A classical result of Aleksandrov allows us to estimate the size of a convex function u at a point x in a bounded domain Ω in terms of the distance from x to the boundary of Ω if ∫ Ω detD 2u dx < ∞. This estimate plays a prominent role in the existence and regularity theory of the Monge-Ampère equation. Jerison proved an extension of Aleksandrov’s result that provides a similar estimate, in som...

We consider a generalised complex Monge-Ampère equation on a compact Kähler manifold and treat it using the method of continuity. For complex surfaces we prove an existence result. We also prove that (for three-folds and a related real PDE in a ball in R3) as long as the Hessian is bounded below by a pre-determined constant (whilst moving along the method of continuity path), a smooth solution ...

We study fully nonlinear partial differential equations involving the determinant of the Hessian matrix of the unknown function with respect to a family of vector fields that generate a Carnot group. We prove a comparison theorem among viscosity suband supersolutions, for subsolutions uniformly convex with respect to the vector fields.

A classical result of Aleksandrov allows one to estimate the size of a convex function u at a point x in a bounded domain Ω in terms of the distance from x to the boundary of Ω if R Ω det Du dx < ∞. This estimate plays a prominent role in the existence and regularity theory of the Monge-Ampère equation. Jerison proved an extension of Aleksandrov’s result that provides a similar estimate, in som...

The rapid ascent to 4560 m caused minor symptoms of acute mountain sickness such as headache, nausea, and insomnia in all participants. We did not, however, find any influence of this short term exposure to high altitude on the morphology of the red cells. Our results are in contrast to the morphological changes observed by Rowles and Williams at a similar altitude (4600 m) during an expedition...

Low pressure, low oxygen concentration, and intense ultraviolet (UV) radiation in high-altitude environments, can cause oxidative stress which can trigger mountain sickness. A recent study demonstrated that hydrogen gas with a good permeability in biological membranes can treat various disorders by exerting its selective anti-oxidation and anti-inflammatory effects, indicating that hydrogen the...

Many high mountaineers die of acute mountain sickness in the high mountains of Nepal during their adventure. Most of them administer acetaminophen (AP) as an analgesic. Various cells with higher expression of cytochrome P-450 (CYP) would metabolize AP and lead to cellular impairment. CYP is a major enzyme system in the metabolism of xenobiotics in different organs of the human being including i...