Г. Л. Литвинов, В. П. Маслов, Г. Б. Шпиз, “Идемпотентный функциональный анализ. Алгебраический подход”, Матем. заметки, 69:5 (2001), 758–797; Math. Notes, 69:5 (2001), 696

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Матем. заметки, 2001, том 69, выпуск 5, страницы 758–797 (Mi mz539)

Эта публикация цитируется в 104 научных статьях (всего в 104 статьях)

Математические обзоры

Идемпотентный функциональный анализ. Алгебраический подход

Г. Л. Литвинов^a, В. П. Маслов^b, Г. Б. Шпиз^a

^a Международный центр "Софус Ли"
^b Московский государственный университет им. М. В. Ломоносова, физический факультет

Аннотация: Статья посвящена идемпотентному функциональному анализу, который является “абстрактной” версией идемпотентного анализа, разработанного В. П. Масловым и его сотрудниками. Дан обзор основных идей идемпотентного анализа. Обсуждается соответствие между концепциями и теоремами традиционного функционального анализа и его идемпотентной версии в духе принципа соответствия Н. Бора в квантовой теории. Представлен алгебраический подход к идемпотентному функциональному анализу. Основные понятия и результаты сформулированы в алгебраических терминах; при этом существенно, что операция идемпотентного сложения может быть определена для бесконечного множества слагаемых. Исследованы идемпотентные аналоги основных принципов линейного анализа и результатов об общем виде линейного функционала и о скалярных произведениях на идемпотентных пространствах.
Библиография: 81 название.

DOI: https://doi.org/10.4213/mzm539

Полный текст: PDF файл (428 kB)

Список литературы: PDF файл HTML файл

Англоязычная версия:
Mathematical Notes, 2001, 69:5, 696–729

Реферативные базы данных:

УДК: 519
Поступило: 17.05.2000

Образец цитирования: Г. Л. Литвинов, В. П. Маслов, Г. Б. Шпиз, “Идемпотентный функциональный анализ. Алгебраический подход”, Матем. заметки, 69:5 (2001), 758–797; Math. Notes, 69:5 (2001), 696–729

Цитирование в формате AMSBIB

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Litvinov G., Maslov V., Shpiz G., “Idempotent (Asymptotic) Mathematics and the Representation Theory”, Asymptotic Combinatorics with Applications to Mathematical Physics, NATO Science Series, Series II: Mathematics, Physics and Chemistry, 77, eds. Malyshev V., Vershik A., Springer, 2002, 267–278
С. Н. Сергеев, “Алгоритмическая сложность одной задачи идемпотентно выпуклой геометрии”, Матем. заметки, 74:6 (2003), 896–901 ; S. N. Sergeev, “Algorithmic Complexity of a Problem of Idempotent Convex Geometry”, Math. Notes, 74:6 (2003), 848–852
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Katsov, Y, “On flat semimodules over semirings”, Algebra Universalis, 51:2–3 (2004), 287
Gaubert, S, “Rational semimodules over the max-plus semiring and geometric approach to discrete event systems”, Kybernetika, 40:2 (2004), 153
Cohen, G, “Duality and separation theorems in idempotent semimodules”, Linear Algebra and Its Applications, 379 (2004), 395
Kim, KH, “Inclines and incline matrices: a survey”, Linear Algebra and Its Applications, 379 (2004), 457
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Г. Л. Литвинов, “Деквантование Маслова, идемпотентная и тропическая математика: краткое введение”, Теория представлений, динамические системы, комбинаторные и алгоритмические методы. XIII, Зап. научн. сем. ПОМИ, 326, ПОМИ, СПб., 2005, 145–182 ; G. L. Litvinov, “The Maslov dequantization, idempotent and tropical mathematics: a brief introduction”, J. Math. Sci. (N. Y.), 140:3 (2007), 426–444
McEneaney W.M., “Curse-of-Dimensionality Free Method for Bellman PDEs with Hamiltonian Written as Maximum of Quadratic Forms”, 2005 44th IEEE Conference on Decision and Control & European Control Conference, Vols 1-8, IEEE Conference on Decision and Control - Proceedings, IEEE, 2005, 42–47
Akian M., Gaubert S., Lakhoua A., “The Max-Plus Finite Element Method for Optimal Control Problems: Further Approximation Results”, 2005 44th IEEE Conference on Decision and Control & European Control Conference, Vols 1-8, IEEE Conference on Decision and Control - Proceedings, IEEE, 2005, 4505–4510
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Litvinov G., “The Maslov Dequantization, Idempotent and Tropical Mathematics: a Very Brief Introduction”, Idempotent Mathematics and Mathematical Physics, Contemporary Mathematics, 377, ed. Litvinov G. Maslov V., Amer Mathematical Soc, 2005, 1–17
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Cohen G., Gaubert S., Quadrat J., Singer I., “Max-Plus Convex Sets and Functions”, Idempotent Mathematics and Mathematical Physics, Contemporary Mathematics Series, 377, eds. Litvinov G., Maslov V., Amer Mathematical Soc, 2005, 105–129
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McEneaney W.M., “Curse-of-Dimensionality Free Method for Bellman PDEs with Semiconvex Hamiltonians”, Proceedings of the 45th IEEE Conference on Decision and Control, Vols 1-14, IEEE Conference on Decision and Control, IEEE, 2006, 967–972
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