A note on smooth forms on analytic spaces

We prove that any smooth mapping between reduced analytic spaces induces a natural pullback operation on smooth differential forms.

Intrinsic non-uniqueness of the acoustic full waveform inverse problem

<p>In the context of seismic imaging, the full waveform inversion (FWI) is more and more popular. Because of its lower numerical cost, the acoustic approximation is often used, especially at the exploration geophysics scale, both for tests and for real data. Moreover, some research domains such as helioseismology face true acoustic medium for which FWI can be useful. In this work, we show that the general acoustic inverse problem based on limited frequency band data is intrinsically non-unique, making any general acoustic FWI impossible. Our work is based on two tools: particle relabelling and homogenization. On the one hand, the particle relabelling method shows it is possible to deform a true medium based on a smooth mapping into a new one without changing the signal recorded at seismic stations. This is a potentially strong source of non-uniqueness for an inverse problem based a seismic data. Nevertheless, in the elastic case, the deformed medium loses the elastic tensor minor symmetries and, in the acoustic case, it implies density anisotropy. It is therefore not a source of non-uniqueness for elastic or isotropic acoustic inverse problems, but it is for the anisotropic acoustic case. On the other hand, the homogenization method shows that any fine-scale medium can be up-scaled into an effective medium without changing the waveforms in a limited frequency band. The effective media are in general anisotropic, both in the elastic and acoustic cases, even if the true media are isotropic at a fine scale. It implies that anisotropy is in general present and needs to be inverted. Therefore, acoustic anisotropy can not be avoided in general. We conclude, based on a particle relabelling and homogenization arguments, that the acoustic FWI solution is in general non-unique. We show, in 2-D numerical FWI examples based on the Gauss-Newton iterative scheme, the effects of this non-uniqueness in the local optimization context. We numerically confirm that the acoustic FWI is in general non-unique and that finding a physical solution is not possible.</p>

Knots connected by wide ribbons

A ribbon is a smooth mapping (possibly self-intersecting) of an annulus [Formula: see text] in 3-space having constant width [Formula: see text]. Given a regular parametrization [Formula: see text], and a smooth unit vector field [Formula: see text] based along [Formula: see text], for a knot [Formula: see text], we may define a ribbon of width [Formula: see text] associated to [Formula: see text] and [Formula: see text] as the set of all points [Formula: see text], [Formula: see text]. For large [Formula: see text], ribbons, and their outer edge curves, may have self-intersections. In this paper, we analyze how the knot type of the outer ribbon edge [Formula: see text] relates to that of the original knot [Formula: see text]. Generically, as [Formula: see text], there is an eventual constant knot type. This eventual knot type is one of only finitely many possibilities which depend just on the vector field [Formula: see text]. The particular knot type within the finite set depends on the parametrized curves [Formula: see text], [Formula: see text], and their interactions. We demonstrate a way to control the curves and their parametrizations so that given two knot types [Formula: see text] and [Formula: see text], we can find a smooth ribbon of constant width connecting curves of these two knot types.

Diffeomorphism groups of tame Cantor sets and Thompson-like groups

The group of ${\mathcal{C}}^{1}$-diffeomorphisms of any sparse Cantor subset of a manifold is countable and discrete (possibly trivial). Thompson’s groups come out of this construction when we consider central ternary Cantor subsets of an interval. Brin’s higher-dimensional generalizations $nV$ of Thompson’s group $V$ arise when we consider products of central ternary Cantor sets. We derive that the ${\mathcal{C}}^{2}$-smooth mapping class group of a sparse Cantor sphere pair is a discrete countable group and produce this way versions of the braided Thompson groups.

TOPOLOGICAL QUANTUM NUMBERS AND CURVATURE — EXAMPLES AND APPLICATIONS

Using the idea of the degree of a smooth mapping between two manifolds of the same dimension we present here the topological (homotopical) classification of the mappings between spheres of the same dimension, vector fields, monopole and instanton solutions. Starting with a review of the elements of Riemannian geometry we also present an original elementary proof of the Gauss–Bonnet theorem and also one of the Poincaré–Hopf theorem.

SCREEN CONFORMAL SUBMERSIONS BETWEEN LIGHTLIKE MANIFOLDS AND SEMI-RIEMANNIAN MANIFOLDS AND THEIR HARMONICITY

In this paper, we introduce a new submersion ϕ : M → N, namely screen conformal submersion, between a lightlike manifold M and a semi-Riemann manifold N. We give examples and show that the lightlike manifold M is shear free under a condition if ϕ : M → N is a screen conformal submersion. Also we define special screen submersions: radical and screen homothetic submersions. We obtain that the radical homothetic submersion ϕ : M → N implies that M is a Reinhart lightlike manifold. Moreover we define and study the lightlike versions of O'Neill's tensors for a horizontally conformal submersion and show that these tensors have different properties from the Riemannian case. Using these tensors, we investigate the relationships between the curvatures of base and total manifolds. Finally, since the trace of a smooth mapping is not meaningful on the radical part of a lightlike manifold, we introduce lightlike harmonic map between lightlike manifolds and semi-Riemannian manifolds, supported by an example. We also give a characterization for a screen conformal submersion to be lightlike harmonic.