It is currently a well-established fact that the dynamics of interacting fluid compartments of the central nervous system (CNS) may play a role in the CNS fluid physiology and pathology of a number of neurological disorders, including neurodegenerative diseases associated with accumulation of waste products in the brain. However, the mechanisms and routes of waste clearance from the brain are still unclear. One of the main components of this interacting cerebral fluids dynamics is blood flow. In the last decades, mathematical modeling and fluid dynamics simulations have become a valuable complementary tool to experimental approaches, contributing to a deeper understanding of the circulatory physiology and pathology. However, modeling blood flow in the brain remains a challenging and demanding task, due to the high complexity of cerebral vascular networks and the difficulties that consequently arise to describe and reproduce the blood flow dynamics in these vascular districts. The first part of this work is devoted to the development of efficient numerical strategies for blood flow simulations in complex vascular networks. In cardiovascular modeling, one-dimensional (1D) and lumped-parameter (0D) models of blood flow are nowadays well-established tools to predict flow patterns, pressure wave propagation and average velocities in vascular networks, with a good balance between accuracy and computational cost. Still, the purely 1D modeling of blood flow in complex and large networks can result in computationally expensive simulations, posing the need for extremely efficient numerical methods and solvers. To address these issues, we develop a novel modeling and computational framework to construct hybrid networks of coupled 1D and 0D vessels and to perform computationally efficient and accurate blood flow simulations in such networks. Starting from a 1D model and a family of nonlinear 0D models for blood flow, with either elastic or viscoelastic tube laws, this methodology is based on (i) suitable coupling equations ensuring conservation principles; (ii) efficient numerical methods and numerical coupling strategies to solve 1D, 0D and hybrid junctions of vessels; (iii) model selection criteria to construct hybrid networks, which provide a good trade-off between accuracy in the predicted results and computational cost of the simulations. By applying the proposed hybrid network solver to very complex and large vascular networks, we show how this methodology becomes crucial to gain computational efficiency when solving networks and models where the heterogeneity of spatial and/or temporal scales is relevant, still ensuring a good level of accuracy in the predicted results. Hence, the proposed hybrid network methodology represents a first step towards a high-performance modeling and computational framework to solve highly complex networks of 1D-0D vessels, where the complexity does not only depend on the anatomical detail by which a network is described, but also on the level at which physiological mechanisms and mechanical characteristics of the cardiovascular system are modeled. Then, in the second part of the thesis, we focus on the modeling and simulation of cerebral blood flow, with emphasis on the venous side. We develop a methodology that, departing from the high-resolution MRI data obtained from a novel in-vivo microvascular imaging technique of the human brain, allows to reconstruct detailed subject-specific cerebral networks of specific vascular districts which are suitable to perform blood flow simulations. First, we extract segmentations of cerebral districts of interest in a way that the arterio-venous separation is addressed and the continuity and connectivity of the vascular structures is ensured. Equipped with these segmentations, we propose an algorithm to extract a network of vessels suitable and good enough, i.e. with the necessary properties, to perform blood flow simulations. Here, we focus on the reconstruction of detailed venous vascular networks, given that the anatomy and patho-physiology of the venous circulation is of great interest from both clinical and modeling points of view. Then, after calibration and parametrization of the MRI-reconstructed venous networks, blood flow simulations are performed to validate the proposed methodology and assess the ability of such networks to predict physiologically reasonable results in the corresponding vascular territories. From the results obtained we conclude that this work represents a proof-of-concept study that demonstrates that it is possible to extract subject-specific cerebral networks from the novel high-resolution MRI data employed, setting the basis towards the definition of an effective processing pipeline for detailed blood flow simulations from subject-specific data, to explore and quantify cerebral blood flow dynamics, with focus on venous blood drainage.

The main bulk of this Thesis is focused on the response of cell membranes due to chemical and mechanical stimuli. Henceforth, it is mainly devoted to deduce how the key aspect of the cell response activated by chemical signaling can be predicted by a simplified energetics, making use of both theoretical models and numerical simulations. The a ention is focused on cell membranes embedding G protein-coupled receptors (GPRCs). By analyzing the behavior of cell mem- branes, one can isolate three main contributions in order to model their respon- se: (1) diffusion of receptors and transporters embedded in the lipid membrane; (2) conformational changes of the receptors; (3) membrane elasticity. Moreover, the interplay between TM confomational changes and lateral pressure of the lipid membrane against such TMs is introduced. The chemical potential of the receptor-ligand compound, deduced as the variational derivative of such energy, is compared with the one calculated by accounting for the work done by the lateral pressure. The result yields a relationship between the conformational field, the mechanical field (interpreted as either the thickness change or the areal change) and the distribution of the compounds receptor-ligand. The analysis of such resulting constitutive equation among those three quantities shows that, essentially, the reason why ligand-GPRCs compounds prefer to live on lipid ra is a necessity involving the interplay between the work performed by the lateral pressure and the need of TMs to change their conformation during ligand binding. Henceforth, mechanobiology gives a justification to the experimental findings of Kobilka and Lei ovitz, Chemistry Nobel Prizes 2012.

