{"id":"W2055009503","doi":"10.1103/physrevd.80.081301","title":"Invariant joint distribution of a stationary random field and its derivatives: Euler characteristic and critical point counts in 2 and 3D","year":2009,"lang":"en","type":"article","venue":"Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D, Particles, fields, gravitation, and cosmology","topic":"Geometry and complex manifolds","field":"Mathematics","cited_by":52,"is_retracted":false,"has_abstract":true,"ca_institutions":"University of Alberta","funders":"","keywords":"Mathematics; Maxima and minima; Hessian matrix; Random field; Isotropy; Invariant (physics); Euler characteristic; Joint probability distribution; Mathematical analysis; Euler's formula; Computation; Excursion; Statistical physics; Applied mathematics; Physics; Mathematical physics; Algorithm; Statistics","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":["metaepi_narrow"],"consensus_categories":[],"category_scores_codex":[0.0006487237,0.0003399581,0.001067111,0.00007220229,0.0001780551,0.00004428905,0.0001070538,0.0001369481,0.000002807923],"category_scores_gemma":[0.003054787,0.0003045773,0.00009453765,0.0003238622,0.0003041481,0.0002313979,0.00007451793,0.0003482706,0.000007974209],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.00001687743,"about_ca_system_score_gemma":0.00004906908,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.00003199766,"about_ca_topic_score_gemma":0.00002143192,"domain_scores_codex":[0.9973348,0.0005700179,0.0008888996,0.0005685166,0.0002335692,0.0004042045],"domain_scores_gemma":[0.9964557,0.002487731,0.000294234,0.0002497945,0.0002828596,0.0002296442],"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":[0.0002519931,0.0009031625,0.007191218,0.003707563,0.00009161502,0.00003017009,0.001853708,0.000001190881,0.003042323,0.9562703,0.002022406,0.02463438],"study_design_scores_gemma":[0.0009856462,0.0007762727,0.02786686,0.0008530125,0.0002606128,0.00006032312,0.0001579243,0.0001965126,0.0003140658,0.9678977,0.0003170217,0.0003140883],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.9736359,0.01631443,0.002095622,0.007048408,0.00006330944,0.0007622883,0.00004551432,0.00002767087,0.000006851768],"genre_scores_gemma":[0.9806794,0.01563495,0.000542802,0.002869316,0.00007517072,0.0001173119,0.00006076855,0.00001487508,0.000005431121],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.02432029,"threshold_uncertainty_score":0.9999406,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.03441163148901913,"score_gpt":0.3436290336155591,"score_spread":0.30921740212654,"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."}}