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 $n$-dimensional Riemannian manifold has Ricci curvature ${\rm Ric}(M)\geq (n-1)k$ for some $k\in \mathbb R$, 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 $k$. 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

Geometric inequalities on Heisenberg groups

We establish geometric inequalities in the sub-Riemannian setting of the Heisenberg group $\mathbb H^n$. 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 $\mathbb H^n$ 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

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 $n$ 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

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 $n$-dimensional Euclidean space, $n \geq 2$, 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

Objects and polymorphism in system programming languages: a new approach

A low-level data structure always has a predefined representation which does not fit into an object of traditional object-oriented languages, where explicit type tag denotes its dynamic type. This is the main reason why the advanced features of object-oriented programming cannot be fully used at the lowest level. On the other hand, the hierarchy of low-level data structures is very similar to class-trees, but instead of an explicit tag-field the value of the object determines its dynamic type. Another peculiar requirement in system programming is that some classes have to be polymorphic by-value with their ancestor: objects must fit into the space of a superclass instance. In our paper we show language constructs which enable the system programmer to handle all data structures as objects, and exploit the advantages of object-oriented programming even at the lowest level. Our solution is based on Predicate Dispatching, but adopted to the special needs of system programming. The techniques we show also allow fo r some classes to be polymorphic by-value with their super. We also describe how to implement these features without losing modularity