Absolute Hodge cycles and Deligne's principle b

I cicli di Hodge assoluti sono stati utilizzati da Deligne per dividere la congettura di Hodge in due sotto-congetture. La prima dice che tutte le classi di Hodge su una varietà complessa proiettiva liscia sono assolute, la seconda che le classi assolute sono algebriche. Deligne ha dato risposta affermativa alla prima sottocongettura nel caso delle varietà abeliane. La dimostrazione si basa su due teoremi, conosciuti rispettivamente come Principio A e Principio B. In questo lavoro vengono presentate la teoria delle classi di Hodge assolute e la dimostrazione del Principio B.

Regularization methods for the solution of a nonlinear least-squares problem in tomography

In this work we study a polyenergetic and multimaterial model for the breast image reconstruction in Digital Tomosynthesis, taking into consideration the variety of the materials forming the object and the polyenergetic nature of the X-rays beam. The modelling of the problem leads to the resolution of a high-dimensional nonlinear least-squares problem that, due to its nature of inverse ill-posed problem, needs some kind of regularization. We test two main classes of methods: the Levenberg-Marquardt method (together with the Conjugate Gradient method for the computation of the descent direction) and two limited-memory BFGS-like methods (L-BFGS). We perform some experiments for different values of the regularization parameter (constant or varying at each iteration), tolerances and stop conditions. Finally, we analyse the performance of the several methods comparing relative errors, iterations number, times and the qualities of the reconstructed images.

Atropisomeric xanthines: Synthesis, stereodynamics and absolute configuration

During the thesis period a new class of atropisomeric xanthine derivatives has been studied. We decided to focus our attention on these purine bases because of their various biological activities, that could play an important role in the discovery of new bioactive atropisomers. The synthesized compounds bear an Aryl-N chiral axis in position 1 of the xanthine scaffold, around which the rotation is prevented by the presence of bulky ortho substituents. Through a retro synthetic analysis we synthesized three atropisomeric structures bearing in position 1 of the purine scaffold respectively an o-tolyl, o-nitrophenyl and a 1-naphthyl group. The conformational studies by DFT simulations showed that the interconversion energy barrier between the two available skewed conformations is higher enough to obtain thermally stable atropisomers. After the separation of the atropisomers, the experimental energy of interconversion was investigated by means of kinetic studies following the thermal racemization process using an enantioselective HPLC column. The absolute configuration of each atropisomer was assigned by experimental ECD analysis and TD-DFT simulations of the ECD spectra.