{"id":"W4412448311","doi":"10.1016/j.jde.2025.113623","title":"Birkhoff center and statistical behavior of competitive dynamical systems","year":2025,"lang":"en","type":"article","venue":"Journal of Differential Equations","topic":"Mathematical Dynamics and Fractals","field":"Mathematics","cited_by":0,"is_retracted":false,"has_abstract":false,"ca_institutions":"","funders":"National Key Research and Development Program of China; National Natural Science Foundation of China; Canadian Anesthesiologists' Society","keywords":"Mathematics; Center (category theory); Mathematical economics; Statistics","routes":{"ca_aff":false,"ca_fund":true,"ca_venue":false,"about_ca":false,"invisible_to_affiliation_only":true},"retraction":null,"screen":null,"direct_labels":[],"prediction":{"model_version":"codex-gemma-dda1882f352a","candidate_categories":[],"consensus_categories":[],"category_scores_codex":[0.0001853537,0.0001003586,0.0004166111,0.0001604323,0.00004549446,0.0000432703,0.0001063152,0.00006808816,0.00008798291],"category_scores_gemma":[0.000656782,0.00007597476,0.0000931348,0.0000854301,0.0001072337,0.00005873156,0.00005108729,0.0001769252,0.000001286239],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.00003387391,"about_ca_system_score_gemma":0.00005433994,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.000005229084,"about_ca_topic_score_gemma":0.000007758969,"domain_scores_codex":[0.9986954,0.0000871618,0.0007612135,0.00007885593,0.000262595,0.00011479],"domain_scores_gemma":[0.9979494,0.001255258,0.0003517761,0.0001089449,0.0002553939,0.0000792944],"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":[0.00002027256,0.0006687748,0.0004440123,0.0001862905,0.00009894016,0.000004859171,0.00007793822,0.00000232793,0.000965328,0.9969837,0.0001071781,0.0004403668],"study_design_scores_gemma":[0.00503642,0.0007701935,0.02975476,0.002112198,0.002223953,0.0001187968,0.00131116,0.1931269,0.0004657716,0.764282,0.0002893087,0.0005085082],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"methods","genre_gemma":"empirical","genre_scores_codex":[0.2969616,0.00003077267,0.7012554,0.00009293675,0.0002609143,0.000162229,0.0001326564,0.000005266579,0.001098237],"genre_scores_gemma":[0.9951195,0.000009554273,0.004622506,0.00000735553,0.00003559028,0.000005676257,0.000007743707,0.000008353032,0.0001837389],"genre_candidate":"empirical","genre_consensus":null,"teacher_disagreement_score":0.6981578,"threshold_uncertainty_score":0.3098161,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.02792239244122619,"score_gpt":0.3337287446897302,"score_spread":0.3058063522485041,"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."}}