Differential forms on Carnot groups

The main goal of this thesis is to understand and link together some of the early works by Michel Rumin and Pierre Julg. The work is centered around the so-called Rumin complex, which is a construction in subRiemannian geometry. A Carnot manifold is a manifold endowed with a horizontal distribution. If further a metric is given, one gets a subRiemannian manifold. Such data arise in different contexts, such as: - formulation of the second principle of thermodynamics; - optimal control; - propagation of singularities for sums of squares of vector fields; - real hypersurfaces in complex manifolds; - ideal boundaries of rank one symmetric spaces; - asymptotic geometry of nilpotent groups; - modelization of human vision. Differential forms on a Carnot manifold have weights, which produces a filtered complex. In view of applications to nilpotent groups, Rumin has defined a substitute for the de Rham complex, adapted to this filtration. The presence of a filtered complex also suggests the use of the formal machinery of spectral sequences in the study of cohomology. The goal was indeed to understand the link between Rumin's operator and the differentials which appear in the various spectral sequences we have worked with: - the weight spectral sequence; - a special spectral sequence introduced by Julg and called by him Forman's spectral sequence; - Forman's spectral sequence (which turns out to be unrelated to the previous one). We will see that in general Rumin's operator depends on choices. However, in some special cases, it does not because it has an alternative interpretation as a differential in a natural spectral sequence. After defining Carnot groups and analysing their main properties, we will introduce the concept of weights of forms which will produce a splitting on the exterior differential operator d. We shall see how the Rumin complex arises from this splitting and proceed to carry out the complete computations in some key examples. From the third chapter onwards we will focus on Julg's paper, describing his new filtration and its relationship with the weight spectral sequence. We will study the connection between the spectral sequences and Rumin's complex in the n-dimensional Heisenberg group and the 7-dimensional quaternionic Heisenberg group and then generalize the result to Carnot groups using the weight filtration. Finally, we shall explain why Julg required the independence of choices in some special Rumin operators, introducing the Szego map and describing its main properties.

Mimicking the one-dimensional marginal distributions of stochastic diffusions

In the large maturity limit, we compute explicitly the Local Volatility surface for Heston, through Dupire’s formula, with Fourier pricing of the respective derivatives of the call price. Than we verify that the prices of European call options produced by the Heston model, concide with those given by the local volatility model where the Local Volatility is computed as said above.

Asymmetric drift and halo flattening in disk galaxies

The main result in this work is the solution of the Jeans equations for an axisymmetric galaxy model containing a baryonic component (distributed according to a Miyamoto-Nagai profile) and a dark matter halo (described by the Binney logarithmic potential). The velocity dispersion, azimuthal velocity and some other interesting quantities such as the asymmetric drift are studied, along with the influence of the model parameters on these (observable) quantities. We also give an estimate for the velocity of the radial flow, caused by the asymmetric drift. Other than the mathematical beauty that lies in solving a model analytically, the interest of this kind of results can be mainly found in numerical simulations that study the evolution of gas flows. For example, it is important to know how certain parameters such as the shape (oblate, prolate, spherical) of a dark matter halo, or the flattening of the baryonic matter, or the mass ratio between dark and baryonic matter, have an influence on observable quantities such as the velocity dispersion. In the introductory chapter, we discuss the Jeans equations, which provide information about the velocity dispersion of a system. Next we will consider some dynamical quantities that will be useful in the rest of the work, e.g. the asymmetric drift. In Chapter 2 we discuss in some more detail the family of galaxy models we studied. In Chapter 3 we give the solution of the Jeans equations. Chapter 4 describes and illustrates the behaviour of the velocity dispersion, as a function of the several parameters, along with asymptotic expansions. In Chapter 5 we will investigate the behaviour of certain dynamical quantities for this model. We conclude with a discussion in Chapter 6.

A linear O(N) model: a functional renormalization group approach for flat and curved space

In questa tesi sono state applicate le tecniche del gruppo di rinormalizzazione funzionale allo studio della teoria quantistica di campo scalare con simmetria O(N) sia in uno spaziotempo piatto (Euclideo) che nel caso di accoppiamento ad un campo gravitazionale nel paradigma dell'asymptotic safety. Nel primo capitolo vengono esposti in breve alcuni concetti basilari della teoria dei campi in uno spazio euclideo a dimensione arbitraria. Nel secondo capitolo si discute estensivamente il metodo di rinormalizzazione funzionale ideato da Wetterich e si fornisce un primo semplice esempio di applicazione, il modello scalare. Nel terzo capitolo è stato studiato in dettaglio il modello O(N) in uno spaziotempo piatto, ricavando analiticamente le equazioni di evoluzione delle quantità rilevanti del modello. Quindi ci si è specializzati sul caso N infinito. Nel quarto capitolo viene iniziata l'analisi delle equazioni di punto fisso nel limite N infinito, a partire dal caso di dimensione anomala nulla e rinormalizzazione della funzione d'onda costante (approssimazione LPA), già studiato in letteratura. Viene poi considerato il caso NLO nella derivative expansion. Nel quinto capitolo si è introdotto l'accoppiamento non minimale con un campo gravitazionale, la cui natura quantistica è considerata a livello di QFT secondo il paradigma di rinormalizzabilità dell'asymptotic safety. Per questo modello si sono ricavate le equazioni di punto fisso per le principali osservabili e se ne è studiato il comportamento per diversi valori di N.

Coded slotted aloha with multipacket reception over block fading channels (for satellite networks)

Random access (RA) protocols are normally used in a satellite networks for initial terminal access and are particularly effective since no coordination is required. On the other hand, contention resolution diversity slotted Aloha (CRDSA), irregular repetition slotted Aloha (IRSA) and coded slotted Aloha (CSA) has shown to be more efficient than classic RA schemes as slotted Aloha, and can be exploited also when short packets transmissions are done over a shared medium. In particular, they relies on burst repetition and on successive interference cancellation (SIC) applied at the receiver. The SIC process can be well described using a bipartite graph representation and exploiting tools used for analyze iterative decoding. The scope of my Master Thesis has been to described the performance of such RA protocols when the Rayleigh fading is taken into account. In this context, each user has the ability to correctly decode a packet also in presence of collision and when SIC is considered this may result in multi-packet reception. Analysis of the SIC procedure under Rayleigh fading has been analytically derived for the asymptotic case (infinite frame length), helping the analysis of both throughput and packet loss rates. An upper bound of the achievable performance has been analytically obtained. It can be show that in particular channel conditions the throughput of the system can be greater than one packets per slot which is the theoretical limit of the Collision Channel case.