Quantum Integrability in Non-Linear Sigma Models related to Gauge/String Correspondences

The Thermodynamic Bethe Ansatz analysis is carried out for the extended-CP^N class of integrable 2-dimensional Non-Linear Sigma Models related to the low energy limit of the AdS_4xCP^3 type IIA superstring theory. The principal aim of this program is to obtain further non-perturbative consistency check to the S-matrix proposed to describe the scattering processes between the fundamental excitations of the theory by analyzing the structure of the Renormalization Group flow. As a noteworthy byproduct we eventually obtain a novel class of TBA models which fits in the known classification but with several important differences. The TBA framework allows the evaluation of some exact quantities related to the conformal UV limit of the model: effective central charge, conformal dimension of the perturbing operator and field content of the underlying CFT. The knowledge of this physical quantities has led to the possibility of conjecturing a perturbed CFT realization of the integrable models in terms of coset Kac-Moody CFT. The set of numerical tools and programs developed ad hoc to solve the problem at hand is also discussed in some detail with references to the code.

Non-Linear Pricing in Imperfectly Competitive Markets

This thesis is dedicated to the analysis of non-linear pricing in oligopoly. Non-linear pricing is a fairly predominant practice in most real markets, mostly characterized by some amount of competition. The sophistication of pricing practices has increased in the latest decades due to the technological advances that have allowed companies to gather more and more data on consumers preferences. The first essay of the thesis highlights the main characteristics of oligopolistic non-linear pricing. Non-linear pricing is a special case of price discrimination. The theory of price discrimination has to be modified in presence of oligopoly: in particular, a crucial role is played by the competitive externality that implies that product differentiation is closely related to the possibility of discriminating. The essay reviews the theory of competitive non-linear pricing by starting from its foundations, mechanism design under common agency. The different approaches to model non-linear pricing are then reviewed. In particular, the difference between price and quantity competition is highlighted. Finally, the close link between non-linear pricing and the recent developments in the theory of vertical differentiation is explored. The second essay shows how the effects of non-linear pricing are determined by the relationship between the demand and the technological structure of the market. The chapter focuses on a model in which firms supply a homogeneous product in two different sizes. Information about consumers' reservation prices is incomplete and the production technology is characterized by size economies. The model provides insights on the size of the products that one finds in the market. Four equilibrium regions are identified depending on the relative intensity of size economies with respect to consumers' evaluation of the good. Regions for which the product is supplied in a single unit or in several different sizes or in only a very large one. Both the private and social desirability of non-linear pricing varies across different equilibrium regions. The third essay considers the broadband internet market. Non discriminatory issues seem the core of the recent debate on the opportunity or not of regulating the internet. One of the main questions posed is whether the telecom companies, owning the networks constituting the internet, should be allowed to offer quality-contingent contracts to content providers. The aim of this essay is to analyze the issue through a stylized two-sided market model of the web that highlights the effects of such a discrimination over quality, prices and participation to the internet of providers and final users. An overall welfare comparison is proposed, concluding that the final effects of regulation crucially depend on both the technology and preferences of agents.

Application-oriented Mixed Integer Non-Linear Programming

In the most recent years there is a renovate interest for Mixed Integer Non-Linear Programming (MINLP) problems. This can be explained for different reasons: (i) the performance of solvers handling non-linear constraints was largely improved; (ii) the awareness that most of the applications from the real-world can be modeled as an MINLP problem; (iii) the challenging nature of this very general class of problems. It is well-known that MINLP problems are NP-hard because they are the generalization of MILP problems, which are NP-hard themselves. However, MINLPs are, in general, also hard to solve in practice. We address to non-convex MINLPs, i.e. having non-convex continuous relaxations: the presence of non-convexities in the model makes these problems usually even harder to solve. The aim of this Ph.D. thesis is to give a flavor of different possible approaches that one can study to attack MINLP problems with non-convexities, with a special attention to real-world problems. In Part 1 of the thesis we introduce the problem and present three special cases of general MINLPs and the most common methods used to solve them. These techniques play a fundamental role in the resolution of general MINLP problems. Then we describe algorithms addressing general MINLPs. Parts 2 and 3 contain the main contributions of the Ph.D. thesis. In particular, in Part 2 four different methods aimed at solving different classes of MINLP problems are presented. Part 3 of the thesis is devoted to real-world applications: two different problems and approaches to MINLPs are presented, namely Scheduling and Unit Commitment for Hydro-Plants and Water Network Design problems. The results show that each of these different methods has advantages and disadvantages. Thus, typically the method to be adopted to solve a real-world problem should be tailored on the characteristics, structure and size of the problem. Part 4 of the thesis consists of a brief review on tools commonly used for general MINLP problems, constituted an integral part of the development of this Ph.D. thesis (especially the use and development of open-source software). We present the main characteristics of solvers for each special case of MINLP.

