{"query": "Optimal transport — the geometry of moving probability", "count": 20, "results": [{"id": "card_n_d59ca677ea24", "title": "Optimal transport — the geometry of moving probability", "shelf": "science", "surface": "secular", "snippet": "Benamou–Brenier: the distance between two distributions is the least kinetic energy of a\nfluid that carries one to the other, subject to the continuity equation. Jordan–Kinderlehrer–Otto\n(1998): the F"}, {"id": "card_c_583b49254af6", "title": "Fluid probability dynamics — one continuity  ↔ Optimal transport — the geometry of moving p", "shelf": "connections", "surface": null, "snippet": "geometry (optimal transport: distance = least kinetic energy to carry one distribution to another; Fokker–Planck is the gradient flow of free energy in Wasserstein space)  — a concord the card itself "}, {"id": "card_c_e655b3fcef54", "title": "Optimal transport instantiates Fluid probability dynamics", "shelf": "connections", "surface": null, "snippet": "The geometry of moving probability — Fokker–Planck as a Wasserstein gradient flow."}, {"id": "card_n_8530c72a2201", "title": "Fluid probability dynamics — one continuity equation across physics, geometry, and ML", "shelf": "science", "surface": "secular", "snippet": "One equation wears many clothes. Probability is a conserved fluid: the continuity equation\ndρ/dt + ∇·J = 0 (current J = ρv) says density is never created or destroyed — it only flows. The\nsame skeleto"}, {"id": "card_src_word_descriptive_geometry", "title": "descriptive geometry", "shelf": "dictionary", "surface": "secular", "snippet": "descriptive geometry: (noun) the geometry of properties that remain invariant under projection — syn: projective geometry"}, {"id": "card_src_word_projective_geometry", "title": "projective geometry", "shelf": "dictionary", "surface": "secular", "snippet": "projective geometry: (noun) the geometry of properties that remain invariant under projection — syn: descriptive geometry"}, {"id": "card_src_book_57355", "title": "The Foundations of Mathematics: A Contribution to the Philosophy of Geometry — Paul Carus", "shelf": "gutenberg", "surface": "secular", "snippet": "The Foundations of Mathematics: A Contribution to the Philosophy of Geometry, by Paul Carus. Subjects: Geometry -- Foundations. Read the full text (public domain): https://www.gutenberg.org/ebooks/573"}, {"id": "card_src_book_17384", "title": "The Foundations of Geometry — David Hilbert", "shelf": "gutenberg", "surface": "secular", "snippet": "The Foundations of Geometry, by David Hilbert. Subjects: Geometry -- Foundations. Read the full text (public domain): https://www.gutenberg.org/ebooks/17384"}, {"id": "card_src_word_analytic_geometry", "title": "analytic geometry", "shelf": "dictionary", "surface": "secular", "snippet": "analytic geometry: (noun) the use of algebra to study geometric properties; operates on symbols defined in a coordinate system — syn: analytical geometry, coordinate geometry"}, {"id": "card_src_word_analytical_geometry", "title": "analytical geometry", "shelf": "dictionary", "surface": "secular", "snippet": "analytical geometry: (noun) the use of algebra to study geometric properties; operates on symbols defined in a coordinate system — syn: analytic geometry, coordinate geometry"}, {"id": "card_src_word_coordinate_geometry", "title": "coordinate geometry", "shelf": "dictionary", "surface": "secular", "snippet": "coordinate geometry: (noun) the use of algebra to study geometric properties; operates on symbols defined in a coordinate system — syn: analytic geometry, analytical geometry"}, {"id": "card_src_book_37681", "title": "The Teaching of Geometry — David Eugene Smith", "shelf": "gutenberg", "surface": "secular", "snippet": "The Teaching of Geometry, by David Eugene Smith. Subjects: Geometry -- Study and teaching. Read the full text (public domain): https://www.gutenberg.org/ebooks/37681"}, {"id": "card_src_book_52091", "title": "An essay on the foundations of geometry — Bertrand Russell", "shelf": "gutenberg", "surface": "secular", "snippet": "An essay on the foundations of geometry, by Bertrand Russell. Subjects: Geometry -- Foundations. Read the full text (public domain): https://www.gutenberg.org/ebooks/52091"}, {"id": "card_src_word_conditional_probability", "title": "conditional probability", "shelf": "dictionary", "surface": "secular", "snippet": "conditional probability: (noun) the probability that an event will occur given that one or more other events have occurred — syn: contingent probability"}, {"id": "card_src_word_contingent_probability", "title": "contingent probability", "shelf": "dictionary", "surface": "secular", "snippet": "contingent probability: (noun) the probability that an event will occur given that one or more other events have occurred — syn: conditional probability"}, {"id": "card_src_book_26373", "title": "The Elements of non-Euclidean Geometry — Julian Lowell Coolidge", "shelf": "gutenberg", "surface": "secular", "snippet": "The Elements of non-Euclidean Geometry, by Julian Lowell Coolidge. Subjects: Geometry, Non-Euclidean. Read the full text (public domain): https://www.gutenberg.org/ebooks/26373"}, {"id": "card_src_word_affine_geometry", "title": "affine geometry", "shelf": "dictionary", "surface": "secular", "snippet": "affine geometry: (noun) the geometry of affine transformations"}, {"id": "card_src_word_fractal_geometry", "title": "fractal geometry", "shelf": "dictionary", "surface": "secular", "snippet": "fractal geometry: (noun) (mathematics) the geometry of fractals"}, {"id": "card_src_word_plane_geometry", "title": "plane geometry", "shelf": "dictionary", "surface": "secular", "snippet": "plane geometry: (noun) the geometry of 2-dimensional figures"}, {"id": "card_src_word_solid_geometry", "title": "solid geometry", "shelf": "dictionary", "surface": "secular", "snippet": "solid geometry: (noun) the geometry of 3-dimensional space"}]}