Search results for "Mg"
showing 10 items of 492 documents
Nowhere differentiable intrinsic Lipschitz graphs
2021
We construct intrinsic Lipschitz graphs in Carnot groups with the property that, at every point, there exist infinitely many different blow-up limits, none of which is a homogeneous subgroup. This provides counterexamples to a Rademacher theorem for intrinsic Lipschitz graphs.
The annular decay property and capacity estimates for thin annuli
2016
We obtain upper and lower bounds for the nonlinear variational capacity of thin annuli in weighted $\mathbf{R}^n$ and in metric spaces, primarily under the assumptions of an annular decay property and a Poincar\'e inequality. In particular, if the measure has the $1$-annular decay property at $x_0$ and the metric space supports a pointwise $1$-Poincar\'e inequality at $x_0$, then the upper and lower bounds are comparable and we get a two-sided estimate for thin annuli centred at $x_0$, which generalizes the known estimate for the usual variational capacity in unweighted $\mathbf{R}^n$. Most of our estimates are sharp, which we show by supplying several key counterexamples. We also character…
Indecomposable sets of finite perimeter in doubling metric measure spaces
2020
We study a measure-theoretic notion of connectedness for sets of finite perimeter in the setting of doubling metric measure spaces supporting a weak $(1,1)$-Poincar\'{e} inequality. The two main results we obtain are a decomposition theorem into indecomposable sets and a characterisation of extreme points in the space of BV functions. In both cases, the proof we propose requires an additional assumption on the space, which is called isotropicity and concerns the Hausdorff-type representation of the perimeter measure.
Removable sets for intrinsic metric and for holomorphic functions
2019
We study the subsets of metric spaces that are negligible for the infimal length of connecting curves; such sets are called metrically removable. In particular, we show that every totally disconnected set with finite Hausdorff measure of codimension 1 is metrically removable, which answers a question raised by Hakobyan and Herron. The metrically removable sets are shown to be related to other classes of "thin" sets that appeared in the literature. They are also related to the removability problems for classes of holomorphic functions with restrictions on the derivative.
Assouad dimension, Nagata dimension, and uniformly close metric tangents
2013
We study the Assouad dimension and the Nagata dimension of metric spaces. As a general result, we prove that the Nagata dimension of a metric space is always bounded from above by the Assouad dimension. Most of the paper is devoted to the study of when these metric dimensions of a metric space are locally given by the dimensions of its metric tangents. Having uniformly close tangents is not sufficient. What is needed in addition is either that the tangents have dimension with uniform constants independent from the point and the tangent, or that the tangents are unique. We will apply our results to equiregular subRiemannian manifolds and show that locally their Nagata dimension equals the to…
Billion-Fold Enhancement in Sensitivity of Nuclear Magnetic Resonance Spectroscopy for Magnesium Ions in Solution
2014
Beta-nuclear magnetic resonance (NMR) spectroscopy is highly sensitive compared to conventional NMR spectroscopy, and may be applied for several elements across the periodic table. Beta-NMR has previously been successfully applied in the fields of nuclear and solid-state physics. In this work, beta-NMR is applied, for the first time, to record an NMR spectrum for a species in solution. 31Mg b-NMR spectra are measured for as few as 10^7 magnesium ions in ionic liquid (EMIM-Ac) within minutes, as a prototypical test case. Resonances are observed at 3882.9 and 3887.2 kHz in an external field of 0.3 T. The key achievement of the current work is to demonstrate that beta-NMR is applicable for the…
Noraksti no Kurzemes hercogistes un Livonijas likumu krājumiem, 1606.-1698. gada dokumenti [Mietausche Policeÿ Ordnung von Anno 1606. 5. Septembris]
1698
Nosaukums pirmajā lapā: Mietausche Policeÿ Ordnung von Anno 1606. 5. Septembris
Classification criteria for regular trees
2021
Esitämme säännöllisten puiden parabolisuudelle yhtäpitäviä ehtoja. We give characterizations for the parabolicity of regular trees. peerReviewed
Rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds
2021
We prove that if $M$ is a closed $n$-dimensional Riemannian manifold, $n \ge 3$, with ${\rm Ric}\ge n-1$ and for which the optimal constant in the critical Sobolev inequality equals the one of the $n$-dimensional sphere $\mathbb{S}^n$, then $M$ is isometric to $\mathbb{S}^n$. An almost-rigidity result is also established, saying that if equality is almost achieved, then $M$ is close in the measure Gromov-Hausdorff sense to a spherical suspension. These statements are obtained in the ${\rm RCD}$-setting of (possibly non-smooth) metric measure spaces satisfying synthetic lower Ricci curvature bounds. An independent result of our analysis is the characterization of the best constant in the Sob…