Search results for "Uniqueness"

The Calderón problem for the fractional Schrödinger equation with drift

2020

We investigate the Calder\'on problem for the fractional Schr\"odinger equation with drift, proving that the unknown drift and potential in a bounded domain can be determined simultaneously and uniquely by an infinite number of exterior measurements. In particular, in contrast to its local analogue, this nonlocal problem does \emph{not} enjoy a gauge invariance. The uniqueness result is complemented by an associated logarithmic stability estimate under suitable apriori assumptions. Also uniqueness under finitely many \emph{generic} measurements is discussed. Here the genericity is obtained through \emph{singularity theory} which might also be interesting in the context of hybrid inverse pro…

The fractional Calderón problem: Low regularity and stability

2017

The Calder\'on problem for the fractional Schr\"odinger equation was introduced in the work \cite{GSU}, which gave a global uniqueness result also in the partial data case. This article improves this result in two ways. First, we prove a quantitative uniqueness result showing that this inverse problem enjoys logarithmic stability under suitable a priori bounds. Second, we show that the results are valid for potentials in scale-invariant $L^p$ or negative order Sobolev spaces. A key point is a quantitative approximation property for solutions of fractional equations, obtained by combining a careful propagation of smallness analysis for the Caffarelli-Silvestre extension and a duality argumen…

On some partial data Calderón type problems with mixed boundary conditions

2021

In this article we consider the simultaneous recovery of bulk and boundary potentials in (degenerate) elliptic equations modelling (degenerate) conducting media with inaccessible boundaries. This connects local and nonlocal Calderón type problems. We prove two main results on these type of problems: On the one hand, we derive simultaneous bulk and boundary Runge approximation results. Building on these, we deduce uniqueness for localized bulk and boundary potentials. On the other hand, we construct a family of CGO solutions associated with the corresponding equations. These allow us to deduce uniqueness results for arbitrary bounded, not necessarily localized bulk and boundary potentials. T…

Revised and short versions of the pseudoscientific belief scale

2021

This is the pre-peer reviewed version of the following article: Fasce, A, Avendaño, D, Adrián‐Ventura, J. Revised and short versions of the pseudoscientific belief scale. Appl Cognit Psychol. 2021; 1– 5, which has been published in final form at https://doi.org/10.1002/acp.3811. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions. In this article, we develop the revised and short versions of the pseudoscientific belief scale through two empirical studies (N = 4154). This revision is motivated by the excessive length of the scale, as well as by consistent observations of poor item loadings across several studies…

Selecting the special or choosing the common? A high-powered conceptual replication of Kim and Markus’ (1999) pen study

2022

Kim and Markus (1999) found that 74% of Americans selected a pen with an uncommon (vs. common) color, whereas only 24% of Asians made such a choice, highlighting a pronounced crosscultural difference in the extent to which people opt for originality or make majority-based choices. The present high-powered study (N = 729) conceptually replicates the results from Kim and Markus (1999; Study 3). However, our obtained effect size (r = .12) is significantly weaker than that of the original study (r = .52). Interestingly, given the globalization of mass media and the rapid economic progress of many Asian cultures during the last decades, a larger proportion of Chinese, but not American, participa…

Shape identification in inverse medium scattering problems with a single far-field pattern

2016

Consider time-harmonic acoustic scattering from a bounded penetrable obstacle $D\subset {\mathbb R}^N$ embedded in a homogeneous background medium. The index of refraction characterizing the material inside $D$ is supposed to be Holder continuous near the corners. If $D\subset {\mathbb R}^2$ is a convex polygon, we prove that its shape and location can be uniquely determined by the far-field pattern incited by a single incident wave at a fixed frequency. In dimensions $N \geq 3$, the uniqueness applies to penetrable scatterers of rectangular type with additional assumptions on the smoothness of the contrast. Our arguments are motivated by recent studies on the absence of nonscattering waven…

The Radó–Kneser–Choquet theorem for $p$-harmonic mappings between Riemannian surfaces

2020

In the planar setting the Rad\'o-Kneser-Choquet theorem states that a harmonic map from the unit disk onto a Jordan domain bounded by a convex curve is a diffeomorphism provided that the boundary mapping is a homeomorphism. We prove the injectivity criterion of Rad\'o-Kneser-Choquet for $p$-harmonic mappings between Riemannian surfaces. In our proof of the injecticity criterion we approximate the $p$-harmonic map with auxiliary mappings that solve uniformly elliptic systems. We prove that each auxiliary mapping has a positive Jacobian by a homotopy argument. We keep the maps injective all the way through the homotopy with the help of the minimum principle for a certain subharmonic expressio…

The Homogeneous Poisson Point Process

2008

$$\mathscr {K}$$-Convergence of Finite Volume Solutions of the Euler Equations

2020

We review our recent results on the convergence of invariant domain-preserving finite volume solutions to the Euler equations of gas dynamics. If the classical solution exists we obtain strong convergence of numerical solutions to the classical one applying the weak-strong uniqueness principle. On the other hand, if the classical solution does not exist we adapt the well-known Prokhorov compactness theorem to space-time probability measures that are generated by the sequences of finite volume solutions and show how to obtain the strong convergence in space and time of observable quantities. This can be achieved even in the case of ill-posed Euler equations having possibly many oscillatory s…

La Géométrie pour l'Unicité, la Parcimonie et l'Appariement des Estimateurs Pénalisés

2022

During the talk we will give a necessary and sufficient condition for the uniqueness of a penalized least squares estimator whose penalty term is a polyhedral norm. Our results cover many methods including the OSCAR, SLOPE and LASSO estimators as well as the related method of basis pursuit. The geometrical condition for uniqueness involves how the row span of the design matrix intersects the faces of the dual normunit ball. Theoretical results on sparsity by LASSO and basis pursuit estimators are deduced from this condition via the characterization of accessible sign vectors for these two methods.