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Синтетическая Теория Меры, Вероятности и Оптимального Транспорта
1. Чурилов М. В. SOTT-D. Метод Синтетической Оптимизации через Топос-Теоретическую Динамику:
2. Когезионный Подход к Глобальной Оптимизации Неинтегрируемых Систем и Фундаментальному
3. Переосмыслению Динамической Сложности. Рукопись, 2025.
4. Чурилов М. В. Синтетический Вариационный Анализ. Конструктивная Двойственность,
5. Монотонные Операторы и Геометрия Оптимизации в Гладких Топосах. Рукопись, 2025.
6. Чурилов М. В. Синтетический Негладкий Анализ. Топология Пенона, локали значений и обобщённые
7. производные в гладком топосе. Рукопись, 2026.
8. Kock, A. Synthetic Differential Geometry. 2nd ed. Cambridge University Press, 2006.
9. Moerdijk, I.; Reyes, G. E. Models for Smooth Infinitesimal Analysis. Springer, 1991.
10. Mac Lane, S.; Moerdijk, I. Sheaves in Geometry and Logic. Springer, 1992.
11. Johnstone, P. T. Stone Spaces. Cambridge University Press, 1982.
12. Vickers, S. Topology via Logic. Cambridge University Press, 1989.
13. Gierz, G.; Hofmann, K. H.; Keimel, K.; Lawson, J. D.; Mislove, M.; Scott, D. S. Continuous Lattices and
14. Domains. Cambridge University Press, 2003.
15. Goubault-Larrecq, J. Non-Hausdorff Topology and Domain Theory. Cambridge University Press, 2013.
16. Jones, C. Probabilistic Non-determinism. PhD Thesis, University of Edinburgh, 1990.
17. Bishop, E.; Bridges, D. Constructive Analysis. Springer, 1985.
18. Rockafellar, R. T. Convex Analysis. Princeton University Press, 1970.
19. Rockafellar, R. T.; Wets, R. J.-B. Variational Analysis. Springer, 1998.
20. Brézis, H. Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert.
21. North-Holland, 1973.
22. Bauschke, H. H.; Combettes, P. L. Convex Analysis and Monotone Operator Theory in Hilbert Spaces. 2nd ed.
23. Springer, 2017.
24. Villani, C. Topics in Optimal Transportation. American Mathematical Society, 2003.
25. Villani, C. Optimal Transport: Old and New. Springer, 2009.
26. Santambrogio, F. Optimal Transport for Applied Mathematicians. Birkhäuser, 2015.
27. Ambrosio, L.; Gigli, N.; Savaré, G. Gradient Flows in Metric Spaces and in the Space of Probability Measures.
28. 2nd ed. Birkhäuser, 2008.
29. Benamou, J.-D.; Brenier, Y. A computational fluid mechanics solution to the Monge–Kantorovich mass
30. transfer problem. Numerische Mathematik 84 (2000), 375–393.
31. Jordan, R.; Kinderlehrer, D.; Otto, F. The variational formulation of the Fokker–Planck equation. SIAM
32. Journal on Mathematical Analysis 29 (1998), 1–17.
33. McCann, R. J. A convexity principle for interacting gases. Advances in Mathematics 128 (1997), 153–179.
34. Otto, F. The geometry of dissipative evolution equations: the porous medium equation. Communications in
35. Partial Differential Equations 26 (2001), 101–174.
36. Léonard, C. A survey of the Schrödinger problem and some of its connections with optimal transport.
37. Discrete and Continuous Dynamical Systems A 34 (2014), 1533–1574.
38. Cuturi, M. Sinkhorn distances: Lightspeed computation of optimal transport. In: Advances in Neural
39. Information Processing Systems 26, 2013.
40. Peyré, G.; Cuturi, M. Computational Optimal Transport. Foundations and Trends in Machine Learning 11
41. (2019), 355–607.
42. Evans, L. C.; Gariepy, R. F. Measure Theory and Fine Properties of Functions. Revised edition. CRC Press, 2015.
43. Federer, H. Geometric Measure Theory. Springer, 1969.
44. Heinonen, J. Lectures on Analysis on Metric Spaces. Springer, 2001.
45. Cheeger, J. Differentiability of Lipschitz functions on metric measure spaces. Geometric and Functional
46. Analysis 9 (1999), 428–517.
47. Shanmugalingam, N. Newtonian spaces: an extension of Sobolev spaces to metric measure spaces. Revista
48. Matemática Iberoamericana 16 (2000), 243–279.
49. Ambrosio, L.; Di Marino, S. Equivalent definitions of BV space and of total variation on metric measure
50. spaces. Journal of Functional Analysis 266 (2014), 4150–4188.
51. Fritz, T. A synthetic approach to Markov kernels, conditional independence and theorems on sufficient
52. statistics. Advances in Mathematics 370 (2020), 107239.
53. Lawvere, F. W. Toward the description in a smooth topos of the dynamically possible motions and
54. deformations of a continuous body. Cahiers de Topologie et Géométrie Différentielle Catégoriques 21 (1980),
55. no. 4, 377–392.
56. Carrillo, J. A.; McCann, R. J.; Villani, C. Kinetic equilibration rates for granular media and related equations:
57. entropy dissipation and mass transportation estimates. Revista Matemática Iberoamericana 19 (2003), 971–
58. 1018.