Design and Computation of Warped Time-Frequency Transforms

In this work we introduce an analytical approach for the frequency warping transform. Criteria for the design of operators based on arbitrary warping maps are provided and an algorithm carrying out a fast computation is defined. Such operators can be used to shape the tiling of time-frequency plane in a flexible way. Moreover, they are designed to be inverted by the application of their adjoint operator. According to the proposed mathematical model, the frequency warping transform is computed by considering two additive operators: the first one represents its nonuniform Fourier transform approximation and the second one suppresses aliasing. The first operator is known to be analytically characterized and fast computable by various interpolation approaches. A factorization of the second operator is found for arbitrary shaped non-smooth warping maps. By properly truncating the operators involved in the factorization, the computation turns out to be fast without compromising accuracy.

An equilibrium approach to modelling social interaction

The aim of this work is to put forward a statistical mechanics theory of social interaction, generalizing econometric discrete choice models. After showing the formal equivalence linking econometric multinomial logit models to equilibrium statical mechanics, a multi- population generalization of the Curie-Weiss model for ferromagnets is considered as a starting point in developing a model capable of describing sudden shifts in aggregate human behaviour. Existence of the thermodynamic limit for the model is shown by an asymptotic sub-additivity method and factorization of correlation functions is proved almost everywhere. The exact solution for the model is provided in the thermodynamical limit by nding converging upper and lower bounds for the system's pressure, and the solution is used to prove an analytic result regarding the number of possible equilibrium states of a two-population system. The work stresses the importance of linking regimes predicted by the model to real phenomena, and to this end it proposes two possible procedures to estimate the model's parameters starting from micro-level data. These are applied to three case studies based on census type data: though these studies are found to be ultimately inconclusive on an empirical level, considerations are drawn that encourage further refinements of the chosen modelling approach, to be considered in future work.

La concezione epistemica dell'analiticità: un dibattito in corso

Il lavoro è una riflessione sugli sviluppi della nozione di definizione nel recente dibattito sull'analiticità. La rinascita di questa discussione, dopo le critiche di Quine e un conseguente primo abbandono della concezione convenzionalista carnapiana ha come conseguenza una nuova concezione epistemica dell'analiticità. Nella maggior parte dei casi le nuove teorie epistemiche, tra le quali quelle di Bob Hale e Crispin Wright (Implicit Definition and the A priori, 2001) e Paul Boghossian (Analyticity, 1997; Epistemic analyticity, a defence, 2002, Blind reasoning, 2003, Is Meaning Normative ?, 2005) presentano il comune carattere di intendere la conoscenza a priori nella forma di una definizione implicita (Paul Horwich, Stipulation, Meaning, and Apriority, 2001). Ma una seconda linea di obiezioni facenti capo dapprima a Horwich, e in seguito agli stessi Hale e Wright, mettono in evidenza rispettivamente due difficoltà per la definizione corrispondenti alle questioni dell'arroganza epistemica e dell'accettazione (o della stipulazione) di una definizione implicita. Da questo presupposto nascono diversi tentativi di risposta. Da un lato, una concezione della definizione, nella teoria di Hale e Wright, secondo la quale essa appare come un principio di astrazione, dall'altro una nozione della definizione come definizione implicita, che si richiama alla concezione di P. Boghossian. In quest'ultima, la definizione implicita è data nella forma di un condizionale linguistico (EA, 2002; BR, 2003), ottenuto mediante una fattorizzazione della teoria costruita sul modello carnapiano per i termini teorici delle teorie empiriche. Un'analisi attenta del lavoro di Rudolf Carnap (Philosophical foundations of Physics, 1966), mostra che la strategia di scomposizione rappresenta una strada possibile per una nozione di analiticità adeguata ai termini teorici. La strategia carnapiana si colloca, infatti, nell'ambito di un tentativo di elaborazione di una nozione di analiticità che tiene conto degli aspetti induttivi delle teorie empiriche

Application of innovative methods of source apportionment in air contamination assessment

