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One of the classical models in mathematical biology is the Lotka-Volterra competition model, describing the dynamics of two populations competing for resources. Two possible regimes in this system are given by their coexistence or extinction of a weaker population. In a distributed system with diffusive spatial coupling, travelling fronts occur, corresponding to transitions between stationary states. In this work we will consider the competition model extended by extra interaction between involved populations which is given by chemotactic coupling, namely, assuming that one species produces a chemical agent which causes the taxis of another species. It is known that in a one-species model (i.e. Keller-Segel model) production of chemoattractor results in formation of stationary periodic (or Turing-type) patterns. In this work, we will investigate conditions for the formation of stationary periodic patterns in a two-species competition model with chemotaxis. We show that in this system periodic patterns can emerge in the course of Turing-type instability (classical way) or from a stable steady state, corresponding to the extinction of one of the species, due to a finite, or over-threshold, amplitude disturbance. We study the characteristics of emerging periodic pattern, such as its amplitude and wavelength, by means of Fourier analysis. We also perform computational simulations to verify our analytical results.

Computational approaches to predicting mental health conditions in social media have been substantially explored in the past years. Multiple surveys have been published on this topic, providing the community with comprehensive accounts of the research in this area. Among all mental health conditions, depression is the most widely studied due to its worldwide prevalence. The COVID-19 global pandemic, starting in early 2020, has had a great impact on mental health worldwide. Harsh measures employed by governments to slow the spread of the virus (e.g., lockdowns) and the subsequent economic downturn experienced in many countries have significantly impacted people's lives and mental health. Studies have shown a substantial increase of above 50% in the rate of depression in the population. In this context, we present a survey on natural language processing (NLP) approaches to modeling depression in social media, providing the reader with a post-COVID-19 outlook. This survey contributes to the understanding of the impacts of the pandemic on modeling depression in social media. We outline how state-of-the-art approaches and new datasets have been used in the context of the COVID-19 pandemic. Finally, we also discuss ethical issues in collecting and processing mental health data, considering fairness, accountability, and ethics.

Mathematics has long been conveyed through natural language, primarily for human understanding. With the rise of mechanized mathematics and proof assistants, there is a growing need to understand informal mathematical text, yet most existing benchmarks focus solely on English, overlooking other languages. This paper introduces RoMath, a Romanian mathematical reasoning benchmark suite comprising three datasets: RoMath-Baccalaureate, RoMath-Competitions and RoMath-Synthetic, which cover a range of mathematical domains and difficulty levels, aiming to improve non-English language models and promote multilingual AI development. By focusing on Romanian, a low-resource language with unique linguistic features, RoMath addresses the limitations of Anglo-centric models and emphasizes the need for dedicated resources beyond simple automatic translation. We benchmark several open-weight language models, highlighting the importance of creating resources for underrepresented languages. We make the code and dataset available., Comment: 4 Figures, 12 Tables

For any $p \in ( 1, +\infty)$, we give a new inequality for the first nontrivial Neumann eigenvalue $\mu _ p (\Omega, \varphi)$ of the $p$-Laplacian on a convex domain $\Omega \subset \mathbb{R}^N$ with a power-concave weight $\varphi$. Our result improves the classical estimate in terms of the diameter, first stated in a seminal paper by Payne and Weinberger: we add in the lower bound an extra term depending on the second largest John semi-axis of $\Omega$ (equivalent to a power of the width in the special case $N = 2$). The power exponent in the extra term is sharp, and the constant in front of it is explicitly tracked, thus enlightening the interplay between space dimension, nonlinearity and power-concavity. Moreover, we attack the stability question: we prove that, if $\mu _ p (\Omega, \varphi)$ is close to the lower bound, then $\Omega$ is close to a thin cylinder, and $\varphi$ is close to a function which is constant along its axis. As intermediate results, we establish a sharp $L^ \infty$ estimate for the associated eigenfunctions, and we determine the asymptotic behaviour of $\mu _ p (\Omega, \varphi)$ for varying weights and domains, including the case of collapsing geometries., Comment: 1 figure

