{"id":"W2075293395","doi":"10.1016/j.laa.2009.03.028","title":"The case of equality in the Dobrushin–Deutsch–Zenger bound","year":2009,"lang":"en","type":"article","venue":"Linear Algebra and its Applications","topic":"Graph theory and applications","field":"Mathematics","cited_by":4,"is_retracted":false,"has_abstract":false,"ca_institutions":"University of Regina","funders":"","keywords":"Mathematics; Stochastic matrix; Markov chain; Combinatorics; Ergodicity; Eigenvalues and eigenvectors; Upper and lower bounds; Random walk; Stationary distribution; Chain (unit); Random matrix; Matrix (chemical analysis); Continuous-time Markov chain; Rate of convergence; Discrete mathematics; Mathematical analysis; Markov property; Markov model","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":[0.002038891,0.0005533071,0.001510019,0.002037749,0.003412901,0.004458652,0.001647691,0.003132318,0.009258814],"category_scores_gemma":[0.007809173,0.0005217377,0.0007889828,0.002674798,0.005850696,0.009121232,0.004554038,0.004414175,0.001423472],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.001539809,"about_ca_system_score_gemma":0.0008137344,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.001958297,"about_ca_topic_score_gemma":0.0009944499,"domain_scores_codex":[0.998314,0.0004777022,0.00008689761,0.0003562377,0.0003875962,0.0003775744],"domain_scores_gemma":[0.9977842,0.001201335,0.0002278292,0.000329935,0.0002329477,0.0002238963],"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":[0.000006527136,0.000002484126,0.00002491791,0.000007582854,0.0000019944,0.00002020573,0.00003477033,0.0001665454,0.00006717374,0.9984301,0.0003965577,0.0008410947],"study_design_scores_gemma":[0.00000274927,0.000001136456,0.0000275435,0.000003877507,0.000001432903,0.0000190441,0.00001459857,0.0009591002,0.00005310267,0.998252,0.000661645,0.000003669196],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"other","genre_gemma":"empirical","genre_scores_codex":[0.236726,0.008470407,0.2999483,0.02006264,0.00152033,0.0000393792,0.0004896347,0.0002408524,0.4325025],"genre_scores_gemma":[0.9606221,0.001611664,0.01602777,0.001435111,0.0008497035,0.00004854824,0.0001630732,0.0001060805,0.01913598],"genre_candidate":"empirical","genre_consensus":null,"teacher_disagreement_score":0.009258814,"threshold_uncertainty_score":0.03097379,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.04209091367552172,"score_gpt":0.3445068681542274,"score_spread":0.3024159544787057,"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."}}