{"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":"codex-gemma-dda1882f352a","candidate_categories":[],"consensus_categories":[],"category_scores_codex":[0.0002922434,0.0001859452,0.0004740158,0.0003250578,0.0001382162,0.000008893337,0.0009687641,0.00005511848,0.00004498942],"category_scores_gemma":[0.0004484753,0.0001839888,0.0001070143,0.001141465,0.000148779,0.0005277218,0.0003424619,0.0001801775,0.000001142108],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.0001271224,"about_ca_system_score_gemma":0.0001212668,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.0001749753,"about_ca_topic_score_gemma":0.0001446095,"domain_scores_codex":[0.9979648,0.00007463111,0.0007869569,0.000343662,0.0005192251,0.0003107263],"domain_scores_gemma":[0.9978213,0.0009106435,0.000424206,0.0006604649,0.00009851078,0.0000848424],"domain_codex":null,"domain_gemma":null,"domain_candidate":null,"domain_consensus":null,"study_design_codex":"simulation_or_modeling","study_design_gemma":"simulation_or_modeling","study_design_scores_codex":[0.00001923017,0.0001850196,0.000253912,0.00006845251,0.0000245816,0.000008137015,0.000824314,0.5009299,0.0002687413,0.496663,0.00001539491,0.0007393206],"study_design_scores_gemma":[0.001096863,0.0003532434,0.0006937238,0.0001429226,0.00004153604,0.00001493049,0.0003795993,0.9370602,0.002596667,0.05422139,0.002977173,0.0004216863],"study_design_candidate":"simulation_or_modeling","study_design_consensus":"simulation_or_modeling","genre_codex":"methods","genre_gemma":"empirical","genre_scores_codex":[0.003584255,0.0006269711,0.993611,0.0001709578,0.0003059321,0.0001807345,0.00007057912,0.00006090077,0.001388607],"genre_scores_gemma":[0.8383373,0.0001780248,0.161225,0.00004839952,0.00001532387,0.00009868963,0.00000889863,0.0000198744,0.00006844674],"genre_candidate":"methods","genre_consensus":null,"teacher_disagreement_score":0.834753,"threshold_uncertainty_score":0.7502845,"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."}}