{"id":"W2015490407","doi":"10.1016/j.aim.2009.02.010","title":"Pseudo-Riemannian geodesics and billiards","year":2009,"lang":"en","type":"article","venue":"Advances in Mathematics","topic":"Advanced Differential Geometry Research","field":"Physics and Astronomy","cited_by":87,"is_retracted":false,"has_abstract":false,"ca_institutions":"University of Toronto","funders":"","keywords":"Geodesic; Mathematics; Geodesic map; Euclidean space; Symplectic geometry; Pure mathematics; Manifold (fluid mechanics); Riemannian geometry; Dynamical billiards; Euclidean geometry; Solving the geodesic equations; Space (punctuation); Riemannian manifold; Exponential map (Riemannian geometry); Mathematical analysis; Geometry; Sectional curvature; Scalar curvature; Computer science","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":[0.0007421839,0.001073797,0.000884834,0.00277326,0.001750068,0.003124128,0.001219516,0.002521183,0.006092374],"category_scores_gemma":[0.002226791,0.0005537278,0.0007577448,0.001812085,0.004119866,0.005017004,0.00160845,0.002695173,0.0008558917],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.001488245,"about_ca_system_score_gemma":0.0005401377,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.001656147,"about_ca_topic_score_gemma":0.00141674,"domain_scores_codex":[0.999633,0.0001088211,0.00002554311,0.00006841013,0.0001217749,0.00004255142],"domain_scores_gemma":[0.9993713,0.0001801569,0.0001127715,0.00009074255,0.0001258891,0.0001189865],"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":[0.000003301068,0.000002763193,0.00002763595,0.000008266483,0.000001448585,0.0000195372,0.00006277769,0.0001966633,0.0001335488,0.9981565,0.0004729972,0.0009146548],"study_design_scores_gemma":[0.000003530909,0.000004415287,0.0001087224,0.000003813824,0.00000160155,0.00005851874,0.00003420178,0.001096428,0.00005463803,0.9946973,0.003931758,0.000005035597],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.4558769,0.03549742,0.1818565,0.02123494,0.004072763,0.0001197738,0.0007368602,0.0005116394,0.3000932],"genre_scores_gemma":[0.8575107,0.009657982,0.02803691,0.001368524,0.002701263,0.00009151721,0.0005331833,0.0002067021,0.09989329],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.006092374,"threshold_uncertainty_score":0.02038103,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.01137825571900157,"score_gpt":0.3192691883422871,"score_spread":0.3078909326232855,"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."}}