Search results for "Carn"
showing 10 items of 371 documents
Carleman estimates for sub-Laplacians on Carnot groups
2022
In this note, we establish a new Carleman estimate with singular weights for the sub-Laplacian on a Carnot group $\mathbb G$ for functions satisfying the discrepancy assumption in (2.16) below. We use such an estimate to derive a sharp vanishing order estimate for solutions to stationary Schr\"odinger equations.
Sard property for the endpoint map on some Carnot groups
2016
In Carnot-Caratheodory or sub-Riemannian geometry, one of the major open problems is whether the conclusions of Sard's theorem holds for the endpoint map, a canonical map from an infinite-dimensional path space to the underlying finite-dimensional manifold. The set of critical values for the endpoint map is also known as abnormal set, being the set of endpoints of abnormal extremals leaving the base point. We prove that a strong version of Sard's property holds for all step-2 Carnot groups and several other classes of Lie groups endowed with left-invariant distributions. Namely, we prove that the abnormal set lies in a proper analytic subvariety. In doing so we examine several characterizat…
A Cornucopia of Carnot groups in Low Dimensions
2022
Abstract Stratified groups are those simply connected Lie groups whose Lie algebras admit a derivation for which the eigenspace with eigenvalue 1 is Lie generating. When a stratified group is equipped with a left-invariant path distance that is homogeneous with respect to the automorphisms induced by the derivation, this metric space is known as Carnot group. Carnot groups appear in several mathematical contexts. To understand their algebraic structure, it is useful to study some examples explicitly. In this work, we provide a list of low-dimensional stratified groups, express their Lie product, and present a basis of left-invariant vector fields, together with their respective left-invaria…
Infinitesimal Hilbertianity of Weighted Riemannian Manifolds
2018
AbstractThe main result of this paper is the following: anyweightedRiemannian manifold$(M,g,\unicode[STIX]{x1D707})$,i.e., a Riemannian manifold$(M,g)$endowed with a generic non-negative Radon measure$\unicode[STIX]{x1D707}$, isinfinitesimally Hilbertian, which means that its associated Sobolev space$W^{1,2}(M,g,\unicode[STIX]{x1D707})$is a Hilbert space.We actually prove a stronger result: the abstract tangent module (à la Gigli) associated with any weighted reversible Finsler manifold$(M,F,\unicode[STIX]{x1D707})$can be isometrically embedded into the space of all measurable sections of the tangent bundle of$M$that are$2$-integrable with respect to$\unicode[STIX]{x1D707}$.By following the…
Sets with constant normal in Carnot groups: properties and examples
2019
We analyze subsets of Carnot groups that have intrinsic constant normal, as they appear in the blowup study of sets that have finite sub-Riemannian perimeter. The purpose of this paper is threefold. First, we prove some mild regularity and structural results in arbitrary Carnot groups. Namely, we show that for every constant-normal set in a Carnot group its sub-Riemannian-Lebesgue representative is regularly open, contractible, and its topological boundary coincides with the reduced boundary and with the measure-theoretic boundary. We infer these properties from a cone property. Such a cone will be a semisubgroup with nonempty interior that is canonically associated with the normal directio…
Regularity of sets with constant horizontal normal in the Engel group
2012
In the Engel group with its Carnot group structure we study subsets of locally finite subRiemannian perimeter and possessing constant subRiemannian normal. We prove the rectifiability of such sets: more precisely we show that, in some specific coordinates, they are upper-graphs of entire Lipschitz functions (with respect to the Euclidean distance). However we find that, when they are written as intrinsic horizontal upper-graphs with respect to the direction of the normal, then the function defining the set might even fail to be continuous. Nevertheless, we can prove that one can always find other horizontal directions for which the set is the intrinsic horizontal upper-graph of a function t…
Area of intrinsic graphs and coarea formula in Carnot Groups
2020
AbstractWe consider submanifolds of sub-Riemannian Carnot groups with intrinsic $$C^1$$ C 1 regularity ($$C^1_H$$ C H 1 ). Our first main result is an area formula for $$C^1_H$$ C H 1 intrinsic graphs; as an application, we deduce density properties for Hausdorff measures on rectifiable sets. Our second main result is a coarea formula for slicing $$C^1_H$$ C H 1 submanifolds into level sets of a $$C^1_H$$ C H 1 function.
Palazzo Abatellis, Matteo Carnilivari e i suoi collaboratori
2015
Il contributo esamina l'attività dell'architetto Matteo Carnelivari e quella dei suoi collaboratori nella Sicilia di fine Quattrocento. Sullo sfondo del cantiere di palazzo Abatellis e con l'ausilio della documentazione, la ricerca indaga la figura di Carnilivari, la sua formazione, le relazioni con la committenza e con i collaboratori locali e stranieri, i modelli desunti dall'architettura spagnola del XV secolo.
The alien vascular flora of Stromboli and Vulcano (Aeolian Islands, Italy)
2021
The updated censuses of the alien flora of the islands of Stromboli and Vulcano, Aeolian Islands in the Central Tyrrhenian Sea, are presented here. They currently consist of 40 and 57 specific and subspecific taxa respectively. As many as 16 and 18 taxa respectively are new in comparison with the last censuses in 2016. Mesembryanthemum × vascosilvae, commonly cultivated in both islands, is reported for the first time as naturalized alien to Italy. Passiflora incarnata from Stromboli and Pelargonium graveolens from Vulcano are reported for the first time as casual aliens to Sicily. The evolution over time of these florulas, as inferred from the main floristic works published on these islands…
Walnut Consumption, Plasma Metabolomics, and Risk of Type 2 Diabetes and Cardiovascular Disease
2021
BACKGROUND: Walnut consumption is associated with lower risk of type 2 diabetes (T2D) and cardiovascular disease (CVD). However, it is unknown whether plasma metabolites related to walnut consumption are also associated with lower risk of cardiometabolic diseases. OBJECTIVES: The study aimed to identify plasma metabolites associated with walnut consumption and evaluate the prospective associations between the identified profile and risk of T2D and CVD. METHODS: The discovery population included 1833 participants at high cardiovascular risk from the PREvención con DIeta MEDiterránea (PREDIMED) study with available metabolomics data at baseline. The study population included 57% women (baseli…