Recursive Moving Frames for Lie Pseudo-Groups

Olver, Peter J.; Valiquette, Francis

doi:10.1007/s00025-018-0818-5

Cited by 8 publications

(6 citation statements)

References 55 publications

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“…One then iterates this procedure by recursively increasing the jet (derivative) order, until (in favorable cases) one has eliminated all the pseudo-group parameters; the resulting non-phantom normalized invariants form a complete system of functionally independent differential invariants for the action. For details on the recursive procedure, see [20,27]. Now, the formulae for the prolonged pseudo-group transformations, i.e., the lifted invariants, can become exceedingly complicated, and the practical implementation of the moving frame algorithm may be beyond even powerful modern symbolic computer algebra systems.…”

Section: Preliminary Materialsmentioning

confidence: 99%

“…The choice of cross-section or, equivalently, normal form will specify or "normalize" expressions for the group parameters that serve to prescribe the equivariant moving frame map. More generally, through a recursive procedure, [27], one can introduce a succession of partial normal forms (partial cross-sections) which can be used to normalize more and more of the group parameters. Substitution of the results into the prolonged transformation formulae can be regarded as an "invariantization" process that maps functions, differential forms, differential operators, etc., to their (partially) invariant counterparts.…”

Section: Introductionmentioning

confidence: 99%

“… 27). where the second version is obtained by substituting the formulae (4.25) for µ Z , µ Z into the first, soG = E + A D + B D, H = F + 2 Re (C D).In particular, according to (4.24), (4.26) and the assumption that σ = 0,H| p = F| p = − (γ + 1)(α + β + γ − 2) σ = 0,and hence the lifted invariant polynomial H is nonzero in some neighborhood of p. Thus, we can use (4.27) to normalize the Maurer-Cartan form Re…”

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confidence: 99%

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Normal forms, moving frames, and differential invariants for nondegenerate hypersurfaces in C^2

Olver¹,

Sabzevari²,

Valiquette³

2022

Preprint

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We use the method of equivariant moving frames to revisit the problem of normal forms and equivalence of nondegenerate real hypersurfaces M ⊂ C 2 under the pseudo-group action of holomorphic transformations. The moving frame recurrence formulae allow us to systematically and algorithmically recover the results of Chern and Moser for hypersurfaces that are either non-umbilic at a point p ∈ M or umbilic in an open neighborhood of it. In the former case, the coefficients of the normal form expansion, when expressed as functions of the jet of the hypersurface at the point, provide a complete system of functionally independent differential invariants that can be used to solve the equivalence problem. We prove that under a suitable genericity condition, the entire algebra of differential invariants for such hypersurfaces can be generated, through the operators of invariant differentiation, by a single real differential invariant of order 7. We then apply moving frames to construct new convergent normal forms for nondegenerate real hypersurfaces at singularly umbilic points, namely those umbilic points where the hypersurface is not identically umbilic around them.

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Section: Preliminary Materialsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Normal forms, moving frames, and differential invariants for nondegenerate hypersurfaces in C^2

Olver¹,

Sabzevari²,

Valiquette³

2022

Preprint

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“…Therefore, submanifolds where the rank of the Lie matrix is not maximal occur where a moving frame does not exist. In those situations it is still possible to construct partial moving frames, [72,75,84]. Intuitively, a partial moving frame is the G-equivariant map that one obtains when some of the group parameters cannot be normalized during the normalization procedure.…”

Section: Moving Frame Approachmentioning

confidence: 99%

Symmetry-Preserving Numerical Schemes

Bihlo

Valiquette

2017

Symmetries and Integrability of Difference Equations

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In these lectures we review two procedures for constructing finite difference numerical schemes that preserve symmetries of differential equations. The first approach is based on Lie's infinitesimal symmetry generators, while the second method uses the novel theory of equivariant moving frames. The advantages of both techniques are discussed and illustrated with the Schwarzian differential equation, the Korteweg-de Vries equation and Burgers' equation. Numerical simulations are presented and innovative techniques for obtaining better invariant numerical schemes are introduced. New research directions and open problems are indicated at the end of these notes.

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“…The Taylor coefficients of a normal form capture the values of all invariants at the origin, hence they effectively characterise the geometry of the problem. In several articles, Olver and his collaborators gave what they call the recurrence formulae to express the higher order differential invariants as derivatives of certain primary invariants [10,11,12]. This is not the focus of the paper for now, but we will show in our future publications how normal forms can be used to find all the homogeneous models classified by Hsu-Kamran in [5].…”

Section: Introductionmentioning

confidence: 99%

Normal Forms of second order Ordinary Differential Equations $y_{xx}=J(x,y,y_{x})$ under Fibre-Preserving Maps

Foo¹,

Heyd²,

Merker³

2021

Preprint

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We study the equivalence problem of classifying second order ordinary differential equations y xx = J(x, y, y x ) modulo fibre-preserving point transformations x −→ ϕ(x), y −→ ψ(x, y) by using Moser's method of normal forms. We first compute a basis of the Lie algebra g {yxx=0} of fibre-preserving symmetries of y xx = 0. In the formal theory of Moser's method, this Lie algebra is used to give an explicit description of the set of normal forms N , and we show that the set is an ideal in the space of formal power series. We then show the existence of the normal forms by studying flows of suitable vector fields with appropriate corrections by the Cauchy-Kovalevskaya theorem. As an application, we show how normal forms can be used to prove that the identical vanishing of Hsu-Kamran primary invariants directly imply that the second order differential equation is fibre-preserving point equivalent to y xx = 0.

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Recursive Moving Frames for Lie Pseudo-Groups

Cited by 8 publications

References 55 publications

Normal forms, moving frames, and differential invariants for nondegenerate hypersurfaces in C^2

Normal forms, moving frames, and differential invariants for nondegenerate hypersurfaces in C^2

Symmetry-Preserving Numerical Schemes

Normal Forms of second order Ordinary Differential Equations $y_{xx}=J(x,y,y_{x})$ under Fibre-Preserving Maps

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