{"id":"W2727121541","doi":"10.5539/jmr.v9n4p1","title":"Solving Liouville-type Problems on Manifolds with Poincaré-Sobolev Inequality by Broadening q-Energy from Finite to Infinite","year":2017,"lang":"en","type":"article","venue":"Journal of Mathematics Research","topic":"Geometric Analysis and Curvature Flows","field":"Mathematics","cited_by":0,"is_retracted":false,"has_abstract":true,"ca_institutions":"","funders":"City University of New York","keywords":"Mathematics; Sobolev space; Type (biology); Ricci curvature; Generalization; Sobolev inequality; Pure mathematics; Extension (predicate logic); Mathematical analysis; Curvature; Ricci-flat manifold; Scalar curvature; Geometry","routes":{"ca_aff":false,"ca_fund":false,"ca_venue":true,"about_ca":false,"invisible_to_affiliation_only":true},"retraction":null,"screen":null,"direct_labels":[],"prediction":{"model_version":"codex-gemma-dda1882f352a","candidate_categories":["metaresearch"],"consensus_categories":[],"category_scores_codex":[0.005947152,0.0003284838,0.0009651655,0.0009765778,0.0007025112,0.0009562931,0.001571556,0.0002248138,0.0003413024],"category_scores_gemma":[0.008799564,0.0002238887,0.0002283394,0.0008896571,0.0001159596,0.0004066456,0.0004209406,0.00119747,0.00006832723],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.0001612003,"about_ca_system_score_gemma":0.0001951482,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.0002329069,"about_ca_topic_score_gemma":0.0001455876,"domain_scores_codex":[0.9944463,0.0003217218,0.001233256,0.0003454486,0.002967172,0.0006860842],"domain_scores_gemma":[0.9917151,0.003309652,0.001368168,0.001353609,0.001812468,0.0004410009],"domain_codex":null,"domain_gemma":null,"domain_candidate":null,"domain_consensus":null,"study_design_codex":"not_applicable","study_design_gemma":"theoretical_or_conceptual","study_design_scores_codex":[0.00348143,0.01718793,0.02454405,0.004950568,0.01126568,0.002401454,0.04804968,0.006782645,0.04703259,0.173768,0.6089463,0.0515897],"study_design_scores_gemma":[0.01529963,0.01772672,0.01008673,0.02231261,0.001919622,0.000489447,0.02160026,0.04361166,0.02118036,0.7283131,0.1122761,0.005183747],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":null,"genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.9238792,0.0005687077,0.05476139,0.002347365,0.000274336,0.0004649253,0.00003889597,0.0000416041,0.01762355],"genre_scores_gemma":[0.9636233,0.0001659653,0.03331096,0.00009434084,0.000375843,0.00001059086,0.000005547103,0.00007989116,0.002333544],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.5545451,"threshold_uncertainty_score":0.9995497,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.1630054101184059,"score_gpt":0.4008207787637601,"score_spread":0.2378153686453542,"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."}}