In this work, new tools in atmospheric pollutant sampling and analysis were applied in order to go deeper in source apportionment study. The project was developed mainly by the study of atmospheric emission sources in a suburban area influenced by a municipal solid waste incinerator (MSWI), a medium-sized coastal tourist town and a motorway. Two main research lines were followed. For what concerns the first line, the potentiality of the use of PM samplers coupled with a wind select sensor was assessed. Results showed that they may be a valid support in source apportionment studies. However, meteorological and territorial conditions could strongly affect the results. Moreover, new markers were investigated, particularly focusing on the processes of biomass burning. OC revealed a good biomass combustion process indicator, as well as all determined organic compounds. Among metals, lead and aluminium are well related to the biomass combustion. Surprisingly PM was not enriched of potassium during bonfire event. The second research line consists on the application of Positive Matrix factorization (PMF), a new statistical tool in data analysis. This new technique was applied to datasets which refer to different time resolution data. PMF application to atmospheric deposition fluxes identified six main sources affecting the area. The incinerator’s relative contribution seemed to be negligible. PMF analysis was then applied to PM2.5 collected with samplers coupled with a wind select sensor. The higher number of determined environmental indicators allowed to obtain more detailed results on the sources affecting the area. Vehicular traffic revealed the source of greatest concern for the study area. Also in this case, incinerator’s relative contribution seemed to be negligible. Finally, the application of PMF analysis to hourly aerosol data demonstrated that the higher the temporal resolution of the data was, the more the source profiles were close to the real one.

Permutation classes, sorting algorithms, and lattice paths

A permutation is said to avoid a pattern if it does not contain any subsequence which is order-isomorphic to it. Donald Knuth, in the first volume of his celebrated book "The art of Computer Programming", observed that the permutations that can be computed (or, equivalently, sorted) by some particular data structures can be characterized in terms of pattern avoidance. In more recent years, the topic was reopened several times, while often in terms of sortable permutations rather than computable ones. The idea to sort permutations by using one of Knuth’s devices suggests to look for a deterministic procedure that decides, in linear time, if there exists a sequence of operations which is able to convert a given permutation into the identical one. In this thesis we show that, for the stack and the restricted deques, there exists an unique way to implement such a procedure. Moreover, we use these sorting procedures to create new sorting algorithms, and we prove some unexpected commutation properties between these procedures and the base step of bubblesort. We also show that the permutations that can be sorted by a combination of the base steps of bubblesort and its dual can be expressed, once again, in terms of pattern avoidance. In the final chapter we give an alternative proof of some enumerative results, in particular for the classes of permutations that can be sorted by the two restricted deques. It is well-known that the permutations that can be sorted through a restricted deque are counted by the Schrӧder numbers. In the thesis, we show how the deterministic sorting procedures yield a bijection between sortable permutations and Schrӧder paths.

Embedded representations of social interactions

Social interactions have been the focus of social science research for a century, but their study has recently been revolutionized by novel data sources and by methods from computer science, network science, and complex systems science. The study of social interactions is crucial for understanding complex societal behaviours. Social interactions are naturally represented as networks, which have emerged as a unifying mathematical language to understand structural and dynamical aspects of socio-technical systems. Networks are, however, highly dimensional objects, especially when considering the scales of real-world systems and the need to model the temporal dimension. Hence the study of empirical data from social systems is challenging both from a conceptual and a computational standpoint. A possible approach to tackling such a challenge is to use dimensionality reduction techniques that represent network entities in a low-dimensional feature space, preserving some desired properties of the original data. Low-dimensional vector space representations, also known as network embeddings, have been extensively studied, also as a way to feed network data to machine learning algorithms. Network embeddings were initially developed for static networks and then extended to incorporate temporal network data. We focus on dimensionality reduction techniques for time-resolved social interaction data modelled as temporal networks. We introduce a novel embedding technique that models the temporal and structural similarities of events rather than nodes. Using empirical data on social interactions, we show that this representation captures information relevant for the study of dynamical processes unfolding over the network, such as epidemic spreading. We then turn to another large-scale dataset on social interactions: a popular Web-based crowdfunding platform. We show that tensor-based representations of the data and dimensionality reduction techniques such as tensor factorization allow us to uncover the structural and temporal aspects of the system and to relate them to geographic and temporal activity patterns.

