We show that products of sufficiently thick Cantor sets generate trees in the plane with constant distance between adjacent vertices. Moreover, we prove that the set of choices for this distance has non-empty interior. We allow our trees to be countably infinite, which further distinguishes this work from previous results on patterns in fractal sets. This builds on the authors’ previous work on graphs and distance sets over products of Cantor sets of sufficient Newhouse thickness.

In this paper we prove that the set $\{|x^1-x^2|,\dots,|x^k-x^{k+1}|\,{:}\,x^i\in E\}$ has non-empty interior in $\mathbb{R}^k$ when $E\subset \mathbb{R}^2$ is a Cartesian product of thick Cantor sets $K_1,K_2\subset\mathbb{R}$. We also prove more general results where the distance map $|x-y|$ is replaced by a function $\phi(x,y)$ satisfying mild assumptions on its partial derivatives. In the process, we establish a nonlinear version of the classic Newhouse Gap Lemma, and show that if $K_1,K_2, \phi$ are as above then there exists an open set S so that $\bigcap_{x \in S} \phi(x,K_1\times K_2)$ has non-empty interior.

Out-of-network air ambulance bills are a pernicious and financially devastating type of surprise medical bill. Courts have broadly interpreted the Airline Deregulation Act to preempt most state attempts to regulate air ambulance billing abuses, so a federal solution is ultimately needed. However, in the absence of a federal fix, states have experimented with a variety of approaches that may survive preemption and provide some protections for their citizens

Our objective is to map dynamic provinces and investigate dynamic changes in Jakobshavn Isbræ, Greenland. We use an approach that combines structural glaciology and remote-sensing data analysis, facilitated by mathematical characterization of generalized spatial surface roughness that provides parameters related to ice dynamics, deformation and interaction of the ice with bed topography. The approach is applied to derive time series of elevation and roughness changes and to attribute changes during rapid retreat. Different dynamic types of fast- and slow-moving ice can be mapped from ICESat Geoscience Laser Altimeter System data (2003–09) and Airborne Topographic Mapper data, using spatial roughness characterization, validated with ASTER and bed-topographic data. Results of comparative analysis of elevation changes and roughness changes of Jakobshavn south ice stream indicate (1) surface lowering of 10–15 m a-1 between 2004 and 2009 and (2) no change in surface roughness and dynamic types. These findings are consistent with a front retreat as part of a fjord-glacier cycle or following warming of fjord water and with climatic warming, but not with an internal dynamic acceleration as a cause of the observed changes during rapid retreat. Relationships to changes in basal water pressure are discussed. All glaciodynamic changes appear to have initiated near the front and propagated up-glacier.

Equilibrium in the market for real assets requires that the price of those assets be bid up to reflect the tax shields they can offer to levered firms. Thus, there must be an equality between the market values of real assets and the values of optimally levered firms. The standard measure of the advantage to leverage compares the values of levered and unlevered assets, and can be misleading and difficult to interpret. We show that a meaningful measure of the advantage to debt is the extra rate of return, net of a market premium for bankruptcy risk, earned by a levered firm relative to an otherwise-identical unlevered firm. We construct an option valuation model to calculate such a measure and present extensive simulation results. We use this model to compute optimal debt maturities, show how this approach can be used for capital budgeting, and discuss its implications for the comparison of bankruptcy costs versus tax shields.