Linear spaces with probability measures, ultraproducts and contiguity

Mushtari, D. Kh.; Haliullin, S. G.

doi:10.1134/s1995080214020097

Cited by 6 publications

(2 citation statements)

References 3 publications

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“…This is why the construction of the ultraproduct goes through some changes. The details can be found in works [1], [2], [4], [6], [8]. Definition 3.2.…”

Section: Ultraproductsmentioning

confidence: 99%

Ultraproducts of quantum mechanical systems

Khaliullin

2022

Ufimsk. Mat. Zh.

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The study of ultraproducts for various spaces is motivated by an interest in methods of non-standard mathematical analysis, which operates on infinitesimal (or infinitely large) sequences as if they were numbers. Onone hand, a space obtained as a set-theoretic ultraproduct of a sequence of spaces becomes very «rich». On the other hand, it loses some attractive properties of factors. In particular, it has no a natural Hausdorff topology generated by its factors, and the natural 𝜎-algebra of its measurable subsets is not countably generated.If a space «is embedded» into its ultrapower with the preservation of required properties, then the usage of the ultraproduct technique gives some advantages in proving many «standard» assertions.In order to preserve various properties of factors, we need to change the construction of an ultraproduct. For example, by changing the construction of an ultraproduct, it becomes possible to preserve the Hausdorff topology, the structure of a normed space, the structure of operator algebras, von Neumann algebras, and so on.In this paper we discuss the stochastic properties of the so-called quantum mechanical systems in a rather abstract form. Such systems (structures) arise in probability theory, in the theory of operator algebras and in the theory of topological vector spaces. The ultraproducts for sequences of such structures are also defined, and certain properties of these ultraproducts are investigated.The notion of an observable on an event structure is an analogue of a random variable defined on a probability space. An observable is naturally given in the ultraproduct of quantum mechanical systems which is defined in the present paper. We study its probabilistic characteristics. Moreover, ultraproducts of quantum logics are also considered within the framework of ultraproducts for quantum mechanical systems.

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“…This is why the construction of the ultraproduct goes through some changes. The details can be found in works [1], [2], [4], [6], [8]. Definition 3.2.…”

Section: Ultraproductsmentioning

confidence: 99%

Ultraproducts of quantum mechanical systems

Khaliullin

2022

Ufimsk. Mat. Zh.

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“…The infinite-dimensional generalizations of Propositions 1 and 2 were achieved during the 60s and the 70s, and are effective for nuclear spaces only. See the introduction of [44] for a general discussion on this subject.…”

Section: Definitionmentioning

confidence: 99%

Gaussian and sparse processes are limits of generalized Poisson processes

Fageot

Uhlmann

Unser

2020

Applied and Computational Harmonic Analysis

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The theory of sparse stochastic processes offers a broad class of statistical models to study signals. In this framework, signals are represented as realizations of random processes that are solution of linear stochastic differential equations driven by white Lévy noises. Among these processes, generalized Poisson processes based on compound-Poisson noises admit an interpretation as random L-splines with random knots and weights. We demonstrate that every generalized Lévy process-from Gaussian to sparse-can be understood as the limit in law of a sequence of generalized Poisson processes. This enables a new conceptual understanding of sparse processes and suggests simple algorithms for the numerical generation of such objects.

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Contiguity and Entire Separability of States on von Neumann Algebras

Haliullin

2017

Int J Theor Phys

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Linear spaces with probability measures, ultraproducts and contiguity

Cited by 6 publications

References 3 publications

Ultraproducts of quantum mechanical systems

Ultraproducts of quantum mechanical systems

Gaussian and sparse processes are limits of generalized Poisson processes

Contiguity and Entire Separability of States on von Neumann Algebras

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