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.

Structural and Interaction Studies of Bacterial Polysaccharides by NMR Spectroscopy

An introduction to bacterial polysaccharides and the methods for structural determination are described in the first two parts of the thesis. In a structural elucidation of bacterial polysaccharides NMR experiments are important as is component analysis. A short description of immunochemical methods such as enzyme immunoassays is included. Two NMR techniques used for interaction studies, trNOE and STD NMR, are also discussed. The third part of the thesis discusses and summarizes the results from the included papers. The structures of the exopolysaccharides produced by two lactic acid bacteria are determined by one- and two dimensional NMR experiments. One is a heteropolysaccharide produced by Streptococcus thermophilus and the other a homopolysaccharide produced by Propionibacterium freudenreichii. The structure of an acidic polysaccharide from a marine bacterium with two serine residues in the repeating unit is also investigated. The structural and immunological relationship between two O-antigenic polysaccharides from Escherichia coli strain 180/C3 and O5 is discussed and investigated. Finally, interaction studies of an octasaccharide derived from the Salmonella enteritidis O-antigen and a bacteriophage are described which were performed with NMR experiments.