A Riemann-Hilbert Approach to the Kissing Polynomials

Motivated by the numerical treatment of highly oscillatory integrals, this thesis studies a family of polynomials known as the Kissing Polynomials through Riemann-Hilbert techniques. The Kissing Polynomials are a family of non-Hermitian orthogonal polynomials, which are orthogonal with respect to the complex weight function $\exp(i\omega z)$ over the interval $[-1,1]$, where $\omega>0$. Although they have already been used to derive complex quadrature rules, there remain two main questions which this thesis addresses. The first is the existence of such polynomials; the second is the behavior of these polynomials throughout the complex plane. The first two chapters of this thesis provide the necessary background needed for the main results presented in the later chapters. In the first chapter, the connection between the numerical integration of highly oscillatory integrals and the Kissing Polynomials is established. Furthermore, we present the theory of non-Hermitian orthogonal polynomials and provide a more detailed description of the results in this thesis. The second chapter is a review on the formulation of the Kissing Polynomials as a solution to a matrix valued Riemann-Hilbert problem. This formulation is crucial to establishing both the existence of the Kissing Polynomials and its properties throughout the complex plane. Moreover, we also provide an overview of the powerful non-commutative steepest descent technique developed by Deift and Zhou in the mid 1990s used to compute the asymptotics for oscillatory Riemann-Hilbert problems, which will be used extensively in Chapters 4 and 5. In Chapter 3, we utilize the Riemann-Hilbert approach of Fokas, Its, and Kitaev to establish our first main result: the existence of the even degree Kissing polynomials for all values of $\omega>0$. First, we use the Riemann-Hilbert problem to show that the Kissing Polynomials satisfy a certain linear ordinary differential equation. Then, using standard results on differential equations, along with previous results on the Kissing Polynomials found in the literature, we are able to provide the desired result. In Chapter 4, we turn our attention to the behavior of the Kissing Polynomials as both the degree $n$ and parameter $\omega$ become large. To achieve this, we formulate this problem in terms of varying-weight Kissing polynomials, whose asymptotics can be handled with the Deift-Zhou steepest descent analysis. Now, the weight function depends now on $n$, the degree of the underlying polynomial. We are able to provide uniform asymptotics of the Kissing Polynomials as both $n$ and $\omega$ go to infinity at a linear rate such that the ratio $\omega/n >t_c$, where $t_c$ is a to be specified critical value. In Chapter 5, we generalize the results of Chapter 4 and study polynomials which are orthogonal with respect to the varying, complex weight, $\exp(n s z)$, over the interval $[-1,1]$, where now $s\in\mathbb{C}$. We will see that there are curves in the $s$-plane, called breaking curves, which separate regions corresponding to differing asymptotic behavior of the polynomials. In this chapter, we provide the large $n$ behavior of the recurrence coefficients associated to these polynomials. Finally, we also study the behavior of these recurrence coefficients as the parameter $s$ approaches a breaking curve in a specified double scaling limit.PhD Studentship: Cantab Capital Institute for the Mathematics of Informatio

Extracting partial decay rates of helium from complex rotation: autoionizing resonances of the one-dimensional configurations

Partial autoionization rates of doubly excited one-dimensional helium in the collinear Zee and eZe configuration are obtained by means of the complex rotation method. The approach presented here relies on a projection of back-rotated resonance wave functions onto singly ionized $\textrm{He}^{+}$ channel wave functions and the computation of the corresponding particle fluxes. In spite of the long-range nature of the Coulomb potential between the electrons and the nucleus, an asymptotic region where the fluxes are stationary is clearly observed. Low-lying doubly excited states are found to decay predomintantly into the nearest single-ionization continuum. This approach paves the way for a systematic analysis of the decay rates observed in higher-dimensional models, and of the role of electronic correlations and atomic structure in recent photoionization experiments

