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Computational research on mental health disorders from written texts covers an interdisciplinary area between natural language processing and psychology. A crucial aspect of this problem is prevention and early diagnosis, as suicide resulted from depression being the second leading cause of death for young adults. In this work, we focus on methods for detecting the early onset of depression from social media texts, in particular from Reddit. To that end, we explore the eRisk 2018 dataset and achieve good results with regard to the state of the art by leveraging topic analysis and learned confidence scores to guide the decision process., Comment: Accepted at Seventh Italian Conference on Computational Linguistics CLiC-it 2020

Shapley values underlie one of the most popular model-agnostic methods within explainable artificial intelligence. These values are designed to attribute the difference between a model's prediction and an average baseline to the different features used as input to the model. Being based on solid game-theoretic principles, Shapley values uniquely satisfy several desirable properties, which is why they are increasingly used to explain the predictions of possibly complex and highly non-linear machine learning models. Shapley values are well calibrated to a user's intuition when features are independent, but may lead to undesirable, counterintuitive explanations when the independence assumption is violated. In this paper, we propose a novel framework for computing Shapley values that generalizes recent work that aims to circumvent the independence assumption. By employing Pearl's do-calculus, we show how these 'causal' Shapley values can be derived for general causal graphs without sacrificing any of their desirable properties. Moreover, causal Shapley values enable us to separate the contribution of direct and indirect effects. We provide a practical implementation for computing causal Shapley values based on causal chain graphs when only partial information is available and illustrate their utility on a real-world example., Comment: Accepted at 34th Conference on Neural Information Processing Systems (NeurIPS 2020)

We introduce a new geometric-analytic functional that we analyse in the context of free discontinuity problems. Its main feature is that the geometric term (the length of the jump set) appears with negative sign. This is motivated by searching quantitative inequalities for best constants of Sobolev-Poincar\'e inequalities with trace terms in $\mathbb{R}^n$ which correspond to fundamental eigenvalues associated to semilinear problems for the Laplace operator with Robin boundary conditions. Our method is based on the study of this new, degenerate, functional which involves an obstacle problem in interaction with the jump set. Ultimately, this becomes a mixed free discontinuity/free boundary problem occuring above/at the level of the obstacle, respectively.

We investigate the convergence as $p\searrow1$ of the minimizers of the $W^{s,p}$-energy for $s\in(0,1)$ and $p\in(1,\infty)$ to those of the $W^{s,1}$-energy, both in the pointwise sense and by means of $\Gamma$-convergence. We also address the convergence of the corresponding Euler-Lagrange equations, and the equivalence between minimizers and weak solutions. As ancillary results, we study some regularity issues regarding minimizers of the $W^{s,1}$-energy., Comment: 43 pages

We analyse the asymptotic behaviour of the eigenvalues and eigenvectors of a Steklov problem in a dumbbell domain consisting of two Lipschitz sets connected by a thin tube with vanishing width. All the eigenvalues are collapsing to zero, the speed being driven by some power of the width which multiplies the eigenvalues of a one dimensional problem. In two dimensions of the space, the behaviour is fundamentally different from the third or higher dimensions and the limit problems are of different nature. This phenomenon is due to the fact that only in dimension two the boundary of the tube has not vanishing surface measure., Comment: 29 pages, 3 figures

Information flow, opinion, and epidemics spread over structured networks. When using individual node centrality indicators to predict which nodes will be among the top influencers or spreaders in a large network, no single centrality has consistently good ranking power. We show that statistical classifiers using two or more centralities as input are instead consistently predictive over many diverse, static real-world topologies. Certain pairs of centralities cooperate particularly well in statistically drawing the boundary between the top spreaders and the rest: local centralities measuring the size of a node's neighbourhood benefit from the addition of a global centrality such as the eigenvector centrality, closeness, or the core number. This is, intuitively, because a local centrality may rank highly some nodes which are located in dense, but peripheral regions of the network---a situation in which an additional global centrality indicator can help by prioritising nodes located more centrally. The nodes selected as superspreaders will usually jointly maximise the values of both centralities. As a result of the interplay between centrality indicators, training classifiers with seven classical indicators leads to a nearly maximum average precision function (0.995) across the networks in this study., Comment: 14 pages, 10 figures, 4 supplementary figures

Scientific publishing is the means by which we communicate and share scientific knowledge, but this process currently often lacks transparency and machine-interpretable representations. Scientific articles are published in long coarse-grained text with complicated structures, and they are optimized for human readers and not for automated means of organization and access. Peer reviewing is the main method of quality assessment, but these peer reviews are nowadays rarely published and their own complicated structure and linking to the respective articles is not accessible. In order to address these problems and to better align scientific publishing with the principles of the Web and Linked Data, we propose here an approach to use nanopublications as a unifying model to represent in a semantic way the elements of publications, their assessments, as well as the involved processes, actors, and provenance in general. To evaluate our approach, we present a dataset of 627 nanopublications representing an interlinked network of the elements of articles (such as individual paragraphs) and their reviews (such as individual review comments). Focusing on the specific scenario of editors performing a meta-review, we introduce seven competency questions and show how they can be executed as SPARQL queries. We then present a prototype of a user interface for that scenario that shows different views on the set of review comments provided for a given manuscript, and we show in a user study that editors find the interface useful to answer their competency questions. In summary, we demonstrate that a unified and semantic publication model based on nanopublications can make scientific communication more effective and user-friendly.

