{"id":"W3201270671","doi":"10.1515/acv-2022-0019","title":"Polygons as maximizers of Dirichlet energy or first eigenvalue of Dirichlet-Laplacian among convex planar domains","year":2022,"lang":"en","type":"article","venue":"Advances in Calculus of Variations","topic":"Advanced Mathematical Modeling in Engineering","field":"Computer Science","cited_by":3,"is_retracted":false,"has_abstract":true,"ca_institutions":"University of Ottawa","funders":"Natural Sciences and Engineering Research Council of Canada; Agence Nationale de la Recherche","keywords":"Convexity; Mathematics; Combinatorics; Regular polygon; Dirichlet distribution; Boundary (topology); Laplace operator; Eigenvalues and eigenvectors; Mathematical analysis; Geometry; Boundary value problem; Physics; Quantum mechanics","routes":{"ca_aff":true,"ca_fund":true,"ca_venue":false,"about_ca":false,"invisible_to_affiliation_only":false},"retraction":null,"screen":null,"direct_labels":[],"prediction":{"model_version":"metacan-v3-hybrid-931329e0061c","candidate_categories":[],"consensus_categories":[],"category_scores_codex":[0.001340306,0.001023536,0.0009670804,0.002260753,0.001002879,0.003603618,0.001080053,0.001507087,0.00467277],"category_scores_gemma":[0.006323433,0.0008294897,0.0009842686,0.0009202202,0.002369279,0.002404425,0.00196453,0.001250483,0.0004293486],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.001509121,"about_ca_system_score_gemma":0.0005181521,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.001316738,"about_ca_topic_score_gemma":0.001099714,"domain_scores_codex":[0.9992368,0.0002465739,0.00002821311,0.0001171579,0.0002313319,0.0001399173],"domain_scores_gemma":[0.997721,0.001259756,0.0003472111,0.0001101242,0.0002810131,0.0002808274],"domain_codex":null,"domain_gemma":null,"domain_candidate":null,"domain_consensus":null,"study_design_codex":"theoretical_or_conceptual","study_design_gemma":"theoretical_or_conceptual","study_design_scores_codex":[0.0001618091,0.00006991225,0.002519468,0.0001800025,0.00005118446,0.0004872399,0.0002713087,0.2053279,0.005393831,0.7716311,0.002402528,0.01150377],"study_design_scores_gemma":[0.00002109677,0.00004403446,0.0007477599,0.0000442708,0.00001811132,0.0002217971,0.0002244334,0.6606934,0.001962977,0.3345374,0.001458202,0.00002644612],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"methods","genre_gemma":"empirical","genre_scores_codex":[0.4737282,0.0006301285,0.4792836,0.001292689,0.0001112062,0.0001104875,0.0002824639,0.0001806688,0.0443805],"genre_scores_gemma":[0.9488814,0.0004202953,0.04000363,0.0001318442,0.00009105725,0.0001377342,0.0002245712,0.0002079435,0.009901615],"genre_candidate":"empirical","genre_consensus":null,"teacher_disagreement_score":0.00467277,"threshold_uncertainty_score":0.01563197,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.009071375042931507,"score_gpt":0.2493020744418838,"score_spread":0.2402306993989523,"validation_status":"score_only:v0-immature-baseline","note":"Baseline scores from an immature model (maturity gate not passed). Scores rank; they never assert a category."}}