is indifferent to resonance in the eigenvalues. Some attention will be given to finding approximations to solutions which are themselves flows. Moreover, we will address the writing of one component in terms of another in the case of a planar system. In the second part of this thesis, we will explore the Michaelis-Menten mechanism of a single enzyme-substrate reaction. The focus is an analysis of the planar reduction in phase space or, equivalently, solutions of the scalar reduction. In particular, we will prove the existence and uniqueness of a slow manifold between the horizontal and vertical isoclines. Also, we will determine the concavity of all solutions in the first quadrant. Moreover, we will establish the asymptotic behaviour of all solutions near the origin, which generally is not given by a Taylor series. Finally, we will determine the asymptotic behaviour of the slow manifold at infinity. Additionally, we will study the planar reduction. In particular, we will find non-trivial bounds on the length of the pre-steady-state period, determine the asymptotic behaviour of solutions as time tends to infinity, and determine bounds on the solutions valid for all time. In the third part of this thesis, we explore the (nonlinear) Lindemann mechanism of unimolecular decay. The analysis will be similar to that for the Michaelis-Menten mechanism with an emphasis on the differences. In the fourth and final part of this thesis, we will present some open problems.

A central theme in phylogenetics is the reconstruction and analysis of evolutionary trees from a given set of data. To determine the optimal search methods for the reconstruction of trees, it is crucial to understand the size and structure of neighbourhoods of trees under tree rearrangement operations. The diameter and size of the immediate neighbourhood of a tree has been well-studied, however little is known about the number of trees at distance two, three or (more generally) k from a given tree. In this thesis we explore previous results on the size of these neighbourhoods under common tree rearrangement operations (NNI, SPR and TBR). We obtain new results concerning the number of trees at distance k from a given tree under the Robinson-Foulds (RF) metric and the Nearest Neighbour Interchange (NNI) operation, and the number of trees at distance two from a given tree under the Subtree Prune and Regraft (SPR) operation. We also obtain an exact count for the number of pairs of binary phylogenetic trees that share a first RF or NNI neighbour.

We develop a new approach to the analysis of pseudodifferential operators with small parameter 'epsilon' in (0,1] on a compact smooth manifold X. The standard approach assumes action of operators in Sobolev spaces whose norms depend on 'epsilon'. Instead we consider the cylinder [0,1] x X over X and study pseudodifferential operators on the cylinder which act, by the very nature, on functions depending on 'epsilon' as well. The action in 'epsilon' reduces to multiplication by functions of this variable and does not include any differentiation. As but one result we mention asymptotic of solutions to singular perturbation problems for small values of 'epsilon'.

Under a class of one dimensional local volatility models, this thesis establishes closed form small time asymptotic formulae for the gradient of the implied volatility, whether or not the options are at the money, and for the at the money Hessian of the implied volatility. Along the way it also partially verifies the statement by Berestycki, Busca and Florent (2004) that the implied volatility admits higher order Taylor series expansions in time near expiry. Both as a prelude to the presentation of these main results and as a highlight of the importance of the no arbitrage condition, this thesis shows in its beginning a Cox-Ingersoll-Ross type stock model where an equivalent martingale measure does not always exist.

Sub-Riemannian geometry is the natural setting for studying dynamical systems, as noise often has a lower dimension than the dynamics it enters. This makes sub-Riemannian geometry an important field of study. In this thesis, we analysis some of the aspects of sub-Riemannian diffusion processes on manifolds. We first focus on studying the small-time asymptotics of sub-Riemannian diffusion bridges. After giving an overview of recent work by Bailleul, Mesnager and Norris on small-time fluctuations for the bridge of a sub-Riemannian diffusion, we show, by providing a specific example, that, unlike in the Riemannian case, small-time fluctuations for sub-Riemannian diffusion bridges can exhibit exotic behaviours, that is, qualitatively different behaviours compared to Brownian bridges. We further extend the analysis by Bailleul, Mesnager and Norris of small-time fluctuations for sub-Riemannian diffusion bridges, which assumes the initial and final positions to lie outside the sub-Riemannian cut locus, to the diagonal and describe the asymptotics of sub-Riemannian diffusion loops. We show that, in a suitable chart and after a suitable rescaling, the small-time diffusion loop measures have a non-degenerate limit, which we identify explicitly in terms of a certain local limit operator. Our analysis also allows us to determine the loop asymptotics under the scaling used to obtain a small-time Gaussian limit for the sub-Riemannian diffusion bridge measures by Bailleul, Mesnager and Norris. In general, these asymptotics are now degenerate and need no longer be Gaussian. We close by reporting on work in progress which aims to understand the behaviour of Brownian motion conditioned to have vanishing $N$th truncated signature in the limit as $N$ tends to infinity. So far, it has led to an analytic proof of the stand-alone result that a Brownian bridge in $\mathbb{R}^d$ from $0$ to $0$ in time $1$ is more likely to stay inside a box centred at the origin than any other Brownian bridge in time $1$.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第22984号 / 理博第4661号 / 新制||理||1669(附属図書館) / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)准教授 藤 定義, 教授 佐々 真一, 教授 早川 尚男 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM