Search results for "Mathematics::Metric Geometry"
showing 10 items of 139 documents
Which measures are projections of purely unrectifiable one-dimensional Hausdorff measures
2008
We give a necessary and sufficient condition for a measure p, on the real line to be an orthogonal projection of XAl for some purely 1-unrectifiable planar set A.
A nicely behaved singular integral on a purely unrectifiable set
2001
We construct an example of a purely 1-unrectifiable AD-regular set E in the plane such that the limit
Sharp dimension free quantitative estimates for the Gaussian isoperimetric inequality
2017
We provide a full quantitative version of the Gaussian isoperimetric inequality: the difference between the Gaussian perimeter of a given set and a half-space with the same mass controls the gap between the norms of the corresponding barycenters. In particular, it controls the Gaussian measure of the symmetric difference between the set and the half-space oriented so to have the barycenter in the same direction of the set. Our estimate is independent of the dimension, sharp on the decay rate with respect to the gap and with optimal dependence on the mass.
Geometric Entropies of Mixing (EOM)
2005
Trigonometric and trigonometric-algebraic entropies are introduced. Regularity increases the entropy and the maximal entropy is shown to result when a regular $n$-gon is inscribed in a circle. A regular $n$-gon circumscribing a circle gives the largest entropy reduction, or the smallest change in entropy from the state of maximum entropy which occurs in the asymptotic infinite $n$ limit. EOM are shown to correspond to minimum perimeter and maximum area in the theory of convex bodies, and can be used in the prediction of new inequalities for convex sets. These expressions are shown to be related to the phase functions obtained from the WKB approximation for Bessel and Hermite functions.
Tangent Measures, Densities, and Singular Integrals
1995
We introduce tangent measures in the sense of David Preiss. We discuss their applications to the density and rectifiability properties of general Borel measures in ℝ n as well as to the behaviour of certain singular integrals with respect to such measures.
A new rotational integral formula for intrinsic volumes in space forms
2010
A new rotational version of Crofton's formula is derived for the intrinsic volumes of a domain Y in a space form. More precisely, a functional is defined on the intersection between Y and a totally geodesic submanifold (plane) through a fixed point, such that the rotational average of this functional is equal to the intrinsic volumes of Y. Particular cases of interest in stereology are considered for the Euclidean case. © 2009 Elsevier Inc. All rights reserved.
Quasihyperbolic boundary conditions and capacity: Uniform continuity of quasiconformal mappings
2005
We prove that quasiconformal maps onto domains which satisfy a suitable growth condition on the quasihyperbolic metric are uniformly continuous when the source domain is equipped with the internal metric. The obtained modulus of continuity and the growth assumption on the quasihyperbolic metric are shown to be essentially sharp. As a tool, we prove a new capacity estimate.
Weak chord-arc curves and double-dome quasisymmetric spheres
2014
Let $\Omega$ be a planar Jordan domain and $\alpha>0$. We consider double-dome-like surfaces $\Sigma(\Omega,t^{\alpha})$ over $\overline{\Omega}$ where the height of the surface over any point $x\in\overline{\Omega}$ equals $\text{dist}(x,\partial\Omega)^{\alpha}$. We identify the necessary and sufficient conditions in terms of $\Omega$ and $\alpha$ so that these surfaces are quasisymmetric to $\mathbb{S}^2$ and we show that $\Sigma(\Omega,t^{\alpha})$ is quasisymmetric to the unit sphere $\mathbb{S}^2$ if and only if it is linearly locally connected and Ahlfors $2$-regular.
Perimeter leakage current in polymer light emitting diodes
2009
Observation of leakage current paths through the device perimeter in standard poly(phenylene vinylene)-based light-emitting devices is reported. Perimeter leakage currents govern the diode performance in reverse and low positive bias and exhibit an ohmic character. Current density correlates with the perimeter-to-area ratio thus indicating that leakage currents are mainly confined on polymer regions in the vicinity of metallic contact limits (device perimeter). © 2008 Elsevier B.V. All rights reserved.
Pauls rectifiable and purely Pauls unrectifiable smooth hypersurfaces
2020
This paper is related to the problem of finding a good notion of rectifiability in sub-Riemannian geometry. In particular, we study which kind of results can be expected for smooth hypersurfaces in Carnot groups. Our main contribution will be a consequence of the following result: there exists a -hypersurface without characteristic points that has uncountably many pairwise non-isomorphic tangent groups on every positive-measure subset. The example is found in a Carnot group of topological dimension 8, it has Hausdorff dimension 12 and so we use on it the Hausdorff measure . As a consequence, we show that any Lipschitz map defined on a subset of a Carnot group of Hausdorff dimension 12, with…