We prove existence and regularity results for weak solutions of the following elliptic system:{−div(|Du|p−2Du)=f(x,u)in Ω,u=0on ∂Ω, where u=(u1,…,um), p>1, and Ω⊂RN is a bounded domain. We also consider the special casef(x,u)=λ|u|p−2u+|u|q−2u, and we prove a classification result. In particular, we show that any least energy solution is of the form (c1ω,…,cmω), where c=(c1,…,cm)∈Sm−1 (the (m−1)-sphere in Rm) and ω is a positive solution of the corresponding scalar equation.
Advanced deep learning techniques are opening up new possibilities for seismic monitoring using distributed acoustic sensing (DAS) technology. In this study, we present a lightweight Transformer-U-Net architecture that combines the fine-grained spatial resolution of U-Net with the global contextual modeling capabilities of Transformers, to accurately pick P- and S-wave arrival times in DAS data. Our model is trained and validated using DAS recordings from the Campi Flegrei caldera in southern Italy, a region that is both geologically complex and seismically active. Even when trained on a limited number of manually labeled events, the proposed approach achieves high phase-picking performance by effectively combining local and global features. The model attains an F1-score of up to 90% for both P and S phases, outperforming state-of-the-art methods. To assess its robustness, we also evaluate the pre-trained model without prior fine-tuning on an independent DAS dataset from Iceland, where it maintains good performance despite different geological settings and acquisition set-up. Finally, we apply the model to several months of continuous DAS recordings, demonstrating its scalability and potential for real-time seismic monitoring.
Green mold, caused by Penicillium digitatum, is the most damaging postharvest citrus fruit disease. Conventionally, it is managed with synthetic fungicides. This study evaluated a cell-free, non-viable, lactic acid bacteria (LAB)-activated lemon peel-derived formulation (LM + LAB), produced by fermenting a lemon peel-based medium with Lactiplantibacillus plantarum N3B2, under two application scenarios. LM + LAB was compared with imazalil (standard synthetic fungicide), the yeast Candida oleophila strain O, a potassium phosphite-based formulation, phosphorous acid, and an untreated control., Sweet orange fruit were treated either before inoculation or 4 h after inoculation with P. digitatum. Fruit were stored for 7 days at 6 degrees C and then at 18 degrees C until 14 days post-inoculation. Disease incidence was assessed at 7, 10, and 14 days, sporulation reduction at 14 days, and fruit quality throughout storage. Imazalil provided the most effective control. LM + LAB significantly reduced disease incidence under both application scenarios. At 14 days, incidence was reduced from 78% to 38% when LM + LAB was applied before inoculation and from 79% to 15% when it was applied 4 h after inoculation. Sporulation was reduced by 43% and 64%, respectively. When applied post-inoculation, LM + LAB was as effective as imazalil and C. oleophila in reducing disease incidence and suppressed sporulation more effectively than the yeast. Potassium phosphite and phosphorous acid were the least effective substances. None of the treatments caused weight loss or significantly affected firmness, total soluble solids, juice pH, or titratable acidity of fruit. Results show LM + LAB is a promising natural product for management of citrus green mold.
Explainable artificial intelligence (XAI) has predominantly focused on generating model-centric explanations that approximate the behavior of black-box models. However, such explanations often overlook a fundamental aspect of interpretability: different users require different explanations depending on their goals, preferences, and cognitive constraints. Although recent work has explored user-centric and personalized explanations, most existing approaches rely on heuristic adaptations or implicit user modeling, lacking a principled framework for representing and learning individual preferences. In this paper, we consider Preference-Based Explainable Artificial Intelligence (PREF-XAI), a novel perspective that reframes explanation as a preference-driven decision problem. Within PREF-XAI, explanations are not treated as fixed outputs, but as alternatives to be evaluated and selected according to user-specific criteria. In the PREF-XAI perspective, here we propose a methodology that combines rule-based explanations with formal preference learning. User preferences are elicited through a ranking of a small set of candidate explanations and modeled via an additive utility function inferred using robust ordinal regression. Experimental results on real-world datasets show that PREF-XAI can accurately reconstruct user preferences from limited feedback, identify highly relevant explanations, and discover novel explanatory rules not initially considered by the user. Beyond the proposed methodology, this work establishes a connection between XAI and preference learning, opening new directions for interactive and adaptive explanation systems.
When decision makers rely on large sets of if-then decision rules to guide complex choices, information overload quickly becomes a barrier to effective use. The Dominance-based Rough Set Approach (DRSA) is a widely used preference learning method whose primary output is a set of if-then decision rules incorporating preference orders over attributes and decision classes. Although effective for knowledge representation and classification, large rule sets can overwhelm decision makers and hinder interpretation. To address this challenge, this paper proposes a new decision graph model for visualising dominance based if-then decision rules. The construction process begins with a preprocessing phase that extracts a binary decision matrix from rule descriptions. To avoid generating nonexistent rules, this matrix may require transformation through column splitting or a combination of column splitting and ordering-based strategies. The transformed matrix is then used to build a directed acyclic decision graph in which nodes represent elementary conditions and decisions, and edges denote their logical relationships. Each path from the root to a decision node corresponds to a complete decision rule. The paper introduces the core concepts of decision graphs and presents algorithms for their construction, linearisation, and exploitation. Both linearisation and exploitation algorithms are based on the Breadth-First Search graph traversal algorithm. The approach is demonstrated using real-world cultural heritage data from Europe, and expert feedback indicates that decision graphs reduce cognitive effort, improve communication, and provide a clear and accessible complement to traditional textual and tabular representations of decision rules.