In this Thesis, we analyze three variational and geometric problems, that extend classical Euclidean issues of the calculus of variations to more general classes of spaces. The results we outline are based on the articles [Ved21; MV21] and on a forthcoming joint work with Nicolussi Golo and Serra Cassano. In the first place, in Chapter 1 we provide a general introduction to metric measure spaces and some of their properties. In Chapter 2 we extend the classical Talenti’s comparison theorem for elliptic equations to the setting of RCD(K,N) spaces: in addition the the generalization of Talenti’s inequality, we will prove that the result is rigid, in the sense that equality forces the space to have a symmetric structure, and stable. Chapter 3 is devoted to the study of the Bernstein problem for intrinsic graphs in the first Heisenberg group H^1: we will show that under mild assumptions on the regularity any stationary and stable solution to the minimal surface equation needs to be intrinsically affine. Finally, in Chapter 4 we study the dimension and structure of the singular set for p-harmonic maps taking values in a Riemannian manifold.

The first aim of this PhD Thesis is devoted to the study of geodesic distances defined on a subdomain of a Carnot group, which are bounded both from above and from below by fixed multiples of the Carnot–Carathéodory distance. Then one shows that the uniform convergence, on compact sets, of these distances can be equivalently characterized in terms of Gamma-convergence of several kinds of variational problems. Moreover, it investigates the relation between the class of intrinsic distances, their metric derivatives and the sub-Finsler convex metrics defined on the horizontal bundle. The second purpose is to obtain the integral representation of some classes of local functionals, depending on a family of vector fields, that satisfy a weak structure assumption. These functionals are defined on degenerate Sobolev spaces and they are assumed to be not translations-invariant. Then one proves some Gamma-compactness results with respect to both the strong topology of L^p and the strong topology of degenerate Sobolev spaces.

In this Thesis we explore, both theoretically and numerically, splitting strategies for a hyperbolic system of one-dimensional (1D) blood flow equations with a passive scalar transport equation. Our analysis involves a two-step framework that includes splitting at the level of partial differential equations (PDEs) and numerical methods for discretizing the ensuing problems. This study is inspired by the original flux splitting approach of Toro and Vázquez-Cendón (2012) originally developed for the conservative Euler equations of compressible gas dynamics. In this approach the flux vector in the conservative case, and the system matrix in the non-conservative one, are split into advection and pressure terms: in this way, two systems of partial differential equations are obtained, the advection system and the pressure system. From the mathematical as well as numerical point of view, a basic problem to be solved is the special Cauchy problem called the Riemann problem. This latter provides an analytical solution to evaluate the performance of the numerical methods and, in our approach, it is of primary importance to build the presented numerical schemes. In the first part of the Thesis a detailed theoretical analysis is presented, involving the exact solution of the Riemann problem for the 1D blood flow equations, depicted for a general constant momentum correction coefficient and a tube law that allows to describe both arteries and veins with continuous or discontinuous mechanical and geometrical properties and an advection equation for a passive scalar transport. In literature, this topic has been already studied only for a momentum correction coefficient equal to one, that is related to the prescribed velocity profile and in this case corresponds to a flat one, i.e. an inviscid fluid. In the case of discontinuous properties, only the subsonic regime is considered. In addition we propose a procedure to compute the obtained exact solution and finally we validate it numerically, by comparing exact solutions to those obtained with well-known, numerical schemes on a carefully designed set of test problems. Furthermore, an analogous theoretical analysis and resolution algorithm are presented for the advection system and the pressure system arising from the splitting at the level of PDEs of the complete system of 1D blood flow equations. It is worth noting that the pressure system, in case of veins, presents a loss of genuine non-linearity resulting in the formation of rarefactions, shocks and compound waves, these latter being a composition of rarefactions and shocks. In the second part of the Thesis we present novel finite volume-type, flux splitting-based, numerical schemes for the conservative 1D blood flow equations and splitting-based numerical schemes for the non-conservative 1D blood flow equations that incorporate an advection equation for a passive scalar transport, considering tube laws that allow to model blood flow in arteries and veins and take into account a general constant momentum correction coefficient. A detailed efficiency analysis is performed in order to showcase the advantages of the proposed methodologies in comparison to standard approaches.

