Search

An algebraic variety is called rationally connected if two generic points can be connected by a curve isomorphic to the projective line. The output of the minimal model program applied to rationally connected variety is variety admitting Mori fiber spaces over a rationally connected base. In this thesis we study the birational geometry of a particular class of rationally connected Mori fiber spaces: Fano fibrations over the projective line. We construct examples of Fano fibrations with a unique Mori fiber space in their birational classes. We prove that these examples are not birational to varieties of Fano type, thus answering the question of Cascini and Gongyo. That is we prove that the classes of rationally connected varieties and varieties of Fano type are not birationally equivalent. To construct the examples we use the techniques of birational rigidity. A Mori fiber space is called birationally rigid if there is a unique Mori fiber space structure in its birational class. The birational rigidity of smooth varieties admitting a del Pezzo fibration of degrees 1 and 2 is a well studied question. Unfortunately it is not enough to study smooth del Pezzo fibrations as there are fibrations which do not have smooth or even smoothable minimal models. In the case of fibrations of degree 2 we know that there is a minimal model with 2-Gorenstein singularities. These singularities are degenerations of the simplest terminal quotient singularity: singular points of the type 1/2(1,1,1). We give first examples of birationally rigid del Pezzo fibrations with 2-Gorenstein singularities. We then apply this result to study finite subgroups of the Cremona group of rank three. We then study the birational geometry of Fano fibrations from a different side. Using the reduction to characteristic 2 method we prove that double covers of Pn-bundles over Pm branched over a divisor of sufficiently high degree are not stably rational. For a del Pezzo fibration Y→P1 of degree 2 such that X is smooth there is a double cover Y→ X, where X is a P2-bundle over P1. In this case a stronger result holds: a very general Y with Pic(Y)≅Z⊕Z is not stably rational. We discuss the proof of this statement.

Today, the Gulf Cooperation Council (the GCC) is often praised to be the most successfully integrated economic/political union after the European Union. A considerable number of scholars devoted their works to this union. However, a large part of the analysis overlooks or does not pay enough attention to the fact that historically the GCC’s main purpose was not economic integration but maintaining regional security by means of a joint effort. Since its establishment in the year 1981 the GCC has faced several issues those became “litmus tests” for these efforts’ success. Before studying these issues closely, the thesis quickly analyzes the background of the union, examining the threats which brought it into being and the first patterns of maintaining regional security. Further on, it discusses the main conflicts and contradictions within the GCC, including the recent Qatari conflict. It continues by analyzing the condition of each country’s army (arms, doctrine, and competency) and evaluating the success of the GCC’s joint effort in Yemen. In this thesis, I argue that that due to the contradictions between the states, lack of flexibility in the mechanisms of coordination and relatively low effectiveness of very expensive armies, the purpose of maintaining regional security solely by means of the GCC has not been reached yet.

The broken ray transform (BRT) is an integral of a function along a union of two rays with a common vertex. Consider an X-ray beam scanning an object of interest. The ray undergoes attenuation and scatters in all directions inside the object. This phenomena may happen repeatedly until the photons either exit the object or are completely absorbed. In our work we assume the single scattering approximation when the intensity of the rays scattered more than once is negligibly small. Among all paths that the scattered rays travel inside the object we pick the one that is a union of two segments with one common scattering point. The intensity of the ray which traveled this path and exited the object can be measured by a collimated detector. The collimated detector is able to measure the intensity of X-rays from the selected direction. The logarithm of such a measurement is the broken ray transform of the attenuation coefficient plus the logarithm of the scattering coefficient at the scattering point (vertex) and a known function of the scattering angle. In this work we consider the reconstruction of X-ray attenuation coefficient distribution in a plane from the measurements on two or three collimated detector arrays. We derive an exact local reconstruction formula for three flat collimated detectors or three curved or pin-hole collimated detectors. We obtain a range condition for the case of three curved or pin-hole detectors and provide a special case of the range condition for three flat detectors. We generalize the reconstruction formula to four and more detectors and find an optimal set of parameters that minimize noise in the reconstruction. We introduce a more accurate scattering model which takes into account energy shifts due to the Compton effect, derive an exact reconstruction formula and develop an iterative reconstruction method for the energy-dependent case. To solve the problem we assume that the radiation source is monoenergetic and the dependence of the attenuation coefficient on energy is linear on an energy interval from the minimal to the maximal scattered energy. %initial radiation energy. We find the parameters of the linear dependence of the attenuation on energy as a function of a point in the reconstruction plane.