Non-Linear Analysis and Design of Synchronous Bearingless Multiphase Permanent Magnet Machines and Drives

A two-dimensional model to analyze the distribution of magnetic fields in the airgap of a PM electrical machines is studied. A numerical algorithm for non-linear magnetic analysis of multiphase surface-mounted PM machines with semi-closed slots is developed, based on the equivalent magnetic circuit method. By using a modular structure geometry, whose the basic element can be duplicated, it allows to design whatever typology of windings distribution. In comparison to a FEA, permits a reduction in computing time and to directly changing the values of the parameters in a user interface, without re-designing the model. Output torque and radial forces acting on the moving part of the machine can be calculated. In addition, an analytical model for radial forces calculation in multiphase bearingless Surface-Mounted Permanent Magnet Synchronous Motors (SPMSM) is presented. It allows to predict amplitude and direction of the force, depending on the values of torque current, of levitation current and of rotor position. It is based on the space vectors method, letting the analysis of the machine also during transients. The calculations are conducted by developing the analytical functions in Fourier series, taking all the possible interactions between stator and rotor mmf harmonic components into account and allowing to analyze the effects of electrical and geometrical quantities of the machine, being parametrized. The model is implemented in the design of a control system for bearingless machines, as an accurate electromagnetic model integrated in a three-dimensional mechanical model, where one end of the motor shaft is constrained to simulate the presence of a mechanical bearing, while the other is free, only supported by the radial forces developed in the interactions between magnetic fields, to realize a bearingless system with three degrees of freedom. The complete model represents the design of the experimental system to be realized in the laboratory.

Modelli non lineari statici e dinamici su dati microeconomici: analisi delle condizioni finanziarie delle famiglie italiane

The thesis studies the economic and financial conditions of Italian households, by using microeconomic data of the Survey on Household Income and Wealth (SHIW) over the period 1998-2006. It develops along two lines of enquiry. First it studies the determinants of households holdings of assets and liabilities and estimates their correlation degree. After a review of the literature, it estimates two non-linear multivariate models on the interactions between assets and liabilities with repeated cross-sections. Second, it analyses households financial difficulties. It defines a quantitative measure of financial distress and tests, by means of non-linear dynamic probit models, whether the probability of experiencing financial difficulties is persistent over time. Chapter 1 provides a critical review of the theoretical and empirical literature on the estimation of assets and liabilities holdings, on their interactions and on households net wealth. The review stresses the fact that a large part of the literature explain households debt holdings as a function, among others, of net wealth, an assumption that runs into possible endogeneity problems. Chapter 2 defines two non-linear multivariate models to study the interactions between assets and liabilities held by Italian households. Estimation refers to a pooling of cross-sections of SHIW. The first model is a bivariate tobit that estimates factors affecting assets and liabilities and their degree of correlation with results coherent with theoretical expectations. To tackle the presence of non normality and heteroskedasticity in the error term, generating non consistent tobit estimators, semi-parametric estimates are provided that confirm the results of the tobit model. The second model is a quadrivariate probit on three different assets (safe, risky and real) and total liabilities; the results show the expected patterns of interdependence suggested by theoretical considerations. Chapter 3 reviews the methodologies for estimating non-linear dynamic panel data models, drawing attention to the problems to be dealt with to obtain consistent estimators. Specific attention is given to the initial condition problem raised by the inclusion of the lagged dependent variable in the set of explanatory variables. The advantage of using dynamic panel data models lies in the fact that they allow to simultaneously account for true state dependence, via the lagged variable, and unobserved heterogeneity via individual effects specification. Chapter 4 applies the models reviewed in Chapter 3 to analyse financial difficulties of Italian households, by using information on net wealth as provided in the panel component of the SHIW. The aim is to test whether households persistently experience financial difficulties over time. A thorough discussion is provided of the alternative approaches proposed by the literature (subjective/qualitative indicators versus quantitative indexes) to identify households in financial distress. Households in financial difficulties are identified as those holding amounts of net wealth lower than the value corresponding to the first quartile of net wealth distribution. Estimation is conducted via four different methods: the pooled probit model, the random effects probit model with exogenous initial conditions, the Heckman model and the recently developed Wooldridge model. Results obtained from all estimators accept the null hypothesis of true state dependence and show that, according with the literature, less sophisticated models, namely the pooled and exogenous models, over-estimate such persistence.