{"id":"W4394779347","doi":"10.1016/j.aim.2024.109640","title":"The Allen-Cahn equation on the complete Riemannian manifolds of finite volume","year":2024,"lang":"en","type":"article","venue":"Advances in Mathematics","topic":"Geometric Analysis and Curvature Flows","field":"Mathematics","cited_by":1,"is_retracted":false,"has_abstract":false,"ca_institutions":"University of Toronto","funders":"National Science Foundation","keywords":"Mathematics; Hypersurface; Allen–Cahn equation; Riemannian manifold; Morse code; Finite volume method; Metric (unit); Minimal volume; Manifold (fluid mechanics); Pure mathematics; Mathematical analysis; Mathematical physics; Ricci curvature; Geometry; Hermitian manifold; Physics","routes":{"ca_aff":true,"ca_fund":false,"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":[0.001668131,0.0001950727,0.0003283155,0.0001543828,0.0001512634,0.0001080995,0.0004805125,0.00007250987,0.0001791867],"category_scores_gemma":[0.001515596,0.0001008044,0.000171006,0.0007833637,0.000120859,0.0001718106,0.00007037104,0.0002844755,0.0001396501],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.00004270615,"about_ca_system_score_gemma":0.00002316778,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.000003171154,"about_ca_topic_score_gemma":0.00007386895,"domain_scores_codex":[0.9981896,0.0001067805,0.0006806062,0.0002090488,0.0005582697,0.0002556686],"domain_scores_gemma":[0.9931486,0.005855318,0.0002435739,0.0006396426,0.00008434885,0.00002850253],"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":[0.000006604515,0.0001296749,0.00005664384,0.0004374099,0.00008485404,0.000006297727,0.002135641,0.0008183055,0.00001559339,0.9864925,0.002016081,0.007800421],"study_design_scores_gemma":[0.00009790144,0.000061559,0.00007771551,0.0003757044,0.00007627314,0.000002813137,0.001685903,0.1262255,0.0000375791,0.793043,0.0781718,0.0001442323],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"methods","genre_gemma":"empirical","genre_scores_codex":[0.08747236,0.04772196,0.6365998,0.01688665,0.002867971,0.004023287,0.000158519,0.0006004556,0.203669],"genre_scores_gemma":[0.9795052,0.001826779,0.01315344,0.0001099867,0.0001359416,0.00008647484,0.00000783163,0.00005261142,0.0051217],"genre_candidate":"empirical","genre_consensus":null,"teacher_disagreement_score":0.8920329,"threshold_uncertainty_score":0.4110686,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.05408008773202515,"score_gpt":0.3126571459262434,"score_spread":0.2585770581942182,"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."}}