La presente tesi studia la nozione di produttività dal punto di vista Classico. In primo luogo, si connette la distinzione tra produttività (productivity) e profittabilità (productiveness) a quella tra il lato della spesa e quello del valore aggiunto dell'economia, vista come un flusso circolare. In secondo luogo, si collegano vari schemi teorici alle strutture empiriche del sistema di contabilità nazionale. Si calcolano quindi sia degli indicatori dei cambiamenti della produttività fisica, utilizzando come unità dell'analisi i subsistemi in crescita, che delle misure del grado di capacità delle singole industrie di generare sovrappiú. Si ottengono ed utilizzano regole di aggregazione e procedure di riduzione al fine di tenere correttamente conto dell'eterogeneità dei mezzi di produzione prodotti. In tutta la tesi, i risultati analitici ottenuti sono corredati da applicazioni empiriche. In larga misura, tale lavoro empirico concerne l'economia italiana (1999-2007); tuttavia, alcuni risultati riguardano un insieme di paesi industrializzati (Germania, Francia, Italia, Giappone, GB e USA) nel decennio 1995-2005. / This is a study on the notion of productivity, viewed from a Classical perspective. First, the distinction between physical productivity and productiveness (i.e. profitability) is connected to the distinction between the expenditure side and value added side of the economy, seen as a circular flow. Second, a mapping of some theoretical frameworks into empirical structures of the System of National Accounts is advanced. Then, indicators of physical productivity changes with the (growing) subsystem as a unit of analysis are obtained, together with measures reflecting the degree of surplus generating capacity at the level of individual industries. Aggregation rules and reduction procedures are devised and applied to deal with the heterogeneous nature of produced means of production. All throughout the study, empirical applications of the analytical results are provided. For the most part, empirical work is referred to the case of Italy (1999-2007), though some results concern a set of advanced industrial economies (Germany, France, Italy, Japan, UK and the US) during the 1995-2005 decade.

Nell’area oggi occupata dall’Università Cattolica del Sacro Cuore di Milano a partire dal 1986 scavi di emergenza hanno messo in luce una porzione del suburbio occidentale di Mediolanum, e in particolare una realtà insediativa suburbana di età imperiale. Un’ampia area di tale insediamento è stata oggetto della presente ricerca che, combinando lo studio dei processi formativi del deposito archeologico e l’analisi quantitativa e statistica degli insiemi di reperti, ha permesso di ricostruire le trasformazioni che ne hanno preceduto e accompagnato la formazione e che ne hanno interessato l’organizzazione spaziale e la distribuzione delle aree di attività. In questo modo è stato possibile comprendere come nel tempo siano avvenuti cambiamenti nei caratteri e nelle finalità della frequentazione dell'area, che sembrano poter gettare nuova luce anche su fenomeni più generali che riguardano l’intero suburbio della città romana. / In the area today occupied by the Università Cattolica del Sacro Cuore di Milano, since 1986, rescue excavations have disclosed a part of Mediolanum’s western suburbium, and in particular an imperial age suburban settlement. A wide area of this settlement has been the object of this research which, combining the study of the formation processes of the archaeological record and the quantitative and statistical analysis of finds assemblages, allowed us to reconstruct the transformations which preceded its formation and those which concerned its spatial organization and activity areas distribution. In this way we have shown that the characteristics and nature of the settlement have considerably changed during the course of time, and these changes seem to shed light even on the general transformations which concern the entire suburbium of the Roman town.

Lo studio esamina la variazione lessicale regionale in forma scritta nell’inglese standard in California. Attraverso ricerche in rete limitate a 336 siti di giornali online con sede in 270 città in California, vengono raccolti i valori di 45 variabili continue di alternanze lessicali e quindi calcolati come proporzioni. Tecniche statistiche di autocorrelazione spaziale globale e locale analizzano i valori. I risultati delle analisi, riportati in 90 mappe, confermano la distribuzione regionale delle variabili in California. Le 45 variabili lessicali sono poi esaminate con tecniche statistiche multivariate per individuare le relazioni linguistiche tra le città della California esaminate. L’analisi fattoriale, che rappresenta il 50,5% della variazione nei dati, evidenzia tre aree nella distribuzione regionale lessicale: nord/sud, urbano/rurale, e aree centrali e basso meridionali/aree alto meridionali e del nord. L’analisi dei cluster gerarchica distingue inoltre sei regioni dialettali principali in California: quella del Nord, quella di Sacramento-Santa Cruz, quella della San Francisco Bay Area, quella centrale, quella alto meridionale, e quella basso meridionale. Cinque mappe multivariate sono fornite nella tesi. La spiegazione dei risultati si basa sia su modelli di insediamento storico che su una spiegazione socio-culturale, che si riflettono nel linguaggio in California. / The study examines regional lexical variation in written Standard California English. The values​of 45 continuous lexical alternation variables are gathered through site-restricted web searches in 336 online newspaper websites based in 270 locations in California and then calculated as proportions. Statistical techniques analyze global and local spatial autocorrelation values. The results of the analysis, reported in 90 maps, confirm the regional distribution of the variables in California. The 45 lexical variables are then analyzed with multivariate techniques to identify the linguistic relations between the surveyed California cities. Factor analysis, which accounts for 50.5% of the variation in the data, highlights three areas in the regional lexical distribution: north/south, urban/rural, central and lower southern/upper southern and northern areas. The hierarchical cluster analysis also distinguishes six major dialect regions in California: the North dialect region, the Sacramento-Santa Cruz dialect region, the San Francisco Bay Area dialect region, the Central dialect region, the Upper Southerns dialect region, and the Lower Southern dialect region. Five multivariate maps are provided in the thesis. The explanation of the results is based both on historical settlement patterns and on a socio-cultural explanation, which are reflected in the language in California.