We consider the minimization property of a Gagliardo-Slobodeckij seminorm which can be seen as the fractional counterpart of the classical problem of functions of least gradient and which is related to the minimization of the nonlocal perimeter functional. We discuss continuity properties for this kind of problem. In particular, we show that, under natural structural assumptions, the minimizers are bounded and continuous in the interior of the ambient domain (and, in fact, also continuous up to the boundary under some mild additional hypothesis). We show that these results are also essentially optimal, since in general the minimizer is not necessarily continuous across the boundary., Comment: 15 pages, 1 figure

We aim to learn the functional co-response group: a group of taxa whose co-response effect (the representative characteristic of the group showing the total topological abundance of taxa) co-responds (associates well statistically) to a functional variable. Different from the state-of-the-art method, we model the soil microbial community as an ecological co-occurrence network with the taxa as nodes (weighted by their abundance) and their relationships (a combination from both spatial and functional ecological aspects) as edges (weighted by the strength of the relationships). Then, we design a method called gFlora which notably uses graph convolution over this co-occurrence network to get the co-response effect of the group, such that the network topology is also considered in the discovery process. We evaluate gFlora on two real-world soil microbiome datasets (bacteria and nematodes) and compare it with the state-of-the-art method. gFlora outperforms this on all evaluation metrics, and discovers new functional evidence for taxa which were so far under-studied. We show that the graph convolution step is crucial to taxa with relatively low abundance (thus removing the bias towards taxa with higher abundance), and the discovered bacteria of different genera are distributed in the co-occurrence network but still tightly connected among themselves, demonstrating that topologically they fill different but collaborative functional roles in the ecological community., Comment: BIOKDD accepted. Note: the first version of this paper is not approved by all authors. the second version is the official version for BIOKDD camera-ready

The fundamental gap conjecture proved by Andrews and Clutterbuck in 2011 provides the sharp lower bound for the difference between the first two Dirichlet Laplacian eigenvalues in terms of the diameter of a convex set in $\mathbb{R}^N$. The question concerning the rigidity of the inequality, raised by Yau in 1990, was left open. Going beyond rigidity, our main result strengthens Andrews-Clutterbuck inequality, by quantifying geometrically the excess of the gap compared to the diameter in terms of flatness. The proof relies on a localized, variational interpretation of the fundamental gap, allowing a dimension reduction via the use of convex partitions \`a la Payne-Weinberger: the result stems by combining a new sharp result for one dimensional Schr\"odinger eigenvalues with measure potentials, with a thorough analysis of the geometry of the partition into convex cells. As a by-product of our approach, we obtain a quantitative form of Payne-Weinberger inequality for the first nontrivial Neumann eigenvalue of a convex set in $\mathbb{R}^N$, thus proving, in a stronger version, a conjecture from 2007 by Hang-Wang., Comment: 5 figures

The main result of the paper shows that the regular $n$-gon is a local minimizer for the first Dirichlet-Laplace eigenvalue among $n$-gons having fixed area for $n \in \{5,6\}$. The eigenvalue is seen as a function of the coordinates of the vertices in $\Bbb R^{2n}$. Relying on fine regularity results of the first eigenfunction in a convex polygon, an explicit a priori estimate is given for the eigenvalues of the Hessian matrix associated to the discrete problem, whose coefficients involve the solutions of some Poisson equations with singular right hand sides. The a priori estimates, in conjunction with certified finite element approximations of these singular PDEs imply the local minimality for $n \in \{5,6\}$. All computations, including the finite element computations, are realized using interval arithmetic.

Biological pattern formation is one of the most intriguing phenomena in nature. Simplest examples of such patterns are represented by travelling waves and stationary periodic patterns which occur during various biological processes including morphogenesis and population dynamics. Formation of these patterns in populations of motile microorganisms such as Dictyostelium discoideum and E. coli have been shown in a number of experimental studies. Conditions for formation of various types of patterns are commonly addressed in mathematical studies of dynamical systems containing diffusive and advection terms. In this work, we do mathematical study of spatio-temporal patterns forming in growing population of chemotactically active bacteria. In particular, we perform linear analysis to find conditions for formation of stationary periodic patterns, and nonlinear (Fourier) analysis to find characteristics, such as amplitude and wavelength, of these patterns. We verify our analytical results by means of numerical simulations.

