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.

Preparation and characterisation of refractory whiskers and selected alumina composites

A whisker is a common name of single crystalline inorganic fibre of small dimensions, typically 0.5-1 μm in diameter and 20-50 μm in length. Whiskers are mainly used as reinforcement of ceramics. This work describes the synthesis and characterisation of new whisker types. Ti0.33Ta0.33Nb0.33CxN1-x, TiB2, B4C, and LaxCe1-xB6 have been prepared by carbothermal vapour–liquid–solid (CTR-VLS) growth mechanisms in the temperature range 900-1800°C, in argon or nitrogen. Generally, carbon and different suitable oxides were used as whisker precursors. The oxides reacted via a carbothermal reduction process. A halogenide salt was added to form gaseous metal halogenides or oxohalogenides and small amount of a transition metal was added to catalyse the whisker growth. In this mechanism, the whisker constituents are dissolved into the catalyst, in liquid phase, which becomes supersaturated. Then a whisker could nucleate and grow out under continuous feed of constituents. The syntheses of TiC, TiB2, and B4C were followed at ordinary synthesis conditions by means of mass spectrometry (MS), thermogravimetry (TG), differential thermal analysis (DTA) and quenching. The main reaction starting temperatures and reaction time for the different mixtures was revealed, and it was found that the temperature inside the crucible during the reactions was up to 100°C below the furnace set-point, due to endothermic nature of the reactions. Quench experiments showed that whiskers were formed already when reaching the temperature plateau, but the yield increased fast with the holding time and reached a maximum after about 20-30 minutes. Growth models for whisker formation have been proposed. Alumina based composites reinforced by (2-5 vol.%) TiCnano and TiNnano and 25 vol.% of carbide, and boride phases (whiskers and particulates of TiC, TiN, TaC, NbC, (Ti,Ta)C, (Ti,Ta,Nb)C, SiC, TiB2 and B4C) have been prepared by a developed aqueous colloidal processing route followed by hot pressing for 90 min at 1700°C, 28 MPa or SPS sintering for 5 minutes at 1200-1600°C and 75 MPa. Vickers indentation measurements showed that the lowest possible sintering temperature is to prefer from mechanical properties point of view. In the TiNnano composites the fracture mode was typically intergranular, while it was transgranular in the SiCnano composites. The whisker and particulate composites have been compared in terms of e.g. microstructure and mechanical properties. Generally, additions of whiskers yielded higher fracture toughness compared to particulates. Composites of commercially available SiC whiskers showed best mechanical properties with a low spread but all the other whisker phases, especially TiB2, exhibited a great potential as reinforcement materials.