scholarly journals Sub-Finsler Horofunction Boundaries of the Heisenberg Group

2021 ◽  
Vol 9 (1) ◽  
pp. 19-52
Author(s):  
Nate Fisher ◽  
Sebastiano Nicolussi Golo
Keyword(s):  
Heisenberg Group ◽  
Distance Function ◽  
Finsler Metrics ◽  
Geometric Description ◽  
Homogeneous Groups ◽  
Asymptotic Cones ◽  
Develop Theory ◽  
Derivatives Of

Abstract We give a complete analytic and geometric description of the horofunction boundary for polygonal sub-Finsler metrics, that is, those that arise as asymptotic cones of word metrics, on the Heisenberg group. We develop theory for the more general case of horofunction boundaries in homogeneous groups by connecting horofunctions to Pansu derivatives of the distance function.

scholarly journals Nearest points to closed sets and directional derivatives of distance functions

1989 ◽  
Vol 39 (2) ◽  
pp. 233-238 ◽  
Author(s):  
Simon Fitzpatrick
Keyword(s):  
Distance Function ◽  
Directional Derivative ◽  
Distance Functions ◽  
Directional Derivatives ◽  
Closed Set ◽  
Closed Sets ◽  
Set Of Points ◽  
Derivatives Of ◽  
Nearest Points ◽  
Dense Set

We investigate the circumstances under which the distance function to a closed set in a Banach space having a one-sided directional derivative equal to 1 or −1 implies the existence of nearest points. In reflexive spaces we show that at a dense set of points outside a closed set the distance function has a directional derivative equal to 1.

Metric normal and distance function in the Heisenberg group

2007 ◽  
Vol 256 (3) ◽  
pp. 661-684 ◽  
Author(s):  
Nicola Arcozzi ◽  
Fausto Ferrari
Keyword(s):  
Heisenberg Group ◽  
Distance Function

scholarly journals Uncertainty relations on nilpotent Lie groups

2017 ◽  
Vol 473 (2201) ◽  
pp. 20170082 ◽  
Author(s):  
Michael Ruzhansky ◽  
Durvudkhan Suragan
Keyword(s):  
Quantum Mechanics ◽  
Heisenberg Group ◽  
Lie Groups ◽  
Coulomb Potential ◽  
Nilpotent Lie Groups ◽  
Uncertainty Relations ◽  
Homogeneous Groups ◽  
Potential Operators ◽  
Nirenberg Inequality ◽  
Uncertainty Inequalities

We give relations between main operators of quantum mechanics on one of most general classes of nilpotent Lie groups. Namely, we show relations between momentum and position operators as well as Euler and Coulomb potential operators on homogeneous groups. Homogeneous group analogues of some well-known inequalities such as Hardy's inequality, Heisenberg–Kennard type and Heisenberg–Pauli–Weyl type uncertainty inequalities, as well as Caffarelli–Kohn–Nirenberg inequality are derived, with best constants. The obtained relations yield new results already in the setting of both isotropic and anisotropic R n , and of the Heisenberg group. The proof demonstrates that the method of establishing equalities in sharper versions of such inequalities works well in both isotropic and anisotropic settings.

scholarly journals The Lusin Theorem and Horizontal Graphs in the Heisenberg Group

2013 ◽  
Vol 1 ◽  
pp. 295-301 ◽  
Author(s):  
Piotr Hajłasz ◽  
Jacob Mirra
Keyword(s):  
Heisenberg Group ◽  
Modulus Of Continuity ◽  
Lipschitz Function ◽  
Measurable Functions ◽  
Derivatives Of

Abstract In this paper we prove that every collection of measurable functions fα , |α| = m, coincides a.e. withmth order derivatives of a function g ∈ Cm−1 whose derivatives of order m − 1 may have any modulus of continuity weaker than that of a Lipschitz function. This is a stronger version of earlier results of Lusin, Moonens-Pfeffer and Francos. As an application we construct surfaces in the Heisenberg group with tangent spaces being horizontal a.e.

On a family of torsional creep problems in Finsler metrics

2020 ◽  
pp. 1-18
Author(s):  
Maria Fărcăşeanu ◽  
Mihai Mihăilescu ◽  
Denisa Stancu-Dumitru
Keyword(s):  
Asymptotic Behavior ◽  
Boundary Value Problems ◽  
Distance Function ◽  
Dirichlet Boundary ◽  
Differential Operators ◽  
General Setting ◽  
Divergence Form ◽  
Finsler Metric ◽  
Asymptotic Behavior Of Solutions ◽  
Finsler Metrics

Abstract The asymptotic behavior of solutions to a family of Dirichlet boundary value problems, involving differential operators in divergence form, on a domain equipped with a Finsler metric is investigated. Solutions are shown to converge uniformly to the distance function to the boundary of the domain, which takes into account the Finsler norm involved in the equation. This implies that a well-known result in the analysis of problems modeling torsional creep continues to hold in this more general setting.

Physical Properties of TCNQ Salts with N-Methyl Derivatives of Pyridine

1982 ◽  
Vol 85 (1) ◽  
pp. 257-263 ◽  
Author(s):  
A. Graja ◽  
M. Przybylski ◽  
B. Butka ◽  
R. Swietlik
Keyword(s):  
Physical Properties ◽  
Methyl Derivatives ◽  
Derivatives Of
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