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Abstract In this paper, we study polar foliations on simply connected symmetric spaces with non-negative curvature. We will prove that all such foliations are isoparametric as defined in [E. Heintze, X. Liu and C. Olmos,Isoparametric submanifolds and a Chevalley-type restriction theorem,Integrable systems, geometry, and topology,American Mathematical Society, Providence 2006, 151–190]. We will also prove a splitting theorem which, when leaves are compact, reduces the study of such foliations to polar foliations in compact simply connected symmetric spaces. Moreover, we will show that solutions to mean curvature flow of regular leaves in such foliations are always ancient solutions. This generalizes part of the results in [X. Liu and C.-L. Terng,Ancient solutions to mean curvature flow for isoparametric submanifolds,Math. Ann. 378 2020, 1–2, 289–315] for mean curvature flows of isoparametric submanifolds in spheres. 

This paper proposes a novel quantity-based demand management system that aims to promote ridesharing. The system sells a time-dependent permit to access a road facility (conceptualized as a bottleneck) by auction but encourages commuters to share permits with each other. The commuters may be assigned one of three roles: solo driver, ridesharing driver, or rider. At the core of this auction-based permit allocation and sharing system (A-PASS) is a trilateral matching problem (TMP) that matches permits, drivers, and riders. Formulated as an integer program, TMP is first shown to be tightly bounded by its linear relaxation. A pricing policy based on the classical Vickrey–Clarke–Groves (VCG) mechanism is then devised to determine the payment of each commuter. We prove that, under the VCG policy, different commuters pay exactly the same price as long as their role and access time are the same. Importantly, by controlling the number of shared rides, any deficit that may arise from the VCG policy can be eliminated. This may be achieved with a relatively small loss to system efficiency, thanks to the revenue generated from selling permits. Results of a numerical experiment suggest A-PASS strongly promotes ridesharing. As sharing increases, all stakeholders are better off: the ridesharing platform receives greater profits, the commuters enjoy higher utility, and society benefits from more efficient utilization of the road infrastructure. 

Abstract A detailed uplift history of the Tibetan Plateau is essential for disentangling the proposed geodynamical models and assessing its impacts on climate and biodiversity. However, when and how the plateau formed remains highly controversial. Here, we present unusual geochemical indicators of marine signatures in the Cenozoic terrestrial strata of the Qaidam Basin, northern Tibetan Plateau, with strong implications for the basin altitude. Our investigations across the basin reveal typical marine alkenones and anomalously high carbonate carbon isotopic values during the mid-Miocene, but not at earlier stages, which are accompanied by a divergent trend in the paired carbonate oxygen and leaf wax hydrogen isotopic records. We infer an incursion of seawater into the Qaidam Basin, thus constraining the mid-Miocene basin altitude close to sea level. Hence, much of the substantial northern plateau uplift afterwards appears to be associated with the outward growth of the Tibetan Plateau.