Statistical Mechanics of Monomer-Dimer Models on Complete and Erdös-Rényi Graphs

Nella tesi sono trattate due famiglie di modelli meccanico statistici su vari grafi: i modelli di spin ferromagnetici (o di Ising) e i modelli di monomero-dimero. Il primo capitolo è dedicato principalmente allo studio del lavoro di Dembo e Montanari, in cui viene risolto il modello di Ising su grafi aleatori. Nel secondo capitolo vengono studiati i modelli di monomero-dimero, a partire dal lavoro di Heilemann e Lieb,con l'intento di dare contributi nuovi alla teoria. I principali temi trattati sono disuguaglianze di correlazione, soluzioni esatte su alcuni grafi ad albero e sul grafo completo, la concentrazione dell'energia libera intorno al proprio valor medio sul grafo aleatorio diluito di Erdös-Rényi.

Non linear response of planar asymmetric systems

The seismic behaviour of one-storey asymmetric structures has been studied since 1970s by a number of researches studies which identified the coupled nature of the translational-to-torsional response of those class of systems leading to severe displacement magnifications at the perimeter frames and therefore to significant increase of local peak seismic demand to the structural elements with respect to those of equivalent not-eccentric systems (Kan and Chopra 1987). These studies identified the fundamental parameters (such as the fundamental period TL normalized eccentricity e and the torsional-to-lateral frequency ratio Ωϑ) governing the torsional behavior of in-plan asymmetric structures and trends of behavior. It has been clearly recognized that asymmetric structures characterized by Ωϑ >1, referred to as torsionally-stiff systems, behave quite different form structures with Ωϑ <1, referred to as torsionally-flexible systems. Previous research works by some of the authors proposed a simple closed-form estimation of the maximum torsional response of one-storey elastic systems (Trombetti et al. 2005 and Palermo et al. 2010) leading to the so called “Alpha-method” for the evaluation of the displacement magnification factors at the corner sides. The present paper provides an upgrade of the “Alpha Method” removing the assumption of linear elastic response of the system. The main objective is to evaluate how the excursion of the structural elements in the inelastic field (due to the reaching of yield strength) affects the displacement demand of one-storey in-plan asymmetric structures. The system proposed by Chopra and Goel in 2007, which is claimed to be able to capture the main features of the non-linear response of in-plan asymmetric system, is used to perform a large parametric analysis varying all the fundamental parameters of the system, including the inelastic demand by varying the force reduction factor from 2 to 5. Magnification factors for different force reduction factor are proposed and comparisons with the results obtained from linear analysis are provided.

Laplace operator on finite graphs and a network diffusion model for the progression of the Alzheimer disease

Nella tesi viene descritto il Network Diffusion Model, ovvero il modello di A. Ray, A. Kuceyeski, M. Weiner inerente i meccanismi di progressione della demenza senile. In tale modello si approssima l'encefalo sano con una rete cerebrale (ovvero un grafo pesato), si identifica un generale fattore di malattia e se ne analizza la propagazione che avviene secondo meccanismi analoghi a quelli di un'infezione da prioni. La progressione del fattore di malattia e le conseguenze macroscopiche di tale processo(tra cui principalmente l'atrofia corticale) vengono, poi, descritte mediante approccio matematico. I risultati teoretici vengono confrontati con quanto osservato sperimentalmente in pazienti affetti da demenza senile. Nella tesi, inoltre, si fornisce una panoramica sui recenti studi inerenti i processi neurodegenerativi e si costruisce il contesto matematico di riferimento del modello preso in esame. Si presenta una panoramica sui grafi finiti, si introduce l'operatore di Laplace sui grafi e si forniscono stime dall'alto e dal basso per gli autovalori. Al fine di costruire una cornice matematica completa si analizza la relazione tra caso discreto e continuo: viene descritto l'operatore di Laplace-Beltrami sulle varietà riemanniane compatte e vengono fornite stime dall'alto per gli autovalori dell'operatore di Laplace-Beltrami associato a tali varietà a partire dalle stime dall'alto per gli autovalori del laplaciano sui grafi finiti.

