{"id":"W1974798866","doi":"10.4153/cjm-2006-016-0","title":"Extremal Metric for the First Eigenvalue on a Klein Bottle","year":2006,"lang":"en","type":"article","venue":"Canadian Journal of Mathematics","topic":"Geometric Analysis and Curvature Flows","field":"Mathematics","cited_by":68,"is_retracted":false,"has_abstract":true,"ca_institutions":"Université de Montréal; McGill University","funders":"Natural Sciences and Engineering Research Council of Canada; Alfred P. Sloan Foundation; Fonds Québécois de la Recherche sur la Nature et les Technologies; National Science Foundation","keywords":"Klein bottle; Mathematics; Torus; Conjecture; Laplace operator; Eigenvalues and eigenvectors; Embedding; Metric (unit); Möbius strip; Combinatorics; Geodesic; Surface (topology); Equivalence of metrics; Infimum and supremum; Equilateral triangle; Pure mathematics; Metric space; Mathematical analysis; Convex metric space; Geometry; Quantum mechanics; Physics","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":"metacan-v3-hybrid-931329e0061c","candidate_categories":[],"consensus_categories":[],"category_scores_codex":[0.000607989,0.0004599802,0.0006758167,0.002211324,0.001242558,0.001903565,0.0006030262,0.001177075,0.003927205],"category_scores_gemma":[0.003731892,0.0003134428,0.0002708242,0.000527956,0.001937636,0.001925349,0.001365865,0.0006831364,0.000262488],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.001591553,"about_ca_system_score_gemma":0.0004520594,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.001010113,"about_ca_topic_score_gemma":0.0008603705,"domain_scores_codex":[0.9994505,0.0001208571,0.00002788958,0.0000866968,0.0001792342,0.0001348238],"domain_scores_gemma":[0.998792,0.0002707566,0.000221497,0.00009225341,0.000231605,0.0003919516],"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":[0.00009767239,0.00005514791,0.003184614,0.0001055927,0.00002569826,0.0004658307,0.0003079402,0.01927296,0.01793361,0.9513634,0.001931608,0.005255937],"study_design_scores_gemma":[0.00003278381,0.0001189411,0.009596101,0.00005204994,0.00001993022,0.0009167136,0.0005268962,0.1745283,0.006813067,0.8041184,0.003162728,0.0001140282],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.9583229,0.0003696854,0.02207021,0.000771308,0.00005778012,0.00002477283,0.000103404,0.000177499,0.01810238],"genre_scores_gemma":[0.9963903,0.00006413312,0.002374616,0.00003424663,0.00001794133,0.00001572917,0.00003678584,0.00001864052,0.001047561],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.003927205,"threshold_uncertainty_score":0.01313782,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.04055072261840174,"score_gpt":0.2584114461582507,"score_spread":0.217860723539849,"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."}}