La Valle d'Aosta, pur non avendo grandi superfici destinate alla viticoltura, è caratterizzata dalla presenza di un ricco patrimonio ampelografico di cui fanno parte dodici varietà autoctone a bacca rossa: Bonda, Cornalin, Crovassa, Fumin, Mayolet, Ner d'Ala, Petit rouge, Premetta, Roussin, Roussin de Morgex, Vien de Nus e Vuillermin. Malgrado i numerosi vitigni menzionati, nella produzione dei vini a D.O.C troviamo in prevalenza il Petit rouge e poche altre varietà che, nell'insieme, rappresentano circa il 34% della produzione totale. Al fine di implementare la presenza di prodotti tipici che esprimano i caratteri distintivi della zona di provenienza sono state approfondite le conoscenze dei vitigni autoctoni per evidenziarne le attitudini e le potenzialità per la produzione di vini rossi di qualità. Allo stesso scopo sono stati valutati gli effetti dell'applicazione di alcune tecniche di vinificazione alternative sull'espressione dei caratteri qualitativi del Petit rouge, le cui uve sono da tempo vinificate in purezza o in assemblaggi nella produzione di vini a D.O.C. / Aosta Valley has a little viticulture surface and it is characterised by a rich ampelographic collection including the following twelve red grape autochthonous varieties: Bonda, Cornalin, Crovassa, Fumin, Mayolet, Ner d'Ala, Petit rouge, Premetta, Roussin, Roussin de Morgex, Vien de Nus e Vuillermin. Only the 34% of the D.O.C. wine production is represented by some of the above mentioned cultivars, in particular the Petit Rouge, despite of the huge number of cultivars. The Petit rouge is used in purity or in miscellaneous in the D.O.C. wine production. In order to implement the presence of typical products, this study was carried out to evaluate the aptitude of Aosta Valley ancient grapes to produce quality red wines. Moreover to improve quality of the most diffused Aosta Valley red wine the effects of some alternative winemaking were evaluated on the Petit rouge characters.

La seguente tesi e’ costituita da tre diversi elaborati, il primo e’ l’estensione di un modello di equilibrio generale a due regioni (Benigno JIE 2004) con l’intento di calcolare i pesi ottimali per l’inflazione dell’area euro utilizzando micro dati sull’eterogeneità delle rigidità dei prezzi in Europa. Il secondo e il terzo elaborato si focalizzano invece sulle crisi d’insolvenza con l’obiettivo di selezionare le variabili che forniscono maggiori informazioni per la previsione della crisi. La metodologia utilizzata e’ l’analisi della transvariazione. Mentre il secondo ‘essay’ si concentra sulla versione univariata, il terzo estende la metodologia al caso multivariato. Il primo analizza le crisi d’insolvenza più severe degli anni ’90, la seconda utilizza invece gli episodi di crisi analizzati da Frankel e Rose (1996). / The present work is made of three different essays, the first is an extension of a two region general equilibrium model (Benigno JIE 2004), with the intent of calculating optimal weights for EU inflation using micro data on the level of price rigidities, the second and the third have as main focus financial and currency country crises, with the task of selecting the most important variables in terms of crisis prediction by means of a descriptive statistics methodology called transvariation analysis. While the second essay focuses on univariate transvariation, the third extends the methodology to a multivariate framework. The last two essays are based on two different datasets. The first studies the most recent deep financial crises of the 1990s and the source is IMF, International Financial Statistics, the second uses a vast sample of currency crisis episodes taken from Frankel and Rose (1996) dataset made of annual data on more than one hundred developed countries from 1971 through 1992 and defining currency crash as a large change of nominal exchange rate that is also a substantial increase in the rate of change of nominal depreciation. The source in this case is World Bank, World Development Indicators.