Simplicial Complexes From Graphs Toward Graph Persistence

Persistent homology is a branch of computational topology which uses geometry and topology for shape description and analysis. This dissertation is an introductory study to link persistent homology and graph theory, the connection being represented by various methods to build simplicial complexes from a graph. The methods we consider are the complex of cliques, of independent sets, of neighbours, of enclaveless sets and complexes from acyclic subgraphs, each revealing several properties of the underlying graph. Moreover, we apply the core ideas of persistence theory in the new context of graph theory, we define the persistent block number and the persistent edge-block number.

Edge-aware Graph Attention Networks: Joint Reasoning On Text and Knowledge Graphs for Biomedical Question Answering

Much of the real-world dataset, including textual data, can be represented using graph structures. The use of graphs to represent textual data has many advantages, mainly related to maintaining a more significant amount of information, such as the relationships between words and their types. In recent years, many neural network architectures have been proposed to deal with tasks on graphs. Many of them consider only node features, ignoring or not giving the proper relevance to relationships between them. However, in many node classification tasks, they play a fundamental role. This thesis aims to analyze the main GNNs, evaluate their advantages and disadvantages, propose an innovative solution considered as an extension of GAT, and apply them to a case study in the biomedical field. We propose the reference GNNs, implemented with methodologies later analyzed, and then applied to a question answering system in the biomedical field as a replacement for the pre-existing GNN. We attempt to obtain better results by using models that can accept as input both node and edge features. As shown later, our proposed models can beat the original solution and define the state-of-the-art for the task under analysis.

Experimental characterization of a planar jet by hot-wire anemometry

The study of turbulence is also nowadays a problem that does not have solution from the mathematical point of view due to the lack of solution to link the mean part of the flow with the fluctuating one. To solve this problem, in the CICLoPE laboratory of Predappio, experiments on different type of jets are performed in order to derive a closure model able to close our mathematical model. One of the most interesting type of jet that could be studied is the planar turbulent free jet which is a two dimensional canonical jet characterized by the self-similarity condition of the velocity profiles. To study this particular jet, a new facility was built. The aim of this project is to characterize the jet at different distances from the nozzle exit, for different values of Reynolds number, to demonstrate that the self-similarity condition is respected. To do that, the evaluation of quantities such as spreading rate, centerline velocity decay and relation between fluctuations and mean part of the flow has to be obtain. All these parameters could be detected thanks to the use of single and X hot-wire anemometry with which it is possible to analyzed the fluctuating behaviour of the flow by associating to an electric signal a physical variable expressed in terms of velocity. To justify the data obtain by the measures, a comparison with results coming from the literature has to be shown.

Quantum Riemannian geometry on graphs for gauge field theory

In questo lavoro estendiamo concetti classici della geometria Riemanniana al fine di risolvere le equazioni di Maxwell sul gruppo delle permutazioni $S_3$. Cominciamo dando la strutture algebriche di base e la definizione di calcolo differenziale quantico con le principali proprietà. Generalizziamo poi concetti della geometria Riemanniana, quali la metrica e l'algebra esterna, al caso quantico. Tutto ciò viene poi applicato ai grafi dando la forma esplicita del calcolo differenziale quantico su $\mathbb{K}(V)$, della metrica e Laplaciano del secondo ordine e infine dell'algebra esterna. A questo punto, riscriviamo le equazioni di Maxwell in forma geometrica compatta usando gli operatori e concetti della geometria differenziale su varietà che abbiamo generalizzato in precedenza. In questo modo, considerando l'elettromagnetismo come teoria di gauge, possiamo risolvere le equazioni di Maxwell su gruppi finiti oltre che su varietà differenziabili. In particolare, noi le risolviamo su $S_3$.

Development and Automatization of a Planar Jet Wind Tunnel for the X-wire Calibration