A generalized two-dimensional cubic Lotka-Volterra model with infinitesimal parameters is studied. Three different cases have been considered, one non-degenerate and two degenerate. The local behavior of the model has been studied in the three cases. Six bifurcation diagrams with thirty different regions have been obtained in the non-degenerate case, respectively, sixteen diagrams with forty regions in the two degenerate cases.

We propose a novel model-selection method for dynamic networks. Our approach involves training a classifier on a large body of synthetic network data. The data is generated by simulating nine state-of-the-art random graph models for dynamic networks, with parameter range chosen to ensure exponential growth of the network size in time. We design a conceptually novel type of dynamic features that count new links received by a group of vertices in a particular time interval. The proposed features are easy to compute, analytically tractable, and interpretable. Our approach achieves a near-perfect classification of synthetic networks, exceeding the state-of-the-art by a large margin. Applying our classification method to real-world citation networks gives credibility to the claims in the literature that models with preferential attachment, fitness and aging fit real-world citation networks best, although sometimes, the predicted model does not involve vertex fitness., Comment: Main text: 13 pages, 4 figures. Supplementary: 12 pages; Textual changes

For an open set $\Om \subset \R^2$ let $\lambda(\Om)$ denote the bottom of the spectrum of the Dirichlet Laplacian acting in $L^2(\Om)$. Let $w_\Om$ be the torsion function for $\Om$, and let $\|.\|_p$ denote the $L^p$ norm. It is shown there exists $\eta>0$ such that $\|w_{\Om}\|_{\infty} \lambda(\Om)\ge 1+\eta$ for any non-empty, open, simply connected set $\Om\subset \R^2$ with $\lb(\Om) >0$. Moreover, if the measure $|\Om|$ of $\Om$ is finite, then $\|w_{\Om}\|_1\lambda(\Om)\le (1-\eta)|\Om|$., Comment: 18 pages, 1 figure

Depression, a prominent contributor to global disability, affects a substantial portion of the population. Efforts to detect depression from social media texts have been prevalent, yet only a few works explored depression detection from user-generated video content. In this work, we address this research gap by proposing a simple and flexible multi-modal temporal model capable of discerning non-verbal depression cues from diverse modalities in noisy, real-world videos. We show that, for in-the-wild videos, using additional high-level non-verbal cues is crucial to achieving good performance, and we extracted and processed audio speech embeddings, face emotion embeddings, face, body and hand landmarks, and gaze and blinking information. Through extensive experiments, we show that our model achieves state-of-the-art results on three key benchmark datasets for depression detection from video by a substantial margin. Our code is publicly available on GitHub., Comment: Accepted at 46th European Conference on Information Retrieval (ECIR 2024)

In this paper, we study the existence of solutions of the equation $(-\Delta)_1^s u=f$ in a bounded open set with Lipschitz boundary $\Omega\subset \Rn$, vanishing on $\Co \Omega$, for some given $s\in (0,1)$, and asymptotics as $p\to 1$ of solutions of $(-\Delta)_p^s u=f$. We obtain existence and convergence by comparing the $L^{\frac{n}{s}}$ norm of $f$ to the sharp fractional Sobolev constant, or, when $f$ is non-negative, the weighted fractional Cheegar constant to $1$ -- in this case, the results are sharp. We further prove that solutions are "flat" on sets of positive Lebesgue measure., Comment: 35 pages

Let A and A' be abelian varieties defined over a number field k such that Hom(A,A') = 0. Under the Generalized Riemann hypothesis for motivic L-functions attached to A and A', we show that there exists a prime p of k of good reduction for A and A' at which the Frobenius traces of A and A' are nonzero and differ by sign, and such that the norm Nm(p) is O(log(2NN')^2), where N and N' respectively denote the absolute conductors of A and A'. Our method extends that of Chen, Park, and Swaminathan who considered the case in which A and A' are elliptic curves., Comment: 9 pages. Includes material formerly appearing in arXiv:2002.08807

This paper presents the methodology and data used for the automatic extraction of the Romanian Academic Word List (Ro-AWL). Academic Word Lists are useful in both L2 and L1 teaching contexts. For the Romanian language, no such resource exists so far. Ro-AWL has been generated by combining methods from corpus and computational linguistics with L2 academic writing approaches. We use two types of data: (a) existing data, such as the Romanian Frequency List based on the ROMBAC corpus, and (b) self-compiled data, such as the expert academic writing corpus EXPRES. For constructing the academic word list, we follow the methodology for building the Academic Vocabulary List for the English language. The distribution of Ro-AWL features (general distribution, POS distribution) into four disciplinary datasets is in line with previous research. Ro-AWL is freely available and can be used for teaching, research and NLP applications.

