{"id":"W1999370902","doi":"10.1016/j.aim.2004.08.004","title":"Smallest singular value of random matrices and geometry of random polytopes","year":2004,"lang":"en","type":"article","venue":"Advances in Mathematics","topic":"Point processes and geometric inequalities","field":"Mathematics","cited_by":263,"is_retracted":false,"has_abstract":false,"ca_institutions":"University of Alberta","funders":"","keywords":"Mathematics; Polytope; Singular value; Random matrix; Combinatorics; Isomorphism (crystallography); Dimension (graph theory); Matrix (chemical analysis); Multivariate random variable; Regular polygon; Convex hull; Random variable; Geometry; Statistics; Eigenvalues and eigenvectors","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":[],"consensus_categories":[],"category_scores_codex":[0.001056188,0.0002289851,0.0009168779,0.000491584,0.00004456071,0.00001921396,0.0002628299,0.0001006543,0.0000351348],"category_scores_gemma":[0.002441969,0.0001859875,0.0001071634,0.0008505156,0.0002078843,0.0002807729,0.0001003283,0.0001302524,0.000001620643],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.0000279805,"about_ca_system_score_gemma":0.0000518208,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.00003593216,"about_ca_topic_score_gemma":0.00001827905,"domain_scores_codex":[0.9980106,0.00004127788,0.001062991,0.0002030725,0.0004068148,0.0002752253],"domain_scores_gemma":[0.9966754,0.002162229,0.0006549587,0.0003214786,0.0001366094,0.00004932358],"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":[0.00029733,0.0009224017,0.002005158,0.01452689,0.0001245226,0.00001810425,0.007221542,0.001108375,0.0006834293,0.9684482,0.000009258109,0.004634782],"study_design_scores_gemma":[0.005533765,0.0001218023,0.00009435933,0.001033701,0.00008857121,0.00002695812,0.002576993,0.0003456739,0.00766902,0.9821396,0.0001293745,0.0002401165],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.889296,0.01964579,0.08765362,0.00006141734,0.0001032487,0.0004284301,0.0000249778,0.00003610737,0.002750424],"genre_scores_gemma":[0.8344231,0.004488026,0.1609031,0.00001470756,0.00003171776,0.00001462119,0.000002169249,0.00002622728,0.00009640724],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.07324945,"threshold_uncertainty_score":0.7584351,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.01830570104178682,"score_gpt":0.3034575945440424,"score_spread":0.2851518935022556,"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."}}