The Hitchin map for one-nodal base curves of compact type

Studying moduli spaces of semistable Higgs bundles (E, \phi) of rank n on a smooth curve C, a key role is played by the spectral curve X (Hitchin), because an important result by Beauville-Narasimhan-Ramanan allows us to study isomorphism classes of such Higgs bundles in terms of isomorphism classes of rank-1 torsion-free sheaves on X. This way, the generic fibre of the Hitchin map, which associates to any semistable Higgs bundle the coefficients of the characteristic polynomial of \phi, is isomorphic to the Jacobian of X. Focusing on rank-2 Higgs data, this construction was extended by Barik to the case in which the curve C is reducible, one-nodal, having two smooth components. Such curve is called of compact type because its Picard group is compact. In this work, we describe and clarify the main points of the construction by Barik and we give examples, especially concerning generic fibres of the Hitchin map. Referring to Hausel-Pauly, we consider the case of SL(2,C)-Higgs bundles on a smooth base curve, which are such that the generic fibre of the Hitchin map is a subvariety of the Jacobian of X, the Prym variety. We recall the description of special loci, called endoscopic loci, such that the associated Prym variety is not connected. Then, letting G be an affine reductive group having underlying Lie algebra so(4,C), we consider G-Higgs bundles on a smooth base curve. Starting from the construction by Bradlow-Schaposnik, we discuss the associated endoscopic loci. By adapting these studies to a one-nodal base curve of compact type, we describe the fibre of the SL(2,C)-Hitchin map and of the G-Hitchin map, together with endoscopic loci. In the Appendix, we give an interpretation of generic spectral curves in terms of families of double covers.

Homological and Hodge-theoretic aspects of matroids and polymatroids

In the first part of this thesis, we study the action of the automorphism group of a matroid on the homology space of the co-independent complex. This representation turns out to be isomorphic, up to tensoring with the sign representation, with that on the homology space associated with the lattice of flats. In the case of the cographic matroid of the complete graph, this result has application in algebraic geometry: indeed De Cataldo, Heinloth and Migliorini use this outcome to study the Hitchin fibration. In the second part, on the other hand, we use ideas from algebraic geometry to prove a purely combinatorial result. We construct a Leray model for a discrete polymatroid with arbitrary building set and we prove a generalized Goresky-MacPherson formula. The first row of the model is the Chow ring of the polymatroid; we prove Poincaré duality, Hard-Lefschetz theorem and Hodge-Riemann relations for the Chow ring.

New perspectives in statistical mechanics and high-dimensional inference

The main purpose of this thesis is to go beyond two usual assumptions that accompany theoretical analysis in spin-glasses and inference: the i.i.d. (independently and identically distributed) hypothesis on the noise elements and the finite rank regime. The first one appears since the early birth of spin-glasses. The second one instead concerns the inference viewpoint. Disordered systems and Bayesian inference have a well-established relation, evidenced by their continuous cross-fertilization. The thesis makes use of techniques coming both from the rigorous mathematical machinery of spin-glasses, such as the interpolation scheme, and from Statistical Physics, such as the replica method. The first chapter contains an introduction to the Sherrington-Kirkpatrick and spiked Wigner models. The first is a mean field spin-glass where the couplings are i.i.d. Gaussian random variables. The second instead amounts to establish the information theoretical limits in the reconstruction of a fixed low rank matrix, the “spike”, blurred by additive Gaussian noise. In chapters 2 and 3 the i.i.d. hypothesis on the noise is broken by assuming a noise with inhomogeneous variance profile. In spin-glasses this leads to multi-species models. The inferential counterpart is called spatial coupling. All the previous models are usually studied in the Bayes-optimal setting, where everything is known about the generating process of the data. In chapter 4 instead we study the spiked Wigner model where the prior on the signal to reconstruct is ignored. In chapter 5 we analyze the statistical limits of a spiked Wigner model where the noise is no longer Gaussian, but drawn from a random matrix ensemble, which makes its elements dependent. The thesis ends with chapter 6, where the challenging problem of high-rank probabilistic matrix factorization is tackled. Here we introduce a new procedure called "decimation" and we show that it is theoretically to perform matrix factorization through it.