Low codimensional intrinsic regular submanifolds in the Heisenberg group H^n

The thesis mainly concerns the study of intrinsically regular submanifolds of low codimension in the Heisenberg group H^n, called H-regular surfaces of low codimension, from the point of view of geometric measure theory. We consider an H-regular surface of H^n of codimension k, with k between 1 and n, parametrized by a uniformly intrinsically differentiable map acting between two homogeneous complementary subgroups of H^n, with target subgroup horizontal of dimension k. In particular the considered submanifold is the intrinsic graph of the parametrization. We extend various results of Ambrosio, Serra Cassano and Vittone, available for the case when k = 1. We prove that the uniform intrinsic differentiability of the parametrizing map is equivalent to the existence and continuity of its intrinsic differential, to the local existence of a suitable approximating family of Euclidean regular maps, and, when the domain and the codomain of the map are orthogonal, to the existence and continuity of suitably defined intrinsic partial derivatives of the function. Successively, we present a series of area formulas, proved in collaboration with V. Magnani. They allow to compute the (2n+2−k)-dimensional spherical Hausdorff measure and the (2n+2−k)-dimensional centered Hausdorff measure of the parametrized H-regular surface, with respect to any homogeneous distance fixed on H^n. Furthermore, we focus on (G,M)-regular sets of G, where G and M are two arbitrary Carnot groups. Suitable implicit function theorems ensure the local existence of an intrinsic parametrization of such a set, at any of its points. We prove that it is uniformly intrinsically differentiable. Finally, we prove a coarea-type inequality for a continuously Pansu differentiable function acting between two Carnot groups endowed with homogeneous distances. We assume that the level sets of the function are uniformly lower Ahlfors regular and that the Pansu differential is everywhere surjective.

A corpus-assisted approach to apology-like expressions in Japanese computer-mediated discourse

This thesis provides a corpus-assisted pragmatic investigation of three Japanese expressions commonly signalled as apologetic, namely gomen, su(m)imasen and mōshiwake arimasen, which can be roughly translated in English with ‘(I’m) sorry’. The analysis is based on a web corpus of 306,670 tokens collected from the Q&A website Yahoo! Chiebukuro, which is examined combining quantitative (statistical) and qualitative (traditional close reading) methods. By adopting a form-to-function approach, the aim of the study is to shed light on three main topics of interest: the pragmatic functions of apology-like expressions, the discursive strategies they co-occur with, and the behaviours that warrant them. The overall findings reveal that apology-like expressions are multifunctional devices whose meanings extend well beyond ‘apology’ alone. These meanings are affected by a number of discursive strategies that can either increase or decrease the perceived (im)politeness level of the speech act to serve interactants’ face needs and communicative goals. The study also identifies a variety of behaviours that people frame as violations, not necessarily because they are actually face-threatening to the receiver, but because doing so is functional to the projection of the apologiser as a moral persona. An additional finding that emerged from the analysis is the pervasiveness of reflexive usages of apology-like expressions, which are often employed metadiscursively to convey, negotiate and challenge opinions on how language should be used. To conclude, the study provides a unique insight into the use of three expressions whose pragmatic meanings are more varied than anticipated. The findings reflect the use of (im)politeness in an online and non-Western context and, hopefully, represent a step towards a more inclusive notion of ‘apologies’ and related speech acts.

Spectral asymptotic properties of semi-regular non-commutative harmonic oscillators

The study carried out in this thesis is devoted to spectral analysis of systems of PDEs related also with quantum physics models. Namely, the research deals with classes of systems that contain certain quantum optics models such as Jaynes-Cummings, Rabi and their generalizations that describe light-matter interaction. First we investigate the spectral Weyl asymptotics for a class of semiregular systems, extending to the vector-valued case results of Helffer and Robert, and more recently of Doll, Gannot and Wunsch. Actually, the asymptotics by Doll, Gannot and Wunsch is more precise (that is why we call it refined) than the classical result by Helffer and Robert, but deals with a less general class of systems, since the authors make an hypothesis on the measure of the subset of the unit sphere on which the tangential derivatives of the X-Ray transform of the semiprincipal symbol vanish to infinity order. Abstract Next, we give a meromorphic continuation of the spectral zeta function for semiregular differential systems with polynomial coefficients, generalizing the results by Ichinose and Wakayama and Parmeggiani. Finally, we state and prove a quasi-clustering result for a class of systems including the aforementioned quantum optics models and we conclude the thesis by showing a Weyl law result for the Rabi model and its generalizations.