{"id":"W2323853349","doi":"10.1142/9789812701732_0022","title":"Schoenberg's Theorem for Positive Definite Functions on Heisenberg Groups","year":2005,"lang":"en","type":"article","venue":"Advances in Analysis","topic":"Mathematical Analysis and Transform Methods","field":"Mathematics","cited_by":1,"is_retracted":false,"has_abstract":false,"ca_institutions":"York University","funders":"","keywords":"Positive-definite matrix; Mathematics; Heisenberg group; Pure mathematics; Physics; Quantum mechanics; 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":"metacan-v3-hybrid-931329e0061c","candidate_categories":[],"consensus_categories":[],"category_scores_codex":[0.001684005,0.00155745,0.001209094,0.002950465,0.001684569,0.002476244,0.001189679,0.001782035,0.007215876],"category_scores_gemma":[0.00311388,0.000473822,0.001310668,0.001532094,0.004074649,0.006754881,0.002060633,0.003514842,0.001676122],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.001014509,"about_ca_system_score_gemma":0.0007087276,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.0007350368,"about_ca_topic_score_gemma":0.0005925687,"domain_scores_codex":[0.999378,0.0001596066,0.00004005851,0.0001215611,0.0002155153,0.00008525628],"domain_scores_gemma":[0.9991865,0.0003805505,0.00008599636,0.00008619917,0.0001736329,0.00008713893],"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":[0.00001455459,0.000007352393,0.00004445484,0.00003400419,0.000007702729,0.00003743605,0.0001399265,0.000661068,0.0007644991,0.9928858,0.001428699,0.003974555],"study_design_scores_gemma":[0.000004224682,0.000005746253,0.0000492144,0.000004257711,0.000002804568,0.00002959005,0.00001466645,0.001677444,0.0002134903,0.9964535,0.001538227,0.000006870651],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"methods","genre_gemma":"empirical","genre_scores_codex":[0.1208046,0.003151025,0.6964906,0.005591677,0.001483454,0.0001139192,0.0005687689,0.000554457,0.1712415],"genre_scores_gemma":[0.792371,0.004126165,0.09934267,0.002252103,0.002226874,0.0004201447,0.0005153959,0.0005482255,0.09819743],"genre_candidate":"empirical","genre_consensus":null,"teacher_disagreement_score":0.007215876,"threshold_uncertainty_score":0.02413952,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.03026463310617735,"score_gpt":0.3677595198058848,"score_spread":0.3374948866997075,"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."}}