In the study of concavity properties of positive solutions to nonlinear elliptic partial differential equations the diffusion and the nonlinearity are typically independent of the space variable. In this paper we obtain new results aiming to get almost concavity results for a relevant class of anisotropic semilinear elliptic problems with spatially dependent source and diffusion., Comment: 16 pages, types corrected, to appear on Asymptotic Analysis

In this work, we present the contribution of the BLUE team in the eRisk Lab task on searching for symptoms of depression. The task consists of retrieving and ranking Reddit social media sentences that convey symptoms of depression from the BDI-II questionnaire. Given that synthetic data provided by LLMs have been proven to be a reliable method for augmenting data and fine-tuning downstream models, we chose to generate synthetic data using ChatGPT for each of the symptoms of the BDI-II questionnaire. We designed a prompt such that the generated data contains more richness and semantic diversity than the BDI-II responses for each question and, at the same time, contains emotional and anecdotal experiences that are specific to the more intimate way of sharing experiences on Reddit. We perform semantic search and rank the sentences' relevance to the BDI-II symptoms by cosine similarity. We used two state-of-the-art transformer-based models (MentalRoBERTa and a variant of MPNet) for embedding the social media posts, the original and generated responses of the BDI-II. Our results show that using sentence embeddings from a model designed for semantic search outperforms the approach using embeddings from a model pre-trained on mental health data. Furthermore, the generated synthetic data were proved too specific for this task, the approach simply relying on the BDI-II responses had the best performance.

We answer the question whether, when forming constellations in the night sky, people in astronomical cultures around the world and through time consistently imagined and assigned the same symbolism to the same (type of) star group. Evidence of semantic parallels has so far been anecdotal. We use two complementary definitions for a star group: (1) a star group in a fixed region of the sky (regardless of its exact star composition), and (2) a star group with a particular shape and brightness (regardless of its location in the sky). Over a dataset of 2003 constellations from 82 astronomical cultures, we find many semantic parallels which are likely naturally induced by the shape and composition of the star pattern. In certain cultural regions, geometric and group symbols are perceived consistently over small and uniformly bright star groups, naturalistic humanoids in large star groups with non-linear minimum spanning tree (MST) and stars inside the convex hull, and reptiles in star groups with low aspect ratio or linear MST. These naturally induced semantics, seemingly endogenous to certain sky patterns, show that there are universal (rather than learnt) patterns behind forming and naming constellations., Comment: Part 2 of arXiv:2110.12329 published in PLOS ONE 17(7): e0272270 (2022). Shares the dataset

Pruning-at-Initialization (PaI) algorithms provide Sparse Neural Networks (SNNs) which are computationally more efficient than their dense counterparts, and try to avoid performance degradation. While much emphasis has been directed towards \emph{how} to prune, we still do not know \emph{what topological metrics} of the SNNs characterize \emph{good performance}. From prior work, we have layer-wise topological metrics by which SNN performance can be predicted: the Ramanujan-based metrics. To exploit these metrics, proper ways to represent network layers via Graph Encodings (GEs) are needed, with Bipartite Graph Encodings (BGEs) being the \emph{de-facto} standard at the current stage. Nevertheless, existing BGEs neglect the impact of the inputs, and do not characterize the SNN in an end-to-end manner. Additionally, thanks to a thorough study of the Ramanujan-based metrics, we discover that they are only as good as the \emph{layer-wise density} as performance predictors, when paired with BGEs. To close both gaps, we design a comprehensive topological analysis for SNNs with both linear and convolutional layers, via (i) a new input-aware Multipartite Graph Encoding (MGE) for SNNs and (ii) the design of new end-to-end topological metrics over the MGE. With these novelties, we show the following: (a) The proposed MGE allows to extract topological metrics that are much better predictors of the accuracy drop than metrics computed from current input-agnostic BGEs; (b) Which metrics are important at different sparsity levels and for different architectures; (c) A mixture of our topological metrics can rank PaI algorithms more effectively than Ramanujan-based metrics. The codebase is publicly available at https://github.com/eliacunegatti/mge-snn., Comment: Accepted at Transactions on Machine Learning Research (TMLR)

