arXiv · 0803.3958
We prove a sharp stability estimate for the problem of reconstructing a symmetric 2-tensor from its integrals along all maximal geodesics on a simple manifold.
Plamen Stefanov. 2008-03-27. A sharp stability estimate in tensor tomography. https://doi.org/10.1088/1742-6596/124/1/012007
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We introduce and study a scalar $q$-deformation of the cross-ratio on $\mathbb P^1(\mathbb Q)$. Our construction is based on the notion of $q$-deformed rational numbers due to Morier-Genoud and the author. The $q$-cross-ratio is invariant under $\mathrm{PSL}(2,\mathbb{Z})$, while elements of determinant $-1$ of $\mathrm{PGL}(2,\mathbb{Z})$ act by $q\mapsto q^{-1}$. A principal result is its relation to $q$-deformed Coxeter friezes associated with rational polygons. The expansion at $q=e^h$ yields an algebraically independent sequence of modular invariants and relative invariants, although this sequence does not separate modular orbits. We compute the first two nonconstant coefficients of this expansion explicitly.
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