Quantizations on the Engel and the Cartan groups

This work aims to develop a global quantization in the concrete settings of two graded nilpotent Lie groups of 3-step; namely of the Engel group and the Cartan group. We provide a preliminary analysis on the structure and the representations of the aforementioned groups, and their corresponding Lie algebras. In addition, the explicit formulas for the difference operators in the two settings are derived, constituting the necessary prerequisites for the constructions of the $\Psi^{m}_{\rho,\delta}$ classes of symbols in both cases. In the case of the Engel group, the relation between the Kohn-Nirenberg quantization and the representations of the Engel group enables us to express operators in this setting in terms of quantization of symbols in the Euclidean space. We illustrate the $L^p-L^q$ boundedness for spectral multipliers theorems with particular examples of operators in both groups, that yield appropriate Sobolev-type inequalities. As a further application of the above, we get some consequences for the $L^p-L^q$ norm for the heat kernel of those particular examples of operators in both settings

Quantizations and Poincare inequalities on graded Lie groups. A study on anharmonic oscillators

The analysis carried out in this dissertation lies in two settings: the Euclidean setting and the non-commutative setting of two stratified Lie groups; namely the Engel and the Cartan group. In the non-commutative direction, following the theory developed by Fischer and Ruzhansky, we set out the prerequisites for the development of the global symbolic calculus in the setting of the below groups. The latter means that explicit formulas for the difference operators in both settings are given, and subsequently the S^m(G) classes of symbols can be concretely described. The group Fourier transform of the sub-Laplacian on the aforementioned groups boils down to an anharmonic oscillator on Rn. This, in turn, hands over the study of anharmonic oscillators, using the Weyl-Hormander calculus, aiming to obtain spectral properties for their negative powers in terms of Schatten-von Neumann classes. Additionally, the associated Weyl-Hormander classes of the degenerate harmonic oscillator are described. Developing the Fourier analysis associated with the eigenfunctions of the, regarded as prototype, anharmonic oscillator, we establish a Hausdorff-Young-Paley inequality, and obtain boundedness results for Fourier multipliers in this setting. In the same spirit, but moving to the non-commutative setting, we apply a generalisation of a classical condition of Hormander on the Lp-Lq boundedness of Fourier multipliers on a locally compact group to spectral multipliers of non-Rockland operators on the Engel and Cartan groups. Conditions on the r-nuclearity of functions of the prototype anharmonic oscillator on modulation spaces are given, and these imply that Lidskii's trace formula that holds true for r >2/3 due to Grothendieck's theory, still holds true for larger values of r. Finally, we prove the Poincare inequality for a family of probability measures, with density depending on a homogeneous norm, on a class of stratifed Lie groups of any stepOpen Acces

Hate is not Binary: Studying Abusive Behavior of #GamerGate on Twitter

Over the past few years, online bullying and aggression have become increasingly prominent, and manifested in many different forms on social media. However, there is little work analyzing the characteristics of abusive users and what distinguishes them from typical social media users. In this paper, we start addressing this gap by analyzing tweets containing a great large amount of abusiveness. We focus on a Twitter dataset revolving around the Gamergate controversy, which led to many incidents of cyberbullying and cyberaggression on various gaming and social media platforms. We study the properties of the users tweeting about Gamergate, the content they post, and the differences in their behavior compared to typical Twitter users. We find that while their tweets are often seemingly about aggressive and hateful subjects, "Gamergaters" do not exhibit common expressions of online anger, and in fact primarily differ from typical users in that their tweets are less joyful. They are also more engaged than typical Twitter users, which is an indication as to how and why this controversy is still ongoing. Surprisingly, we find that Gamergaters are less likely to be suspended by Twitter, thus we analyze their properties to identify differences from typical users and what may have led to their suspension. We perform an unsupervised machine learning analysis to detect clusters of users who, though currently active, could be considered for suspension since they exhibit similar behaviors with suspended users. Finally, we confirm the usefulness of our analyzed features by emulating the Twitter suspension mechanism with a supervised learning method, achieving very good precision and recall.Comment: In 28th ACM Conference on Hypertext and Social Media (ACM HyperText 2017

qq-Poincar{\'e} inequalities on Carnot Groups with a filiform Lie algebra

In this paper we prove (global) $q$-Poincar{\'e} inequalities for probability measures on nilpotent Lie groups with filiform Lie algebra of any length. The probability measures under consideration have a density with respect to the Haar measure given as a function of a suitable homogeneous norm

The heat equation with singular potentials. II: Hypoelliptic case

This work is a continuation of the work arXiv:2004.11255v3. It is 21 pagesIn this paper we consider the heat equation with a strongly singular potential and show that it has a very weak solution. Our analysis is devoted to general hypoelliptic operators and is developed in the setting of graded Lie groups. The current work continues and extends a previous work , where the classical heat equation on $\mathbb R^n$ was considered

Large scale crowdsourcing and characterization of Twitter abusive behavior

In recent years online social networks have suffered an increase in sexism, racism, and other types of aggressive and cyberbullying behavior, often manifesting itself through offensive, abusive, or hateful language. Past scientific work focused on studying these forms of abusive activity in popular online social networks, such as Facebook and Twitter. Building on such work, we present an eight month study of the various forms of abusive behavior on Twitter, in a holistic fashion. Departing from past work, we examine a wide variety of labeling schemes, which cover different forms of abusive behavior. We propose an incremental and iterative methodology that leverages the power of crowdsourcing to annotate a large collection of tweets with a set of abuse-related labels.By applying our methodology and performing statistical analysis for label merging or elimination, we identify a reduced but robust set of labels to characterize abuse-related tweets. Finally, we offer a characterization of our annotated dataset of 80 thousand tweets, which we make publicly available for further scientific exploration.Accepted manuscrip