THE POSTULATE OF THE HISTORICAL LAW THEORY AND CONFLICT OF LAWS: AN ARTICULATION OF AFRICAN (UKELE) COMMUNAL LEGALISM

This essay is titled "Critique the Postulation of the Historical Law Theory and relate it to African Law. The postulation of the historical law school that law emanates from customs through an ordered pattern of systematized progress into a codified system in relation to African law forms the crust of this essay. To achieve this task, this essay adopts a critical method in exposing c postulation of the historical law school and the African Law (keeping in mind the Ukelle communal Law System). This essay questions whether there can be an independent law made or promulgated without targeting a given people or that there can be a people-free law? This essay claims that like the historical law school, laws emanate from their ground norms but insists that unlike the historical law school, laws in Ukelle Traditional System do not necessarily have to submit to through the rigor of systematic and strict evolutionary pattern of progress. Like Herder, this essay avers that there is a unique character with each culture, and as such Ukelle Traditional Law does not have to submit to any universal character of law

Mercury Contamination and Spill-over at Human-Wildlife-Environment Interface

Man’s quest for energy demands that fuel for running machines and cooking is vital for mankind. Oil and coal have served this energy quest for time immemorial. This oil quest has been present in the Albertine Graben since 1920, threatening biodiversity spots, terrestrial wildlife, and aquatic resources. The current book chapter provides insights into the spatial distribution of potentially toxic elements (Mercury) in terrestrial and marine species and the health risk posed to terrestrial and aquatic species due to oil exploitation

Global‐phase portrait and large‐degree asymptotics for the Kissing polynomials

Funder: Comunidad de Madrid; Id: http://dx.doi.org/10.13039/100012818Funder: Consejería de Educación e Investigación; Id: http://dx.doi.org/10.13039/501100010774Funder: Engineering and Physical Sciences Research Council; Id: http://dx.doi.org/10.13039/501100000266Funder: Cantab Capital Institute for the Mathematics of InformationFunder: Cambridge Centre for AnalysisAbstract: We study a family of monic orthogonal polynomials that are orthogonal with respect to the varying, complex‐valued weight function, exp ( n s z ) , over the interval [ − 1 , 1 ] , where s ∈ C is arbitrary. This family of polynomials originally appeared in the literature when the parameter was purely imaginary, that is, s ∈ i R , due to its connection with complex Gaussian quadrature rules for highly oscillatory integrals. The asymptotics for these polynomials as n → ∞ have recently been studied for s ∈ i R , and our main goal is to extend these results to all s in the complex plane. We first use the technique of continuation in parameter space, developed in the context of the theory of integrable systems, to extend previous results on the so‐called modified external field from the imaginary axis to the complex plane minus a set of critical curves, called breaking curves. We then apply the powerful method of nonlinear steepest descent for oscillatory Riemann–Hilbert problems developed by Deift and Zhou in the 1990s to obtain asymptotics of the recurrence coefficients of these polynomials when the parameter s is away from the breaking curves. We then provide the analysis of the recurrence coefficients when the parameter s approaches a breaking curve, by considering double scaling limits as s approaches these points. We see a qualitative difference in the behavior of the recurrence coefficients, depending on whether or not we are approaching the points s = ± 2 or some other points on the breaking curve

Inflammation, wound repair, and fibrosis: reassessing the spectrum of tissue injury and resolution

Estimates from various disease‐specific registries suggest that chronic inflammatory and fibrotic disorders affect a large proportion of the world's population, yet therapies for these conditions are largely ineffective. Recent advances in our collective understanding of mechanisms underlying both physiological and pathological repair of tissue injury are informing new clinical approaches to deal with various human inflammatory and fibrotic diseases. This 2013 Annual Review Issue of The Journal of Pathology offers an up‐to‐date glimpse of ongoing research in the fields of inflammation, wound healing, and tissue fibrosis, and highlights novel pathways and mechanisms that may be exploited to provide newer, more effective treatments to patients worldwide suffering from these conditions.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/95378/1/path4126.pd

De medicina libri octo

Editio secundaFalto de h. calc. con retrato firmada por J. HackiusMarca tip. en port.Sign.: *12, A-Z12, Aa-Bb1

Why honey is effective as a medicine. 1. Its use in modern medicine

Honey has been used as a medicine for thousands of years and its curative properties are well documented. However, modern medicine turned its back on honey and it is only now, with the advent of multi-resistant bacteria, that the antibiotic properties of honey are being rediscovered