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.

Learning Democracy Together in School? : Student and Teacher Attitudes in Bosnia and Herzegovina

The principal aim of this study is to examine attitudes and values, through questionnaires, among students and teachers in the last grade of primary school (grade 8) regarding issues related to authoritarianism, democracy, human rights, children rights, conflict resolution and legislation in Bosnia and Herzegovina. A second aim is to explore and analyze the role of the international community in the democratization and education processes in the light of globalization in this country through secondary sources of data, site visits and observations. Analysis of the student sample reveals suspicion towards democracy, especially when democracy was associated with politics and politicians. When the issue of democracy was de-contextualized from Bosnia and Herzegovina realities in the questionnaire, students showed more positive attitudes towards it. Students generally agreed with very strong authoritarian statements. High achieving students were more democratic, more socially responsible, more tolerant regarding attitudes towards religion, race and disabilities, and less authoritarian compared to low achievers. High achievers felt that they had influence over daily events, and were positive towards social and civil engagement. High achievers viewed politics negatively, but had high scores on the democracy scale. High achievers also agreed to a larger extent that it is acceptable to break the law. The more authoritarian students were somewhat more prone to respond that it is not acceptable to break the law. The major findings from the teacher sample show that teachers who agreed with non-peaceful mediation, and had a non-forgiving and rigid approach to interpersonal conflicts, also agreed with strong authoritarian statements and were less democratic. In general, teachers valued students who behave respectfully, have a good upbringing and are obedient. They were very concerned about the general status of education in society, which they felt was becoming marginalized. Teachers were not happy with the overloaded curricula and they showed an interest in more knowledge and skills to help children with traumatic war experiences. When asked about positive reforms, teachers were highly critical of, and dissatisfied with, the educational situation. Bosnia and Herzegovina is undergoing a transition from a state-planned economy and one party system to a market economy and a multi party system. During this transition, the country has become more involved in the globalization process than ever. Today the country is a semi-protectorate where international authorities intervene when necessary. The International community is attempting to introduce western democracy and some of the many complexities in this process are discussed in this study. Globalization processes imply contradictory demands and pressures on the education system. On one hand, economic liberalization has affected education policies —a closer alignment between education and economic competitiveness. On the other hand, there is a political and ideological globalization process underlying the importance of human rights, and the inclusiveness of education for all children. Students and teachers are caught between two opposing ideals — competition and cooperation.