Characterization of separability for $LF$-spaces

Vidossich, Giovanni

doi:10.5802/aif.293

Cited by 9 publications

(18 citation statements)

References 3 publications

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“…Proof that DpDq is separable : DpDq is an LF-space as the inductive limit of the Fréchet spaces D K i pDq :" tϕ P C 8 pDq : Supppϕq Ă K i u, i P N, where 31], p.131-133). As such, DpDq is separable iff D K i pDq is separable for all i P N [36], which we now show. The Fréchet topology of D K i pDq is the one induced by the usual Fréchet topology of C 8 pDq when D K i pDq is seen as a subspace of C 8 pDq ( [30], section 1.46).…”

Section: Stochastic Processes Under Linear Differential Constraintsmentioning

confidence: 51%

“…As a time-dependent PDE, we show that performing GPR on wave equation data amounts to reconstructing the corresponding initial conditions of the wave equation from incomplete scattered data. This immediately echoes with questions rising from the inverse problem community [24]; the solution we provide in this article entirely falls in the domain of Photo Acoustic Tomography (PAT), which deals with recovering the initial conditions of wave propagation problems for which equation (36) is the archetype. We quote [25] and the many references therein on that topic.…”

Section: State Of the Artmentioning

confidence: 87%

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Stochastic Processes Under Linear Differential Constraints : Application to Gaussian Process Regression for the 3 Dimensional Free Space Wave Equation

Henderson¹,

Noble²,

Roustant³

2021

Preprint

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Let P be a linear differential operator over D Ă R d and U " pU x q xPD a second order stochastic process. In the first part of this article, we prove a new necessary and sufficient condition for all the trajectories of U to verify the partial differential equation (PDE) T pUq " 0. This condition is formulated in terms of the covariance kernel of U. When compared to previous similar results [1], the novelty of this result is that the equality T pUq " 0 is understood in the sense of distributions, which is a functional analysis framework particularly adapted to the study of PDEs. This theorem provides precious insights during the second part of this article, which is dedicated to performing "physically informed" machine learning on data that is solution to the homogeneous 3 dimensional free space wave equation. We perform Gaussian Process Regression (GPR) on this data, which is a kernel based machine learning technique. To do so, we begin by modelling the solution of this PDE as a trajectory drawn from a Gaussian process (GP). We obtain explicit formulas for the covariance kernel of the corresponding stochastic process; this kernel can then be used for GPR. Our theorem states that this kernel, the trajectories of the corresponding GP and the predictions provided by GPR are all solutions to the wave equation in the sense of distributions. We explore two particular cases : the radial symmetry and the point source. In the case of radial symmetry, we derive "fast to compute" GPR formulas; in the case of the point source, we show a direct link between GPR and the classical triangulation method for point source localization used e.g. in GPS systems. We also show that this use of GPR can be interpreted as a new answer to the ill-posed inverse problem of reconstructing initial conditions for the wave equation with finite dimensional data, and also provides a way of estimating physical parameters from this data as in [2]. We finish by showcasing this physically informed GPR on a number of practical examples.

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Section: Stochastic Processes Under Linear Differential Constraintsmentioning

confidence: 51%

Section: State Of the Artmentioning

confidence: 87%

Stochastic Processes Under Linear Differential Constraints : Application to Gaussian Process Regression for the 3 Dimensional Free Space Wave Equation

Henderson¹,

Noble²,

Roustant³

2021

Preprint

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“…Since the class of feathered topological groups includes both locally compact and metrizable groups, Theorem 8 provides a generalization of the results of Comfort and Itzkowitz [19] for locally compact groups and also well-known results for metrizable groups [20,21].…”

Section: Theorem 8 ([4]mentioning

confidence: 80%

Separability of Topological Groups: A Survey with Open Problems

Leiderman

Morris

2018

Axioms

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Separability is one of the basic topological properties. Most classical topological groups and Banach spaces are separable; as examples we mention compact metric groups, matrix groups, connected (finite-dimensional) Lie groups; and the Banach spaces C ( K ) for metrizable compact spaces K; and ℓ p , for p ≥ 1 . This survey focuses on the wealth of results that have appeared in recent years about separable topological groups. In this paper, the property of separability of topological groups is examined in the context of taking subgroups, finite or infinite products, and quotient homomorphisms. The open problem of Banach and Mazur, known as the Separable Quotient Problem for Banach spaces, asks whether every Banach space has a quotient space which is a separable Banach space. This paper records substantial results on the analogous problem for topological groups. Twenty open problems are included in the survey.

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“…Corollary 4.11. ( [15,22]) If a metrizable group G is isomorphic to a subgroup of a separable topological group, then G is separable.…”

Section: Separability Of Strongly Topological Gyrogroupsmentioning

confidence: 99%

Separability in (strongly) topological gyrogroups

Bao

Zhang

2021

Filomat

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Separability is one of the most basic and important topological properties. In this paper, the separability in (strongly) topological gyrogroups is studied. It is proved that every first-countable left ?-narrow strongly topological gyrogroup is separable. Furthermore, it is shown that if a feathered strongly topological gyrogroup G is isomorphic to a subgyrogroup of a separable strongly topological gyrogroup, then G is separable. Therefore, if a metrizable strongly topological gyrogroup G is isomorphic to a subgyrogroup of a separable strongly topological gyrogroup, then G is separable, and if a locally compact strongly topological gyrogroup G is isomorphic to a subgyrogroup of a separable strongly topological gyrogroup, then G is separable.

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Characterization of separability for $LF$-spaces

Cited by 9 publications

References 3 publications

Stochastic Processes Under Linear Differential Constraints : Application to Gaussian Process Regression for the 3 Dimensional Free Space Wave Equation

Stochastic Processes Under Linear Differential Constraints : Application to Gaussian Process Regression for the 3 Dimensional Free Space Wave Equation

Separability of Topological Groups: A Survey with Open Problems

Separability in (strongly) topological gyrogroups

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