Lipschitz-free spaces F(M ) are canonical linearizations of arbitrary complete metric spaces M . More specifically, F(M ) is the unique Banach space that contains an isometric copy of M that is linearly dense, and such that any Lipschitz mapping from M into some Banach space X extends to a bounded linear operator from F(M ) into X. Those spaces are a very powerful tool for studies of the nonlinear geometry of Banach spaces, as they allow the application of well-known classical linear techniques to nonlinear problems. But this effort is only worthwhile if we have sufficient knowledge about the structure of F(M ). The systematic study of Lipschitz-free spaces is rather recent and so the current understanding of their structure is still quite limited. This thesis is framed within the general program of studying the structure of general Lipschitz-free spaces.We start our study by developing some basic tools for the general theory of Lipschitz-free spaces. First we introduce weighting operators and use them to solve Weaver's conjecture that all normal functionals in the bidual F(M ) * * are weak * continuous. Next we prove the intersection theorem, which essentially says that the intersection of Lipschitz-free spaces is again a Lipschitz-free space. That result allows us to develop the concept of support of an element of F(M ), analogous to the support of a measure. Furthermore, we extend the use of these tools to the bidual F(M ) * * and apply them to establish a decomposition of the bidual into spaces of functionals that are "concentrated at infinity" and "separated from infinity", respectively. * * que poden expressar-se com a diferència de dos elements positius, com ara l'existència d'un anàleg de la descomposició de Jordan per a mesures.En segon lloc, estudiem l'estructura extremal de la bola unitat de F(M ) i fem algunes contribucions al programa general consistent en trobar caracteritzacions purament geomètriques de tots els seus elements extremals. Concretament, caracteritzem els punts extrems preservats de la bola, així com aquells punts extrems i exposats que tenen suport finit. A més fem una descripció completa de l'estructura extremal de la part positiva de la bola unitat. La teoria dels suports en F(M ) desenvolupada anteriorment juga un paper crucial en les demostracions d'aquests resultats.viii

Positron Emission Tomography (PET) is a non-invasive nuclear medical imaging modality that makes it possible to observe the distribution of metabolic substances within a patient's body after marking them with radioactive isotopes and arranging an annular scanner around him in order to detect their decays. The main applications of this technique are the detection and tracing of tumors in cancer patients and metabolic studies with small animals.The Electronic Design for Nuclear Applications (EDNA) research group within the Instituto de Instrumentación para Imagen Molecular (I3M) has been involved in the study of high performance PET systems and maintains a small experimental setup with two detector modules. This thesis is framed within the necessity of developing a new data acquisition system (DAQ) for the aforementioned setup that corrects the drawbacks of the existing one. The main objective is to define a DAQ architecture that is completely scalable, modular, and guarantees the mobility and the possibility of reusing its components, so that it admits any extension of modification of the setup and it is possible to export it directly to the configurations used by other groups or experiments. At the same time, this architecture should be compatible with the best possible resolutions attainable at the present instead of imposing artificial limits on system performance. In particular, the new DAQ system should outperform the previous one.As a first step, a general study of DAQ architectures is carried out in the context of experimental setups for PET and other high energy physics applications. On one hand, the conclusion is reached that the desired specifications require early digitization of detector signals, exclusively digital communication between modules, and the absence of a centralized trigger. On the other hand, the necessity of a very precise distributed synchronization scheme between modules becomes apparent, with errors in the order of 100 ps, and operating directly over the data links. A study of the existing methods reveals their severe limitations in terms of achievable precision. A theoretical analysis of the situation is carried out with the goal of overcoming them, and a new synchronization algorithm is proposed that is able to reach the desired resolution while getting rid of the restrictions on clock iii alignment that are imposed by virtually all usual schemes. Since the measurement of clock phase difference plays a crucial role in the proposed algorithm, extensions to the existing methods are defined and analyzed that improve them significantly. The proposed scheme for synchronism is validated using commercial evaluation boards.Taking the proposed synchronization method as a starting point, a DAQ architecture for PET is defined that is composed of two types of module (acquisition and concentration) whose replication makes it possible to arrange a hierarchic system of arbitrary size, and circuit boards are designed and commissioned that implement a realization of the architecture for the...