Lipschitz extensions of maps between Heisenberg groups

Let \H^n be the Heisenberg group of topological dimension $2n+1$. We prove that if $n$ is odd, the pair of metric spaces (\H^n, \H^n) does not have the Lipschitz extension property

Intrinsic curvature of curves and surfaces and a Gauss-Bonnet theorem in the Heisenberg group

We use a Riemannnian approximation scheme to define a notion of intrinsic Gaussian curvature for a Euclidean C2 -smooth surface in the Heisenberg group H away from characteristic points, and a notion of intrinsic signed geodesic curvature for Euclidean C2 -smooth curves on surfaces. These results are then used to prove a Heisenberg version of the Gauss–Bonnet theorem. An application to Steiner’s formula for the Carnot–Carathéodory distance in H is provided

Sharp comparison and maximum principles via horizontal normal mapping in the Heisenberg group

In this paper we solve a problem raised by Gutiérrez and Montanari about comparison principles for H-convex functions on subdomains of Heisenberg groups. Our approach is based on the notion of the sub-Riemannian horizontal normal mapping and uses degree theory for set-valued maps. The statement of the comparison principle combined with a Harnack inequality is applied to prove the Aleksandrov-type maximum principle, describing the correct boundary behavior of continuous H-convex functions vanishing at the boundary of horizontally bounded subdomains of Heisenberg groups. This result answers a question by Garofalo and Tournier. The sharpness of our results are illustrated by examples

Lipschitz extensions of maps between Heisenberg groups

Let \H^n be the Heisenberg group of topological dimension 2n+1 . We prove that if n is odd, the pair of metric spaces (\H^n, \H^n) does not have the Lipschitz extension property

Inégalités d'interpolation géodésique sur les groupes de Heisenberg

In this Note, we present geodesic versions of the Borell–Brascamp–Lieb, Brunn–Minkowski and entropy inequalities on the Heisenberg group Hn. Our arguments use the Riemannian approximation of Hn combined with optimal mass-transportation techniquesDans cette Note, nous présentons des versions géodésiques des inégalités de Borell–Brascamp–Lieb et de Brunn–Minkowski, et des inégalités d'entropie sur le groupe de Heisenberg Hn. Nos démonstrations s'appuient sur l'approximation riemannienne de Hn et sur des techniques de transport optimal

OIF Spaces

A base β of a space X is called an OIF base when every element of B is a subset of only a finite number of other elements of β. We will explore the fundamental properties of spaces having such bases. In particular, we will show that in T2 spaces, strong OIF bases are the same as uniform bases, and that in T3 spaces where all subspaces have OIF bases, compactness, countable compactness, or local compactness will give metrizability

Performanse tri populacije mlađi smuđa (sander lucioperca) iz Mađarske u intenzivnom uzgoju

Zahvaljujući blagom ukusu i niskom sadržaju masti uz poželjan masno-kiselinski sastav smuđ (Sander lucioperca) je cenjen među potrošačima i uz pad izlova i povećanje potražnje dobija sve veću pažnju među uzgajivačima. S obzirom da su osnovni izvori matica smuđa iz ekstenzivnog uzgoja ili divljeg porekla, informacija o podobnosti mlađi pomenutog porekla sa svrhom intenzivnog uzgoja je od izuzetnog značaja. S tom svrhom, naša namera bila je da ovom studijom ocenimo performanse dve grupe mlađi poreklom od matica iz ekstenzivnog uzgoja i jedne grupe mlađi poreklom od divljih matica iz najveće vodene mase u državi, reke Dunav, kao i da ocenimo stepen zadržavanja NMT Alpha vidljivih oznaka u obraz riba. Iako bez statistički značajne razlike, mlađ poreklom iz ekstenzivnog uzgoja ispoljila je brži rast. Ovakav rezultat nagoveštava obimnu selekciju među maticama iz jezerskog uzgoja, posebno uzimajući u obzir stres rukovanja i suboptimalne uslove kvaliteta vode što sezonski karakteriše jezerski uzgoj. Kakogod, treba nagovestiti da je u svakoj grupi zabeležena određena količina iznurenih individua. Konačno, primanje NMT Alpha vidljivih oznaka je bilo visoko u sve tri grupe i bez statistički značajnih razlika