Martingales arising from minimal submanifolds and other geometric contexts

We consider a class of martingales on Cartan-Hadamard manifolds that includes Brownian motion on a minimal submanifold. We give sufficient conditions for such martingales to be transient, extending previous results on the transience of minimal submanifolds. We also give conditions for the almost sure convergence of the angular component (in polar coordinates) of a martingale in this class, including both the negatively pinched case (using earlier results on martingales of bounded dilation), and the radially symmetric case with quadratic decay of the upper curvature bound. Applied to minimal submanifolds, this gives curvature conditions on the ambient Cartan-Hadamard manifold under which any minimal submanifold admits a non-constant, bounded, harmonic function. Though our discussion is primarily motivated by minimal submanifolds, this class of martingales includes diffusions naturally associated to ancient solutions of mean curvature flow and to certain sub-Riemannian structures, and we briefly discuss these contexts as well. Our techniques are elementary, consisting mainly of comparison geometry and Ito's rule.Comment: Accepted version (some mistakes corrected from the previous), to appear in Illinois Journal of Mathematic

An Investigation Utilizing an Electrical Analogue of Cyclic Deicing of Hollow Steel Propellers with Internal Electric Heaters

A study has been made of the heating requirements for the cyclic de?icing of hollow steel propellers fitted with two types of internal electric heaters. Solutions to the transient?teat?flow equations depicting the cyclic de?icing of propellers were obtained by use of an electrical analogy. The study showed the impracticability of using an internal tubular heater and illustrated the advantages of employing an internal shoe?type heater, which distributes the heat more evenly to the blade surface. The importance of minimizing the thermal inertia of the system was demonstrated, and the magnitude of reductions in the total energy requirement made possible through reductions in the heating period was indicated

Health Care Costs and the Arc of Innovation

Health care costs continue their inexorable rise, threatening America’s long-term fiscal stability, competitiveness, and standard of living. Over the past half-century, efforts to rein in spending have uniformly failed. In this Article, we explain why, breaking with standard accounts of regulatory and market dysfunction. We point instead to the nexus of economics, mutual empathy, and social expectations that drives medical innovation and locks in low-value technologies. We show how law reflects and reinforces this nexus and how and why health-policy-makers avert their gaze. Next, we propose to circumvent these barriers instead of surmounting them. Rather than targeting today’s excessive spending, we seek to leverage available legal tools to bend the arc of innovation, away from marginally-beneficial technology and toward high-value advances. To this end, we set forth a novel, value-based approach to pricing and patent protection—one that departs sharply from current practice by rewarding innovators in proportion to the therapeutic benefits new tests and treatments yield. Using cancer therapy as an example, we explain how emerging information technology and large troves of electronic clinical data are opening the way to near-real-time assessment of efficacy. We then show how such assessment can power ongoing adjustment of pricing and patent terms. Finally, we offer a blueprint for how laws governing health care payment and intellectual property can be tailored to realize this value-focused vision. For the reasons we lay out, the transformation of incentives we urge will both slow clinical spending growth and greatly enhance the social value that this spending yields

Who Cares About Patents? Cross-Industry Differences in the Marginal Value of Patent Term

How much do market participants in different industries value a marginal change in patent term (i.e., duration of patent protection)? We explore this research question by measuring the behavioral response of patentees to a rare natural experiment: a change in patent term rules, due to passage of the TRIPS agreement. We find significant heterogeneity in patentee behavior across industries, some of which follows conventional wisdom (patent term is important in pharmaceuticals) and some of which does not (it also appears to matter for some software). Our measure is highly correlated with patent renewal rates across industries, suggesting the marginal value of patent term increases with higher expected profits toward the end of term