On regular ultrafilters, Boolean ultrapowers, and Keisler's order

In this paper we analyse and compare two different notions of regularity for filters on complete Boolean algebras. We also announce two results from a forthcoming paper in preparation, which provide a characterization of Keisler's order in terms of Boolean ultrapowers

Boolean Valued Models, Saturation, Forcing Axioms

This dissertation will focus on Boolean-valued models, giving some insight into the theory of Boolean ultrapowers, and developing the connection with forcing axioms and absoluteness results. This study will be divided into three chapters. The first chapter provides the basic material to understand the subsequent work. Boolean-valued models are well known in set theory for independence results and the development of forcing. In the second chapter of this dissertation, Boolean-valued models are studied from a general point of view. In particular, we give the definition of B-valued model for an arbitrary first-order signature, and we study Boolean ultrapowers as a general model-theoretic technique. A more ambitious third chapter develops the connection with forcing axioms and absoluteness results. From a philosophical point of view, forcing axioms are very appealing. Not only do they imply that the Continuum Hypothesis is false, but also they are particularly successful in deciding many independent statements in mathematics. First, we give an interesting formulation of bounded forcing axioms in terms of absoluteness. Furthermore, we prove that the Axiom of Choice is a “global” forcing axiom, and that also some large cardinal axioms are in fact natural generalizations of forcing axioms

Leggi zero-uno per grafi random e tecniche di amalgamazione

Lo studio dei grafi random, iniziato nel 1960 con un fondamentale articolo di Erdős e Rényi, si è dimostrato un terreno di ricerca molto fertile. Il grafo random G(n, p) è un grafo su n vertici ottenuto congiungendo ciascuna coppia di vertici indipendentemente con probabilità p. Diremo che una successione p: ω → [0,1] soddisfa la legge zero-uno se, per ogni enunciato del primo ordine nel linguaggio dei grafi σ, la probabilità che G(n, p(n)) soddisfi σ tende a 0 oppure a 1 quando n → ∞. I risultati più significativi in questo ambito sono la legge zero-uno di Fagin (1976) e quella di Shelah e Spencer (1988). D’altra parte, le tecniche di amalgamazione hanno creato un gran numero di esempi e controesempi in teoria dei modelli. L’idea è di produrre, a partire da opportune classi di strutture finite, una “struttura limite” numerabile. La costruzione originale, dovuta a Fraïssé, è stata in seguito modificata e generalizzata da Hrushovski e molti altri. Nel 1997, Baldwin e Shelah uniscono queste due linee di ricerca a prima vista incorrelate. Motivati da questo fatto, presenteremo le tecniche di amalgamazione di Fraïssé e di Hrushovski, dimostreremo le due leggi zero-uno enunciate sopra e soprattutto metteremo in evidenza l’interazione tra questi due mondi

Orderings of ultrafilters on Boolean algebras

We study two generalizations of the Rudin-Keisler ordering to ultrafilters on complete Boolean algebras. To highlight the difference between them, we develop new techniques to construct incomparable ultrafilters in this setting. Furthermore, we discuss the relation with Tukey reducibility and prove that, assuming the Continuum Hypothesis, there exist ultrafilters on the Cohen algebra which are RK-equivalent in the generalized sense but Tukey-incomparable, in stark contrast with the classical setting

Universality properties of forcing

The purpose of this paper is to investigate forcing as a tool to construct universal models. In particular, we look at theories of initial segments of the universe and show that any model of a sufficiently rich fragment of those theories can be embedded into a model constructed by forcing. Our results rely on the model-theoretic properties of good ultrafilters, for which we provide a new existence proof on non-necessarily complete Boolean algebras

Regularity of ultrafilters, Boolean ultrapowers, and Keisler’s order

This thesis investigates combinatorial properties of ultrafilters and their model-theoretic significance. Motivated by recent results on Keisler’s order, we develop new tools for the study of Boolean ultrapowers, deepening our understanding of the interplay between set theory and model theory. The main contributions can be summarized as follows. In Chapter 2, we undertake a systematic study of regular ultrafilters on Boolean algebras. In particular, we analyse two different notions of regularity which have appeared in the literature and compare their modeltheoretic properties. We then apply our analysis to the study of cofinal types of ultrafilters; as an application, we answer a question of Brown and Dobrinen by giving two examples of complete Boolean algebras on which all ultrafilters have maximum cofinal type. In conclusion, we discuss the existence of non-regular ultrafilters and prove that, consistently, every decomposable ultrafilter on a complete Boolean algebra is regular. Chapter 3 centres around the study of Keisler’s order. We prove that good ultrafilters on Boolean algebras are precisely the ones which capture the maximum class in Keisler’s order, solving a problem posed by Benda in 1974. We also show that, given a regular ultrafilter on a complete Boolean algebra satisfying a distributivity condition, the saturation of the Boolean ultrapower of a model of a complete theory does not depend on the choice of the particular model, but only on the theory itself. Motivated by this fact, we apply and expand the framework of ‘separation of variables’, recently developed by Malliaris and Shelah, to obtain a new characterization of Keisler’s order via Boolean ultrapowers

