{"id":"W2479356005","doi":"10.1090/surv/187/16","title":"Domain curvature and best constants in the Hardy-Sobolev inequalities","year":2013,"lang":"en","type":"book-chapter","venue":"Mathematical surveys and monographs","topic":"Nonlinear Partial Differential Equations","field":"Mathematics","cited_by":0,"is_retracted":false,"has_abstract":false,"ca_institutions":"University of British Columbia","funders":"","keywords":"Sobolev space; Curvature; Domain (mathematical analysis); Mathematics; Mathematical analysis; Constant curvature; Sobolev inequality; Pure mathematics; Geometry","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":"metacan-v3-hybrid-931329e0061c","candidate_categories":[],"consensus_categories":[],"category_scores_codex":[0.001458712,0.001018874,0.000745285,0.002187956,0.000743811,0.002607447,0.0008900537,0.001121203,0.005045062],"category_scores_gemma":[0.003597362,0.0005337315,0.0005422547,0.00215427,0.002480303,0.004900074,0.001467327,0.004117362,0.001443495],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.001853511,"about_ca_system_score_gemma":0.0007269879,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.001876285,"about_ca_topic_score_gemma":0.001578779,"domain_scores_codex":[0.9995696,0.0001284189,0.00002257665,0.00009241737,0.0001412209,0.0000458048],"domain_scores_gemma":[0.9988828,0.0006165551,0.00007416286,0.0001225094,0.0001949282,0.0001090129],"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":[0.00001553213,0.00001233984,0.0001555268,0.0001005958,0.000007798321,0.00003178802,0.000123932,0.001393782,0.0004530164,0.9694861,0.007349744,0.02086995],"study_design_scores_gemma":[0.000003522521,0.00001162262,0.0004947244,0.00005319454,0.0000090989,0.00007576364,0.00005628992,0.003491819,0.000275009,0.965184,0.03033219,0.00001284919],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"other","genre_gemma":"methods","genre_scores_codex":[0.05219201,0.1517025,0.2856205,0.01536389,0.003859751,0.00006298484,0.0007079643,0.0003074361,0.4901829],"genre_scores_gemma":[0.5808511,0.09658577,0.08529526,0.002553535,0.007101684,0.0001972113,0.0006901161,0.0005144148,0.226211],"genre_candidate":"methods","genre_consensus":null,"teacher_disagreement_score":0.005045062,"threshold_uncertainty_score":0.01687741,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.09292832223434806,"score_gpt":0.3108317710417252,"score_spread":0.2179034488073771,"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."}}