A partial solution of the isoperimetric problem for the Heisenberg group

We provide a partial solution to the isoperimetric problem in the Heisenberg group.Comment: 32 pages, 1 figur

Quantitative uniqueness for elliptic equations with singular lower order terms

We use a Carleman type inequality of Koch and Tataru to obtain quantitative estimates of unique continuation for solutions of second order elliptic equations with singular lower order terms. First we prove a three sphere inequality and then describe two methods of propagation of smallness from sets of positive measure.Comment: 23 pages, v2 small changes are done and some mistakes are correcte

Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domains in the Heisenberg group

We prove geometric $L^p$ versions of Hardy's inequality for the sub-elliptic Laplacian on convex domains $\Omega$ in the Heisenberg group $\mathbb{H}^n$, where convex is meant in the Euclidean sense. When $p=2$ and $\Omega$ is the half-space given by $\langle \xi, \nu\rangle > d$ this generalizes an inequality previously obtained by Luan and Yang. For such $p$ and $\Omega$ the inequality is sharp and takes the form \begin{equation} \int_\Omega |\nabla_{\mathbb{H}^n}u|^2 \, d\xi \geq \frac{1}{4}\int_{\Omega} \sum_{i=1}^n\frac{\langle X_i(\xi), \nu\rangle^2+\langle Y_i(\xi), \nu\rangle^2}{\textrm{dist}(\xi, \partial \Omega)^2}|u|^2\, d\xi, \end{equation} where $\textrm{dist}(\, \cdot\,, \partial \Omega)$ denotes the Euclidean distance from $\partial \Omega$.Comment: 14 page

Evidence for the Concreteness of Abstract Language: A Meta-Analysis of Neuroimaging Studies

The neural mechanisms subserving the processing of abstract concepts remain largely debated. Even within the embodiment theoretical framework, most authors suggest that abstract concepts are coded in a linguistic propositional format, although they do not completely deny the role of sensorimotor and emotional experiences in coding it. To our knowledge, only one recent proposal puts forward that the processing of concrete and abstract concepts relies on the same mechanisms, with the only difference being in the complexity of the underlying experiences. In this paper, we performed a meta-analysis using the Activation Likelihood Estimates (ALE) method on 33 functional neuroimaging studies that considered activations related to abstract and concrete concepts. The results suggest that (1) concrete and abstract concepts share the recruitment of the temporo-fronto-parietal circuits normally involved in the interactions with the physical world, (2) processing concrete concepts recruits fronto-parietal areas better than abstract concepts, and (3) abstract concepts recruit Broca’s region more strongly than concrete ones. Based on anatomical and physiological evidence, Broca’s region is not only a linguistic region mainly devoted to speech production, but it is endowed with complex motor representations of different biological effectors. Hence, we propose that the stronger recruitment of this region for abstract concepts is expression of the complex sensorimotor experiences underlying it, rather than evidence of a purely linguistic format of its processing

Local behavior of p-harmonic Green's functions in metric spaces

We describe the behavior of p-harmonic Green's functions near a singularity in metric measure spaces equipped with a doubling measure and supporting a Poincar\'e inequality