Green bean biofortification for Si through soilless cultivation: Plant response and Si bioaccessibility in pods

Food plants biofortification for micronutrients is a tool for the nutritional value improvement of food. Soilless cultivation systems, with the optimal control of plant nutrition, represent a potential effective technique to increase the beneficial element content in plant tissues. Silicon (Si), which proper intake is recently recommended for its beneficial effects on bone health, presents good absorption in intestinal tract from green bean, a high-value vegetable crop. In this study we aimed to obtain Si biofortified green bean pods by using a Si-enriched nutrient solution in soilless system conditions, and to assess the influence of boiling and steaming cooking methods on Si content, color parameters and Si bioaccessibility (by using an in vitro digestion process) of pods. The Si concentration of pods was almost tripled as a result of the biofortification process, while the overall crop performance was not negatively influenced. The Si content of biofortified pods was higher than unbiofortified also after cooking, despite the cooking method used. Silicon bioaccessibility in cooked pods was more than tripled as a result of biofortification, while the process did not affect the visual quality of the product. Our results demonstrated that soilless cultivation can be successfully used for green bean Si biofortification

Image-Based Monitoring of Cracks: Effectiveness Analysis of an Open-Source Machine Learning-Assisted Procedure

The proper inspection of a cracks pattern over time is a critical diagnosis step to provide a thorough knowledge of the health state of a structure. When monitoring cracks propagating on a planar surface, adopting a single-image-based approach is a more convenient (costly and logistically) solution compared to subjective operators-based solutions. Machine learning (ML)- based monitoring solutions offer the advantage of automation in crack detection; however, complex and time-consuming training must be carried out. This study presents a simple and automated ML-based crack monitoring approach implemented in open sources software that only requires a single image for training. The effectiveness of the approach is assessed conducting work in controlled and real case study sites. For both sites, the generated outputs are significant in terms of accuracy (~1 mm), repeatability (sub-mm) and precision (sub-pixel). The presented results highlight that the successful detection of cracks is achievable with only a straightforward ML-based training procedure conducted on only a single image of the multi-temporal sequence. Furthermore, the use of an innovative camera kit allowed exploiting automated acquisition and transmission fundamental for Internet of Things (IoTs) for structural health monitoring and to reduce user-based operations and increase safety

Fluid seepage evidence in the Adventure Plateau (NW Sicily Channel)

The northwestern region of the Sicilian Channel hosts a great number of morphological highs, the widest of which is the Adventure Plateau which is part of the Sicilian-Maghrebian Fold and Thrust Belt system, formed mainly during the Neogene. The complex morphology of the Adventure Plateau was shaped in the Lower Pliocene also by an extensional phase that produced high-angle normal faults mostly NW-SE to N-S oriented. Through these preferential pathways, magmatic fluids ascended and produced widespread volcanic manifestations. The interpretation of MBES (Multibeam Echosounder), seismic (sparker, airgun), well logs, and ROV (Remote Operative Vehicle) data identified several features related to the presence of fluids in the study area. The morpho-structural analysis showed an NW-SE-oriented fault system and a string of depressions that follows the same trend. The seismic-stratigraphic analysis highlighted various seismic signals located below the depressions, such as bright spots, enhanced reflection zones, and chimneys, which also suggest the presence of fluids that would rise to a few meters’ depth. Since all these fluids’ seepage evidence occur along the fault system, we consider this as the main pathway for vertical fluids migration. Further, to understand the fluid’s origin, a detailed well-log analysis has been performed; this showed the presence of oil traces, at a depth of 250 m, and gas (CO2) at a depth of 500 m. So, also considering the magmatic fluids evidence found by literature data, a possible mixed origin of the fluids, both biogenic and magmatic, should be assumed. Therefore, this work allowed us not only to identify new fluid seepage evidence in the Adventure Plateau but also to provide new insights into the identification of fluid leakage pathways