916 research outputs found
Equality in Borell-Brascamp-Lieb inequalities on curved spaces
By using optimal mass transportation and a quantitative H\"older inequality,
we provide estimates for the Borell-Brascamp-Lieb deficit on complete
Riemannian manifolds. Accordingly, equality cases in Borell-Brascamp-Lieb
inequalities (including Brunn-Minkowski and Pr\'ekopa-Leindler inequalities)
are characterized in terms of the optimal transport map between suitable
marginal probability measures. These results provide several qualitative
applications both in the flat and non-flat frameworks. In particular, by using
Caffarelli's regularity result for the Monge-Amp\`ere equation, we {give a new
proof} of Dubuc's characterization of the equality in Borell-Brascamp-Lieb
inequalities in the Euclidean setting. When the -dimensional Riemannian
manifold has Ricci curvature for some , it turns out that equality in the Borell-Brascamp-Lieb inequality is
expected only when a particular region of the manifold between the marginal
supports has constant sectional curvature . A precise characterization is
provided for the equality in the Lott-Sturm-Villani-type distorted
Brunn-Minkowski inequality on Riemannian manifolds. Related results for (not
necessarily reversible) Finsler manifolds are also presented.Comment: 28 pages (with 1 figure); to appear in Advances in Mathematic
Hausdorff dimension distribution of quasiconformal mappings on the Heisenberg group
We construct quasiconformal mappings on the Heisenberg group which change the Hausdorff dimension of Cantor-type sets in an arbitrary fashion. On the other hand, we give examples of subsets of the Heisenberg group whose Hausdorff dimension cannot be lowered by any quasiconformal mapping. For a general set of a certain Hausdorff dimension we obtain estimates of the Hausdorff dimension of the image set in terms of the magnitude of the quasiconformal distortio
Weak contact equations for mappings into Heisenberg groups
Let k>n be positive integers. We consider mappings from a subset of
k-dimensional Euclidean space R^k to the Heisenberg group H^n with a variety of
metric properties, each of which imply that the mapping in question satisfies
some weak form of the contact equation arising from the sub-Riemannian
structure of the Heisenberg group. We illustrate a new geometric technique that
shows directly how the weak contact equation greatly restricts the behavior of
the mappings. In particular, we provide a new and elementary proof of the fact
that the Heisenberg group H^n is purely k-unrectifiable. We also prove that for
an open set U in R^k, the rank of the weak derivative of a weakly contact
mapping in the Sobolev space W^{1,1}_{loc}(U;R^{2n+1}) is bounded by almost
everywhere, answering a question of Magnani. Finally we prove that if a mapping
from U to H^n is s-H\"older continuous, s>1/2, and locally Lipschitz when
considered as a mapping into R^{2n+1}, then the mapping cannot be injective.
This result is related to a conjecture of Gromov.Comment: 28 page
Geometric inequalities on Heisenberg groups
We establish geometric inequalities in the sub-Riemannian setting of the
Heisenberg group . Our results include a natural sub-Riemannian
version of the celebrated curvature-dimension condition of Lott-Villani and
Sturm and also a geodesic version of the Borell-Brascamp-Lieb inequality akin
to the one obtained by Cordero-Erausquin, McCann and Schmuckenschl\"ager. The
latter statement implies sub-Riemannian versions of the geodesic
Pr\'ekopa-Leindler and Brunn-Minkowski inequalities. The proofs are based on
optimal mass transportation and Riemannian approximation of
developed by Ambrosio and Rigot. These results refute a general point of view,
according to which no geometric inequalities can be derived by optimal mass
transportation on singular spaces.Comment: to appear in Calculus of Variations and Partial Differential
Equations (42 pages, 1 figure
RELATIONSHIP BETWEEN ORGANIZATIONAL CULTURE AND CULTURAL INTELLIGENCE
This article examines one of the key competences of the 21st century, cultural intelligence. In our empirical research studies, we examined the cultural intelligence of full-time university students. We identified the corporate culture they would like to work in, and also examined if there is a correlation between their cultural intelligence and their preference for a particular corporate culture. We found that the majority of student would prefer to be employed in a Clan-type corporate culture. We also identified a correlation between their preferred corporate cultural and their cultural intelligence and its components. Students with a high degree of cultural intelligence would like to work in an adhocracy.Cameron and Quinn, CQS, cultural intelligence, Hungarian university student, OCAI, organizational culture.
Quasiconformal mappings that highly distort dimensions of many parallel lines
We construct a quasiconformal mapping of -dimensional Euclidean space, , that simultaneously distorts the Hausdorff dimension of a nearly
maximal collection of parallel lines by a given amount. This answers a question
of Balogh, Monti, and Tyson.Comment: 12 page
Rectifiability and Lipschitz extensions into the Heisenberg group
Denote by the 2n+1 dimensional Heisenberg group. We show that the pairs and do not have the Lipschitz extension property for k >
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