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

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

Identifying the connection between Roman Conceptions of ‘Pure Air’ and Physical and Mental Health in Pompeian Gardens (c. 150 BC-AD 79): A Multi-Sensory Approach to Ancient Medicine

Different genres of Roman literature commented on the relationship between the condition of the environment and physical and mental health. They often refer to clear, pure, or good air as a beneficial aspect of the environment. Yet, unlike fetid air, they provide few descriptions of what constituted healthy air quality. Moreover, aside from pointing out the association between the environment and bodily condition, the writers also did not explain precisely how the link between the two was made. This paper utilizes a comparative study of ancient literature and the archaeological remains of Roman gardens in Pompeii: archaeobotanical samples, fresco paintings, location, and surviving features. Three questions are addressed in this study: First, how did the Romans identify and define pure? Second, how did air connect to the body? Third, what were the qualities of pure air and how did they benefit the body? Not only was inhalation a means of linking air to the body, but the two were also related through sensory perception. I argue that sight, sound, and olfaction were used to identify the qualities of pure air. Through the sensory process of identification, the beneficial properties of pure air were, in accordance with ancient perceptions of sensory function, taken into the body and affected health. Thus, sensory perception acted as the bridge between the environment and health

A "hair-raising" history of alopecia areata

YesA 3500‐year‐old papyrus from ancient Egypt provides a list of treatments for many diseases including “bite hair loss,” most likely alopecia areata (AA). The treatment of AA remained largely unchanged for over 1500 years. In 30 CE, Celsus described AA presenting as scalp alopecia in spots or the “windings of a snake” and suggested treatment with caustic compounds and scarification. The first “modern” description of AA came in 1813, though treatment still largely employed caustic agents. From the mid‐19th century onwards, various hypotheses of AA development were put forward including infectious microbes (1843), nerve defects (1858), physical trauma and psychological stress (1881), focal inflammation (1891), diseased teeth (1902), toxins (1912) and endocrine disorders (1913). The 1950s brought new treatment developments with the first use of corticosteroid compounds (1952), and the first suggestion that AA was an autoimmune disease (1958). Research progressively shifted towards identifying hair follicle‐specific autoantibodies (1995). The potential role of lymphocytes in AA was made implicit with immunohistological studies (1980s). However, studies confirming their functional role were not published until the development of rodent models (1990s). Genetic studies, particularly genome‐wide association studies, have now come to the forefront and open up a new era of AA investigation (2000s). Today, AA research is actively focused on genetics, the microbiome, dietary modulators, the role of atopy, immune cell types in AA pathogenesis, primary antigenic targets, mechanisms by which immune cells influence hair growth, and of course the development of new treatments based on these discoveries.Alopecia UK

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

Queen Cleopatra and the other 'Cleopatras': their medical legacy.

Cleopatra is a female figure widespread in Greece (especially in Macedonian territory), Egypt and Syria during the Hellenistic era. Ancient women doctors bearing the name Cleopatra have been identified by a systematic search through the ancient Greek, Latin and Egyptian bibliography, including original resources from the first century BC. Fictional and non-fictional figures have been distinguished and their works identified. Queen Cleopatra of Egypt, Galen's physician assistant, the outcast Metrodora, Cleopatra the Alchemist and Cleopatra the Gynaecologist deliver a story of medicine and name-giving that confuses researchers of the past and intrigues those of the present