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The Orlicz Brunn-Minkowski theory originated with the work of Lutwak, Yang, and Zhang in 2010. In this paper, we first introduce the Orlicz addition of convex bodies containing the origin in their interiors, and then extend the Lp Brunn-Minkowski inequality to the Orlicz Brunn-Minkowski inequality. Furthermore, we extend the Lp Minkowski mixed volume inequality to the Orlicz mixed volume inequality by using the Orlicz Brunn-Minkowski inequality.

It is known that the maximum diameter for the rupture-risk assessment of the abdominal aortic aneurysm is a generally good method, but not sufficient. Alternative features obtained with computational modeling may provide additional useful criteria. Though computational approaches are noninvasive, they are often time-consuming because of the high computational complexity. In this paper, we present a highly parallel algorithm for the numerical simulation of unsteady blood flows in the patient-specific abdominal aorta. We model the blood flow with the unsteady incompressible Navier-Stokes equations, and solve the discretized system with a highly scalable domain decomposition method. With this approach, the complete flow field can be obtained in less than an hour, instead of days with older methods. We show experimentally that the proposed method offers accurate solutions of the pressure, the velocity and the wall shear stress, and the parallel efficiency is higher than 70% on a parallel computer with more than 1,000 processor cores.

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The Birkhoff conjecture says that the boundary of a strictly convex integrable billiard table is necessarily an ellipse. In this article, we consider a stronger notion of integrability, namely integrability close to the boundary, and prove a local version of this conjecture: a small perturbation of an ellipse of small eccentricity which preserves integrability near the boundary, is itself an ellipse. This extends the result in Avila et al. (Ann Math 184:527–558, ADK16), where integrability was assumed on a larger set. In particular, it shows that (local) integrability near the boundary implies global integrability. One of the crucial ideas in the proof consists in analyzing Taylor expansion of the corresponding action-angle coordinates with respect to the eccentricity parameter, deriving and studying higher order conditions for the preservation of integrable rational caustics.