{"id":"W1960754904","doi":"10.1090/s0002-9939-00-05686-0","title":"A functional equation arising from ranked additive and separable utility","year":2000,"lang":"en","type":"article","venue":"Proceedings of the American Mathematical Society","topic":"Functional Equations Stability Results","field":"Mathematics","cited_by":11,"is_retracted":false,"has_abstract":true,"ca_institutions":"University of Waterloo","funders":"Natural Sciences and Engineering Research Council of Canada; Hungarian Scientific Research Fund; National Science Foundation","keywords":"Monotonic function; Convexity; Rank (graph theory); Separable space; Mathematics; Functional equation; Mathematical economics; Reduction (mathematics); Applied mathematics; Pure mathematics; Mathematical analysis; Combinatorics; Economics; Differential equation","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":["insufficient_payload"],"consensus_categories":[],"category_scores_codex":[0.0006301801,0.0001877718,0.0004138777,0.00001652173,0.0002491968,0.00005027387,0.0001856934,0.00006259509,0.001359668],"category_scores_gemma":[0.001466994,0.0001372312,0.0002635334,0.0004335942,0.0008262369,0.0002240308,0.00009108574,0.0002313128,0.00002704334],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.00009009294,"about_ca_system_score_gemma":0.00004283143,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.00006507547,"about_ca_topic_score_gemma":0.000001316469,"domain_scores_codex":[0.9983153,0.00002520108,0.0004965346,0.0003450237,0.0005901845,0.00022776],"domain_scores_gemma":[0.997546,0.00153901,0.0003806366,0.0001777805,0.0002764543,0.00008007888],"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":[0.00369813,0.007485055,0.03206166,0.003076512,0.004488545,7.180454e-7,0.04667305,0.0001113944,0.04473076,0.595844,0.1348716,0.1269586],"study_design_scores_gemma":[0.0005123164,0.00005635855,0.01956509,0.0001039541,0.0001570443,0.000004697621,0.001729296,0.02694708,0.002029461,0.9484856,0.0002210147,0.0001880515],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.9890497,0.00001530555,0.002283416,0.001004983,0.0000221488,0.000394168,0.00007482296,0.00007994229,0.007075453],"genre_scores_gemma":[0.9512761,0.00001653234,0.04795654,0.0001837602,0.00007693287,0.00004979909,0.000006736094,0.00002214014,0.0004114026],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.3526417,"threshold_uncertainty_score":0.9995532,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.04184041683842723,"score_gpt":0.2778799811765093,"score_spread":0.2360395643380821,"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."}}