We consider the torsion function for the Dirichlet Laplacian $-\Delta$, and for the Schr\"odinger operator $- \Delta + V$ on an open set $\Omega\subset \R^m$ of finite Lebesgue measure $0<|\Omega|<\infty$ with a real-valued, non-negative, measurable potential $V.$ We investigate the efficiency and the phenomenon of localisation for the torsion function, and their interplay with the geometry of the first Dirichlet eigenfunction., Comment: 33 pages. The published version in Potential Analysis (2022) 57, 571--600 has some typos: Theorem 3(i): the first exponent should read $(m-2)/m$; Example 2 Line 2: .... $B(p_{n+1};cn^{-\beta})$.....; Formula (109): $\kappa^{-1}$ missing after the second inequality

Given two planar, conformal, smooth open sets $\Omega$ and $\omega$, we prove the existence of a sequence of smooth sets $\Omega_n$ which geometrically converges to $\Omega$ and such that the (perimeter normalized) Steklov eigenvalues of $\Omega_n$ converge to the ones of $\omega$. As a consequence, we answer a question raised by Girouard and Polterovich on the stability of the Weinstock inequality and prove that the inequality is genuinely unstable. However, under some a priori knowledge of the geometry related to the oscillations of the boundaries, stability may occur., Comment: typos corrected in section 2 and 4, statement of main result was precised

As well known, harmonic functions satisfy the mean value property, namely the average of the function over a ball is equal to its value at the center. This fact naturally raises the question on whether this is a characterizing feature of balls, namely whether a set for which all harmonic functions satisfy the mean value property is necessarily a ball. This question was investigated by several authors, and was finally elegantly, completely and positively settled by \"Ulk\"u Kuran, with an artful use of elementary techniques. This classical problem has been recently fleshed out by Giovanni Cupini, Nicola Fusco, Ermanno Lanconelli and Xiao Zhong who proved a quantitative stability result for the mean value formula, showing that a suitable "mean value gap" (measuring the normalized difference between the average of harmonic functions on a given set and their pointwise value) is bounded from below by the Lebesgue measure of the "gap" between the set and the ball (and, consequently, by the Fraenkel asymmetry of the set). That is, if a domain "almost" satisfies the mean value property, then it must be necessarily close to a ball. Here we investigate the nonlocal counterparts of these results. In particular we will prove a classification result and a stability result, establishing that: if fractional harmonic functions enjoy a suitable exterior average property for a given domain, then the domain is necessarily a ball, a suitable "nonlocal mean value gap" is bounded from below by an appropriate measure of the difference between the set and the ball. Differently from the classical case, some of our arguments rely on purely nonlocal properties, with no classical counterpart, such as the fact that "all functions are locally fractional harmonic up to a small error".

From the generalized Riemann hypothesis for motivic L-functions, we derive an effective version of the Sato-Tate conjecture for an abelian variety A defined over a number field k with connected Sato-Tate group. By effective we mean that we give an upper bound on the error term in the count predicted by the Sato-Tate measure that only depends on certain invariants of A. We discuss three applications of this conditional result. First, for an abelian variety defined over k, we consider a variant of Linnik's problem for abelian varieties that asks for an upper bound on the least norm of a prime whose normalized Frobenius trace lies in a given interval. Second, for an elliptic curve defined over k with complex multiplication, we determine (up to multiplication by a nonzero constant) the asymptotic number of primes whose Frobenius trace attain the integral part of the Hasse-Weil bound. Third, for a pair of abelian varieties defined over k with no common factors up to k-isogeny, we find an upper bound on the least norm of a prime at which the respective Frobenius traces have opposite sign., Comment: 25 pages; refereed version. A table of notation has been added at the end. \S 5 has been removed; its content will be incorporated into a subsequent paper

The recent literature has intensively studied two classes of nonlocal variational problems, namely the ones related to the minimisation of energy functionals that act on functions in suitable Sobolev-Gagliardo spaces, and the ones related to the minimisation of fractional perimeters that act on measurable sets of the Euclidean space. In this article, we relate these two types of variational problems. In particular, we investigate the connection between the nonlocal minimal surfaces and the minimisers of the $W^{s,1}$-seminorm. In particular, we show that a function is a minimiser for the fractional seminorm if and only if its level sets are minimisers for the fractional perimeter, and that the characteristic function of a nonlocal minimal surface is a minimiser for the fractional seminorm; we also provide an existence result for minimisers of the fractional seminorm, an explicit non-uniqueness example for nonlocal minimal surfaces, and a Yin-Yang result describing the full and void patterns of nonlocal minimal surfaces.

Mean value formulas are of great importance in the theory of partial differential equations: many very useful results are drawn, for instance, from the well known equivalence between harmonic functions and mean value properties. In the nonlocal setting of fractional harmonic functions, such an equivalence still holds, and many applications are now-days available. The nonlinear case, corresponding to the $p$-Laplace operator, has also been recently investigated, whereas the validity of a nonlocal, nonlinear, counterpart remains an open problem. In this paper, we propose a formula for the \emph{nonlocal, nonlinear mean value kernel}, by means of which we obtain an asymptotic representation formula for harmonic functions in the viscosity sense, with respect to the fractional (variational) $p$-Laplacian (for $p\geq 2$) and to other gradient dependent nonlocal operators., Comment: 26 pages