The scope of this study is to design an automatic control system and create an automatic x-wire calibrator for a facility named Plane Air Tunnel; whose exit creates planar jet flow. The controlling power state as well as automatic speed adjustment of the inverter has been achieved. Thus, the wind tunnel can be run with respect to any desired speed and the x-wire can automatically be calibrated at that speed. To achieve that, VI programming using the LabView environment was learned, to acquire the pressure and temperature, and to calculate the velocity based on the acquisition data thanks to a pitot-static tube. Furthermore, communication with the inverter to give the commands for power on/off and speed control was also done using the LabView VI coding environment. The connection of the computer to the inverter was achieved by the proper cabling using DAQmx Analog/Digital (A/D) input/output (I/O). Moreover, the pressure profile along the streamwise direction of the plane air tunnel was studied. Pressure tappings and a multichannel pressure scanner were used to acquire the pressure values at different locations. Thanks to that, the aerodynamic efficiency of the contraction ratio was observed, and the pressure behavior was related to the velocity at the exit section. Furthermore, the control of the speed was accomplished by implementing a closed-loop PI controller on the LabView environment with and without using a pitot-static tube thanks to the pressure behavior information. The responses of the two controllers were analyzed and commented on by giving suggestions. In addition, hot wire experiments were performed to calibrate automatically and investigate the velocity profile of a turbulent planar jet. To be able to analyze the results, the physics of turbulent planar jet flow was studied. The fundamental terms, the methods used in the derivation of the equations, velocity profile, shear stress behavior, and the effect of vorticity were reviewed.

Towards the kinetic resolution of mechanically planar chiral rotaxanes

In this work, we reported the synthesis and characterization of two [2]rotaxanes endowed with a central ammonium group and two triazolium recognition stations on either side, acting as complexation sites for a dibenzo-24-crown-8 ether macrocycle. These mechanically interlocked architectures were obtained through the interlocking of a functionalized achiral macrocycle with Cs symmetry (where the symmetry element is a mirror plane corresponding to plane of the ring) and a C∞v symmetric axle (where a mirror plane and a C∞ principal axis are aligned along the axle length). We took advantage of the reversible acid/base triggered molecular shuttling of the ring between two lateral triazolium units to switch the rotaxanes between prochiral and mechanically planar chiral forms, which exists as two rapidly-interconverting co-conformers. We exploited the reactivity of the central amino group to attach an optically pure chiral substituent, with the goal of demonstrating the enantiomeric nature of the co-conformers and to obtain a non-zero diastereomeric excess in the resulting diastereomeric products through a dynamic kinetic resolution. To this end, two enantiopure reagents were chosen that could perform clean and fast reaction with amines: a sulfonyl chloride and an acyl chloride. Only the acyl chloride successfully produced an amide in high yield with the deprotonated rotaxane. The group added to the central amine station acted as a stopper against the shuttling of the macrocycle along the axis, thus preventing the fast interconversion of the two mechanically planar enantiomers. We analysed the results through static and dynamic NMR spectroscopic techniques by varying temperature and solvent used. Indeed, the presence of diastereomers was recorded alongside the configurational isomers resulting from the slow rotation of the CN-CO bond of the amide moiety, thus paving the way for a dynamic kinetic resolution.

Bio-QA-GNN: Reasoning with Language Models and Knowledge Graphs for Interpretable Biomedical Question Answering

Nowadays the idea of injecting world or domain-specific structured knowledge into pre-trained language models (PLMs) is becoming an increasingly popular approach for solving problems such as biases, hallucinations, huge architectural sizes, and explainability lack—critical for real-world natural language processing applications in sensitive fields like bioinformatics. One recent work that has garnered much attention in Neuro-symbolic AI is QA-GNN, an end-to-end model for multiple-choice open-domain question answering (MCOQA) tasks via interpretable text-graph reasoning. Unlike previous publications, QA-GNN mutually informs PLMs and graph neural networks (GNNs) on top of relevant facts retrieved from knowledge graphs (KGs). However, taking a more holistic view, existing PLM+KG contributions mainly consider commonsense benchmarks and ignore or shallowly analyze performances on biomedical datasets. This thesis start from a propose of a deep investigation of QA-GNN for biomedicine, comparing existing or brand-new PLMs, KGs, edge-aware GNNs, preprocessing techniques, and initialization strategies. By combining the insights emerged in DISI's research, we introduce Bio-QA-GNN that include a KG. Working with this part has led to an improvement in state-of-the-art of MCOQA model on biomedical/clinical text, largely outperforming the original one (+3.63\% accuracy on MedQA). Our findings also contribute to a better understanding of the explanation degree allowed by joint text-graph reasoning architectures and their effectiveness on different medical subjects and reasoning types. Codes, models, datasets, and demos to reproduce the results are freely available at: \url{https://github.com/disi-unibo-nlp/bio-qagnn}.