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.

Chemical Bonding of Hydrocarbons to Metal Surfaces

Using x-ray absorption spectroscopy (XAS), x-ray emission spectroscopy (XES) and x-ray photoelectron spectroscopy (XPS) in combination with density functional theory (DFT) the changes in electronic and geometric structure of hydrocarbons upon adsorption are determined. The chemical bonding is analyzed and the results provide new insights in the mechanisms responsible for dehydrogenation in heterogeneous catalysis. In the case of alkanes, n-octane and methane are studied. XAS and XES show significant changes in the electronic structure upon adsorption. XES shows new adsorption induced occupied states and XAS shows quenching of CH*/Rydberg states in n-octane. In methane the symmetry forbidden gas phase lowest unoccupied molecular orbital becomes allowed due to broken symmetry. New adsorption induced unoccupied features with mainly metal character appear just above the Fermi level in XA spectra of both adsorbed methane and n-octane. These changes are not observed in DFT total energy geometry optimizations. Comparison between experimental and computed spectra for different adsorbate geometries reveals that the molecular structures are significantly changed in both molecules. The C-C bonds in n-octane are shortened upon adsorption and the C-H bonds are elongated in both n-octane and methane. In addition ethylene and acetylene are studied as model systems for unsaturated hydrocarbons. The validity of both the Dewar-Chatt-Duncanson chemisorption model and the alternative spin-uncoupling picture is confirmed, as well as C-C bond elongation and upward bending of the C-H bonds. The bonding of ethylene to Cu(110) and Ni(110) are compared and the results show that the main difference is the amount of back-donation into the molecular π* orbital, which allows the molecule to desorb molecularly from the Cu(110) surface, whereas it is dehydrogenated upon heating on the Ni(110) surface. Acetylene is found to adsorb in two different adsorption sites on the Cu(110) surface at liquid nitrogen temperature. Upon heating the molecules move into one of these sites due to attractive adsorbate-adsorbate interaction and only one adsorbed species is present at room temperature, at which point the molecules start reacting to form benzene. The bonding of the two species is very similar in both sites and the carbon atoms are rehybridized essentially to sp2.