Let $\Omega\subset\mathbb{R}^n$ be an open set with same volume as the unit ball $B$ and let $\lambda_k(\Omega)$ be the $k$-th eigenvalue of the Laplace operator of $\Omega$ with Dirichlet boundary conditions in $\partial\Omega$. In this work, we answer the following question: if $\lambda_1(\Omega)-\lambda_1(B)$ is small, how large can $|\lambda_k(\Omega)-\lambda_k(B)|$ be ? We establish quantitative bounds of the form $|\lambda_k(\Omega)-\lambda_k(B)|\le C (\lambda_1(\Omega)-\lambda_1(B))^\alpha$ with sharp exponents $\alpha$ depending on the multiplicity of $\lambda_k(B)$. We first show that such an inequality is valid with $\alpha=1/2$ for any $k$, improving previous known results and providing the sharpest possible exponent. Then, through the study of a vectorial free boundary problem, we show that one can achieve the better exponent $\alpha=1$ if $\lambda_{k}(B)$ is simple. We also obtain a similar result for the whole cluster of eigenvalues when $\lambda_{k}(B)$ is multiple, thus providing a complete answer to the question above. As a consequence of these results, we obtain the persistence of the ball as minimizer for a large class of spectral functionals which are small perturbations of the fundamental eigenvalue on the one hand, and a full reverse Kohler-Jobin inequality on the other hand, solving an open problem formulated by M. Van Den Berg, G. Buttazzo and A. Pratelli.

Given a non-increasing and radially symmetric kernel in $L ^ 1 _{\rm loc} (\Bbb{R} ^ 2 ; \Bbb{R}_+)$, we investigate counterparts of the classical Hardy-Littlewood and Riesz inequalities when the class of admissible domains is the family of polygons with given area and $N$ sides. The latter corresponds to study the polygonal isoperimetric problem in nonlocal version. We prove that, for every $N \geq 3$, the regular $N$-gon is optimal for Hardy-Littlewood inequality. Things go differently for Riesz inequality: while for $N = 3$ and $N = 4$ it is known that the regular triangle and the square are optimal, for $N\geq 5$ we prove that symmetry or symmetry breaking may occur (i.e. the regular $N$-gon may be optimal or not), depending on the value of $N$ and on the choice of the kernel.

In this paper we study extremal behaviors of the mean to max ratio of the $p$-torsion function with respect to the geometry of the domain. For $p$ larger than the dimension of the space $N$, we prove that the upper bound is uniformly below $1$, contrary to the case $p \in (1,N]$. For $p=+\infty$, in two dimensions, we prove that the upper bound is asymptotically attained by a disc from which is removed a network of points consisting on the vertices of a tiling of the plane with regular hexagons of vanishing size.

Depression detection from user-generated content on the internet has been a long-lasting topic of interest in the research community, providing valuable screening tools for psychologists. The ubiquitous use of social media platforms lays out the perfect avenue for exploring mental health manifestations in posts and interactions with other users. Current methods for depression detection from social media mainly focus on text processing, and only a few also utilize images posted by users. In this work, we propose a flexible time-enriched multimodal transformer architecture for detecting depression from social media posts, using pretrained models for extracting image and text embeddings. Our model operates directly at the user-level, and we enrich it with the relative time between posts by using time2vec positional embeddings. Moreover, we propose another model variant, which can operate on randomly sampled and unordered sets of posts to be more robust to dataset noise. We show that our method, using EmoBERTa and CLIP embeddings, surpasses other methods on two multimodal datasets, obtaining state-of-the-art results of 0.931 F1 score on a popular multimodal Twitter dataset, and 0.902 F1 score on the only multimodal Reddit dataset., Comment: Accepted at ECIR 2023

In this paper we study obstacles immerged in a Stokes flow with Navier boundary conditions. We prove the existence and regularity of an obstacle with minimal drag, among all shapes of prescribed volume and controlled surface area, taking into account that these shapes may naturally develop geometric features of codimension 1. The existence is carried out in the framework of free discontinuity problems and leads to a relaxed solution in the space of special functions of bounded deformation (SBD). In dimension 2, we prove that the solution is classical.