murmur
发表于 2025-3-25 04:59:28
,Name / Herkunft / Lebensumstände / Bildung,We have seen that a measure preserving automorphism . on a probability space (., ., .) is ergodic if and only if for all ., . ∈ .,.. Two properties stronger than ergodieity discovered by Koopman and von Neumann will now be discussed.
PAD416
发表于 2025-3-25 09:15:38
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capsaicin
发表于 2025-3-25 14:58:03
Henry E. Kyburg, Jr. & Isaac LeviLet (., .) be a standard Borel space. A group ., . ∈ ℝ, of Borel automorphisms on (., .) is called a jointly measurable flow, or simply a flow, if
oxidant
发表于 2025-3-25 17:37:03
https://doi.org/10.1007/978-94-009-7718-1Liouville’s theorem has its origin in classical mechanics. In its simplified version it gives a necessary and sufficient condition for a flow of homeomor-phisms on an open subset in ℝ. to be volume preserving. Following K. R. Parthasarathy we give this version first, followed by a discussion of its version in classical mechanics.
偶然
发表于 2025-3-25 22:47:35
,The Poincaré Recurrence Lemma,Let . be a non-empty set. A .-algebra . on . is a non-empty collection of subsets of . which is closed under countable unions and complements. A set together with a .-algebra . is called a Borel space or a Borel structure (., .).
HEED
发表于 2025-3-26 00:12:33
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抱狗不敢前
发表于 2025-3-26 04:42:52
Ergodicity,A measure preserving Borel automorphism . on a probability space (., ., .) is said to be ergodic if for every . ∈ . invariant under ., .(.) = 0 or .(. − .) = 0.
财政
发表于 2025-3-26 11:17:35
Mixing Conditions and Their Characterisations,We have seen that a measure preserving automorphism . on a probability space (., ., .) is ergodic if and only if for all ., . ∈ .,.. Two properties stronger than ergodieity discovered by Koopman and von Neumann will now be discussed.
aggrieve
发表于 2025-3-26 14:49:31
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坚毅
发表于 2025-3-26 16:51:53
Flows and Their Representations,Let (., .) be a standard Borel space. A group ., . ∈ ℝ, of Borel automorphisms on (., .) is called a jointly measurable flow, or simply a flow, if