{"id":"W2167472711","doi":"10.1112/s0024609305004261","title":"ON THE NUMBER OF SUMS AND PRODUCTS","year":2005,"lang":"en","type":"article","venue":"Bulletin of the London Mathematical Society","topic":"Limits and Structures in Graph Theory","field":"Mathematics","cited_by":76,"is_retracted":false,"has_abstract":true,"ca_institutions":"University of British Columbia","funders":"","keywords":"Mathematics; Mathematics Subject Classification; Subject (documents); Set (abstract data type); Finite set; Combinatorics; Discrete mathematics; Mathematical analysis; Computer science; Library science","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":["insufficient_payload"],"consensus_categories":[],"category_scores_codex":[0.0009536892,0.0001593737,0.0003093095,0.000005918744,0.0001810063,0.00001115511,0.0004153206,0.0000986095,0.001140568],"category_scores_gemma":[0.001344841,0.00007208241,0.0002710094,0.00009501523,0.000600064,0.000009580858,0.0002776558,0.0002346751,0.00002091061],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.00001354047,"about_ca_system_score_gemma":0.00001068724,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.0000023159,"about_ca_topic_score_gemma":1.816808e-7,"domain_scores_codex":[0.9987395,0.0001232337,0.0003915461,0.0001702062,0.0003882813,0.0001872459],"domain_scores_gemma":[0.9970556,0.001946796,0.000235216,0.0006524357,0.00007596306,0.00003397625],"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":[0.0000149939,0.0001343007,0.00008960743,0.0003286673,0.00006546072,7.23664e-8,0.001462616,0.000001526391,0.0001745281,0.9407186,0.05674889,0.0002607728],"study_design_scores_gemma":[0.0002778746,0.00002079112,0.0002260519,0.0001467203,0.00006449938,0.00001391806,0.0002016451,0.00007796752,0.008417531,0.9807966,0.009650533,0.0001058675],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.9649127,0.00008371811,0.000146849,0.01730715,0.00007001372,0.000480303,0.000009229896,0.00002709922,0.0169629],"genre_scores_gemma":[0.970776,0.00002820212,0.02512965,0.0008566145,0.0000922597,0.00001411323,2.144738e-7,0.00002693319,0.003076078],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.04709836,"threshold_uncertainty_score":0.9997725,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.01865851947645158,"score_gpt":0.2644217652044575,"score_spread":0.245763245728006,"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."}}