Let X and B be two Riemannian manifolds with π : X → B being a Riemannian submersion. Let H be the corresponding horizontal distribution, which is perpendicular to the tangent bundle of the fibres of π : X → B. Then X (just considered as a differentiable manifold), together with the distribution H, forms a so-called Carnot-Caratheodory space [1], when the Riemannian metric of X is restricted to...

We provide some new Caratheodory-type selection theorems. i.e.. selections for correspondences of two variables which are continuous with respect to one variable and measurable with respect to the other. These results generalize simultaneously Michael’s [2l] continuous selection theorem for lower-semicontinuous correspondences as well as a Caratheodory-type selection theorem of Fryszkowski [IO]...

‎In this essay we introduce and study the notion of dimension for observers via Caratheodory structures and relative probability measures‎. ‎We show that the dimension as a three variables function is an increasing function on observers‎, ‎and decreasing function on the cuts of an observer‎. ‎We find observers with arbitrary non-negative dimensions‎. ‎We show that Caratheodory dimension for obs...

We show that the Heisenberg group, with its Carnot-Caratheodory metric, does not admit a bi-Lipschitz embedding into L1.

Elements of constitutive model formulation for simple materials within a thermodynamic, finite deformation framework are reviewed. Recent developments in addressing the second law and entropy existence are discussed. Caratheodory-based approaches are emphasized for their simplicity and ease of interpretation. An original treatment of finite thermoviscoelasticity is developed in this context. A ...

* Correspondence: tzymath@gmail. com Department of Mathematics and System Science, National University of Defense Technology, Changsha, PR China Abstract This article is devoted to extensions of some existing results about the Caratheodory operator from the function sense to the differential form situation. Similarly as the function sense, we obtain the convergence of sequences of differential ...

In the context of two-path convexity, we study the rank, Helly number, Radon number, Caratheodory number, and hull number for multipartite tournaments. We show the maximum Caratheodory number of a multipartite tournament is 3. We then derive tight upper bounds for rank in both general multipartite tournaments and clone-free multipartite tournaments. We show that these same tight upper bounds ho...