On a Solution to a Global Inverse Problem With Respect to Certain Generalized Symmetrizable Kac-Moody Algebras

Leslie, Joshua A.

doi:10.1142/9789812777089_0003

Cited by 5 publications

(13 citation statements)

References 9 publications

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“…Even more, for close reasons, considering a diffeological group, the tangent space at the neutral element may not carry a smooth Lie bracket. This is the difference between diffeological groups and diffeological Lie groups: in the second framework, one considers only diffeological groups which have a diffeological Lie algebra structure on their tangent space at the identity [11]. This is very difficult to prove that a diffeological group is not a diffeological Lie group and actually there is no such rigorously proven example.…”

Section: Preliminaries On Formal Pseudo-differential Operators In a D...mentioning

confidence: 99%

Diffeologies and generalized Kadomtsev-Petviashvili hierarchies

Eslami-Rad,

Magnot,

Reyes

et al. 2024

Contemporary Mathematics

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Section: Preliminaries On Formal Pseudo-differential Operators In a D...mentioning

confidence: 99%

Diffeologies and generalized Kadomtsev-Petviashvili hierarchies

Eslami-Rad,

Magnot,

Reyes

et al. 2024

Contemporary Mathematics

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“…Now, given an algebraic structure, we can define a corresponding compatible diffeological (resp. Frölicher) structure, see for instance [25]. For example, see [19, pp.…”

Section: Preliminaries On Categories Of Regular Frölicher Lie Groupsmentioning

confidence: 99%

“…Let us concentrate on Frölicher Lie groups, following [29] and [25]. If G is a Frölicher Lie group then, after (i) and (ii) above we have that:…”

Section: Preliminaries On Categories Of Regular Frölicher Lie Groupsmentioning

confidence: 99%

On the Cauchy problem for a Kadomtsev-Petviashvili hierarchy on non-formal operators and its relation with a group of diffeomorphisms

Magnot,

Reyes

2024

Dynamics of Partial Differential Equations

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We establish a rigorous link between infinite-dimensional regular Frölicher Lie groups built out of non-formal pseudodifferential operators and the Kadomtsev-Petviashvili hierarchy. We introduce a (parameter-depending) version of the Kadomtsev-Petviashvili hierarchy on a regular Frölicher Lie group of series of non-formal odd-class pseudodifferential operators. We solve its corresponding Cauchy problem, and we establish a link between the dressing operator of our hierarchy and the action of diffeomorphisms and non-formal Sato-like operators on jet spaces. In appendix, we describe the group of Fourier integral operators in which this correspondence seems to take place. Also, motivated by Mulase's work on the KP hierarchy, we prove a group factorization theorem for this group of Fourier integral operators. Contents 1. Introduction 236 2. Preliminaries on categories of regular Frölicher Lie groups 238 3. Preliminaries on pseudodifferential operators 243 4. The h-KP hierarchy with non-formal odd-class operators 246 5. KP equations and Dif f + (S 1 ) 253 Appendix:the group of Dif f + (S 1 )−pseudodifferential operators 254 References 258

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“…Through this definition, d T x X is intrinsically linked with the tangent space at the identity i T Id X Diff(X) described in [27] for any diffeological group (i.e. group equipped with a diffeology which makes composition and inversion smooth), see e.g.…”

Section: Tangent Spaces Diffeology and Group Of Diffeomorphismsmentioning

confidence: 99%

“…(c) the diffeology P(Diff) coincides with the diffeology made of plots which are locally of the form ev x • p, where x ∈ X and p is a plot of the diffeology of Diff(X). We have that i T Id Diff(X) is a diffeological vector space, following [27]. This relation follows from the differentiation of the multiplication of the group: given two paths…”

Section: Tangent Spaces Diffeology and Group Of Diffeomorphismsmentioning

confidence: 99%

On the differential geometry of numerical schemes and weak solutions of functional equations

Magnot

2020

Nonlinearity

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We exhibit differential geometric structures that arise in numerical methods, based on the construction of Cauchy sequences, that are currently used to prove explicitly the existence of weak solutions to functional equations. We describe the geometric framework, highlight several examples and describe how two well-known proofs fit with our setting. The first one is a reinterpretation of the classical proof of an implicit functions theorem in an ILB setting, for which our setting enables us to state an implicit functions theorem without additional norm estimates, and the second one is the finite element method of the Dirichlet problem where the set of triangulations appear as a smooth set of parameters. In both case, smooth dependence on the set of parameters is established. Before that, we develop the necessary theoretical tools, namely the notion of Cauchy diffeology on spaces of Cauchy sequences and a new generalization of the notion of tangent space to a diffeological space.

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On a Solution to a Global Inverse Problem With Respect to Certain Generalized Symmetrizable Kac-Moody Algebras

Cited by 5 publications

References 9 publications

Diffeologies and generalized Kadomtsev-Petviashvili hierarchies

Diffeologies and generalized Kadomtsev-Petviashvili hierarchies

On the Cauchy problem for a Kadomtsev-Petviashvili hierarchy on non-formal operators and its relation with a group of diffeomorphisms

On the differential geometry of numerical schemes and weak solutions of functional equations

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