Search results for "F12"
showing 10 items of 53 documents
On Radon transforms on compact Lie groups
2016
We show that the Radon transform related to closed geodesics is injective on a Lie group if and only if the connected components are not homeomorphic to $S^1$ nor to $S^3$. This is true for both smooth functions and distributions. The key ingredients of the proof are finding totally geodesic tori and realizing the Radon transform as a family of symmetric operators indexed by nontrivial homomorphisms from $S^1$.
On the non-triviality of the torsion subgroup of the abelianized Johnson kernel
2022
The Johnson kernel is the subgroup of the mapping class group of a closed oriented surface that is generated by Dehn twists along separating simple closed curves. The rational abelianization of the Johnson kernel has been computed by Dimca, Hain and Papadima, and a more explicit form was subsequently provided by Morita, Sakasai and Suzuki. Based on these results, Nozaki, Sato and Suzuki used the theory of finite-type invariants of 3-manifolds to prove that the torsion subgroup of the abelianized Johnson kernel is non-trivial. In this paper, we give a purely 2-dimensional proof of the non-triviality of this torsion subgroup and provide a lower bound for its cardinality. Our main tool is the …
Does complexity explain the structure of trade?
2013
This paper analyzes whether complexity, measured by the number of skilled tasks that are performed in production, explains countries commodity trade structure. We modify the Romalis ( ) model to incorporate advantage differences in complexity across commodities together with differences in the number of mistakes made by workers in the production process in developed and developing countries as a source of comparative advantage. Our model predicts that the share of developed countries in world trade increases with products complexity. Empirical tests confirm this prediction. Moreover, we find that complexity complements the explanation provided by skillintensity on countries commodity trade …
La construcción ideológica de la Restauración en Nueva España (1814-1816)
2020
La vuelta de Fernando VII en 1814 a la península y el golpe de Estado por el que procedió a anular la Constitución de Cádiz no supusieron, a ambos lados del Atlántico, un retorno a la situación previa a la crisis de 1808. Este trabajo examina dicho problema en el virreinato de la Nueva España desde el ámbito de los discursos para el periodo que transcurre entre 1814 y 1816. Para ello se relacionan tres perspectivas. En primer lugar, se confrontan algunos de los distintos proyectos que se plantearon dentro del absolutismo para efectuar la Restauración. Seguidamente, examinamos aquellos componentes teóricos que los intelectuales de la contrarrevolución utilizaron para legitimarla. Este proyec…
Torus computed tomography
2020
We present a new computed tomography (CT) method for inverting the Radon transform in 2D. The idea relies on the geometry of the flat torus, hence we call the new method Torus CT. We prove new inversion formulas for integrable functions, solve a minimization problem associated to Tikhonov regularization in Sobolev spaces and prove that the solution operator provides an admissible regularization strategy with a quantitative stability estimate. This regularization is a simple post-processing low-pass filter for the Fourier series of a phantom. We also study the adjoint and the normal operator of the X-ray transform on the flat torus. The X-ray transform is unitary on the flat torus. We have i…
Unique continuation property and Poincar�� inequality for higher order fractional Laplacians with applications in inverse problems
2020
We prove a unique continuation property for the fractional Laplacian $(-\Delta)^s$ when $s \in (-n/2,\infty)\setminus \mathbb{Z}$. In addition, we study Poincar\'e-type inequalities for the operator $(-\Delta)^s$ when $s\geq 0$. We apply the results to show that one can uniquely recover, up to a gauge, electric and magnetic potentials from the Dirichlet-to-Neumann map associated to the higher order fractional magnetic Schr\"odinger equation. We also study the higher order fractional Schr\"odinger equation with singular electric potential. In both cases, we obtain a Runge approximation property for the equation. Furthermore, we prove a uniqueness result for a partial data problem of the $d$-…
Fourier analysis of periodic Radon transforms
2019
We study reconstruction of an unknown function from its $d$-plane Radon transform on the flat $n$-torus when $1 \leq d \leq n-1$. We prove new reconstruction formulas and stability results with respect to weighted Bessel potential norms. We solve the associated Tikhonov minimization problem on $H^s$ Sobolev spaces using the properties of the adjoint and normal operators. One of the inversion formulas implies that a compactly supported distribution on the plane with zero average is a weighted sum of its X-ray data.
The differential Galois group of the rational function field
2020
We determine the absolute differential Galois group of the field $\mathbb{C}(x)$ of rational functions: It is the free proalgebraic group on a set of cardinality $|\mathbb{C}|$. This solves a longstanding open problem posed by B.H. Matzat. For the proof we develop a new characterization of free proalgebraic groups in terms of split embedding problems, and we use patching techniques in order to solve a very general class of differential embedding problems. Our result about $\mathbb{C}(x)$ also applies to rational function fields over more general fields of coefficients.
Free differential Galois groups
2019
We study the structure of the absolute differential Galois group of a rational function field over an algebraically closed field of characteristic zero. In particular, we relate the behavior of differential embedding problems to the condition that the absolute differential Galois group is free as a proalgebraic group. Building on this, we prove Matzat's freeness conjecture in the case that the field of constants is algebraically closed of countably infinite transcendence degree over the rationals. This is the first known case of the twenty year old conjecture.
Local Asymptotic Normality for Shape and Periodicity in the Drift of a Time Inhomogeneous Diffusion
2017
We consider a one-dimensional diffusion whose drift contains a deterministic periodic signal with unknown periodicity $T$ and carrying some unknown $d$-dimensional shape parameter $\theta$. We prove Local Asymptotic Normality (LAN) jointly in $\theta$ and $T$ for the statistical experiment arising from continuous observation of this diffusion. The local scale turns out to be $n^{-1/2}$ for the shape parameter and $n^{-3/2}$ for the periodicity which generalizes known results about LAN when either $\theta$ or $T$ is assumed to be known.