A curve shortening flow rule for closed embedded plane curves with a prescribed rate of change in enclosed area

Motivated by a problem from fluid mechanics, we consider a generalization of the standard curve shortening flow problem for a closed embedded plane curve such that the area enclosed by the curve is forced to decrease at a prescribed rate. Using formal asymptotic and numerical techniques, we derive possible extinction shapes as the curve contracts to a point, dependent on the rate of decreasing area; we find there is a wider class of extinction shapes than for standard curve shortening, for which initially simple closed curves are always asymptotically circular. We also provide numerical evidence that self-intersection is possible for non-convex initial conditions, distinguishing between pinch-off and coalescence of the curve interior

Posing as the ultimate Scottish Songster: James Hogg’s Songs by the Ettrick Shepherd or Hogg GOLD!

A brief article about the completion of the AHRC-funded James Hogg Songs Project and the place of Hogg's final collection of songs in his work and legacy

Assessing the Financial Health of Medicaid Managed Care Plans and the Quality of Patient Care They Provide

Examines the administrative and medical expenses, quality of care, and financial stability of publicly traded health plans contracted to manage the care of Medicaid beneficiaries by plan characteristics and compared with non-publicly traded plans

How Has the Affordable Care Act Affected Health Insurers' Financial Performance?

Starting in 2014, the Affordable Care Act transformed the market for individual health insurance by changing how insurance is sold and by subsidizing coverage for millions of new purchasers. Insurers, who had no previous experience under these market conditions, competed actively but faced uncertainty in how to price their products. This issue brief uses newly available data to understand how health insurers fared financially during the ACA's first year of full reforms. Overall, health insurers' financial performance began to show some strain in 2014, but the ACA's reinsurance program substantially buffered the negative effects for most insurers. Although a quarter of insurers did substantially worse than others, experience under the new market rules could improve the accuracy of pricing decisions in subsequent years

Corner and finger formation in Hele--Shaw flow with kinetic undercooling regularisation

We examine the effect of a kinetic undercooling condition on the evolution of a free boundary in Hele--Shaw flow, in both bubble and channel geometries. We present analytical and numerical evidence that the bubble boundary is unstable and may develop one or more corners in finite time, for both expansion and contraction cases. This loss of regularity is interesting because it occurs regardless of whether the less viscous fluid is displacing the more viscous fluid, or vice versa. We show that small contracting bubbles are described to leading order by a well-studied geometric flow rule. Exact solutions to this asymptotic problem continue past the corner formation until the bubble contracts to a point as a slit in the limit. Lastly, we consider the evolving boundary with kinetic undercooling in a Saffman--Taylor channel geometry. The boundary may either form corners in finite time, or evolve to a single long finger travelling at constant speed, depending on the strength of kinetic undercooling. We demonstrate these two different behaviours numerically. For the travelling finger, we present results of a numerical solution method similar to that used to demonstrate the selection of discrete fingers by surface tension. With kinetic undercooling, a continuum of corner-free travelling fingers exists for any finger width above a critical value, which goes to zero as the kinetic undercooling vanishes. We have not been able to compute the discrete family of analytic solutions, predicted by previous asymptotic analysis, because the numerical scheme cannot distinguish between solutions characterised by analytic fingers and those which are corner-free but non-analytic