Topics in geometry, analysis and inverse problems

The thesis consists of three independent parts. Part I: Polynomial amoebas We study the amoeba of a polynomial, as de ned by Gelfand, Kapranov and Zelevinsky. A central role in the treatment is played by a certain convex function which is linear in each complement component of the amoeba, which we call the Ronkin function. This function is used in two di erent ways. First, we use it to construct a polyhedral complex, which we call a spine, approximating the amoeba. Second, the Monge-Ampere measure of the Ronkin function has interesting properties which we explore. This measure can be used to derive an upper bound on the area of an amoeba in two dimensions. We also obtain results on the number of complement components of an amoeba, and consider possible extensions of the theory to varieties of codimension higher than 1. Part II: Differential equations in the complex plane We consider polynomials in one complex variable arising as eigenfunctions of certain differential operators, and obtain results on the distribution of their zeros. We show that in the limit when the degree of the polynomial approaches innity, its zeros are distributed according to a certain probability measure. This measure has its support on the union of nitely many curve segments, and can be characterized by a simple condition on its Cauchy transform. Part III: Radon transforms and tomography This part is concerned with different weighted Radon transforms in two dimensions, in particular the problem of inverting such transforms. We obtain stability results of this inverse problem for rather general classes of weights, including weights of attenuation type with data acquisition limited to a 180 degrees range of angles. We also derive an inversion formula for the exponential Radon transform, with the same restriction on the angle

Proposed Thalmann algorithm air diving decompression table for the Swedish Armed Forces

The Swedish Armed Forces (SwAF) air dive tables are under revision. Currently, the air dive table from the U.S. Navy (USN) Diving Manual (DM) Rev. 6 is used with an msw-to-fsw conversion. Since 2017, the USN has been diving according to USN DM rev. 7, which incorporates updated air dive tables derived from the Thalmann Exponential Linear Decompression Algorithm (EL-DCM) with VVAL79 parameters. The SwAF decided to replicate and analyze the USN table development methodology before revising their current tables. The ambition was to potentially find a table that correlates with the desired risk of decompression sickness. New compartmental parameters for the EL-DCM algorithm, called SWEN21B, were developed by applying maximum likelihood methods on 2,953 scientifically controlled direct ascent air dives with known outcomes of decompression sickness (DCS). The targeted probability of DCS for direct ascent air dives was ≤1% overall and ≤1‰ for neurological DCS (CNS-DCS). One hundred fifty-four wet validation dives were performed with air between 18 to 57 msw. Both direct ascent and decompression stop dives were conducted, resulting in incidences of two joint pain DCS (18 msw/59 minutes), one leg numbness CNS-DCS (51 msw/10 minutes with deco-stop), and nine marginal DCS cases, such as rashes and itching. A total of three DCS incidences, including one CNS-DCS, yield a predicted risk level (95% confidence interval) of 0.4-5.6% for DCS and 0.0-3.6% for CNS-DCS. Two out of three divers with DCS had patent foramen ovale. The SWEN21 table is recommended for the SwAF for air diving as it, after results from validation dives, suggests being within the desired risk levels for DCS and CNS-DCS