Flatness of the setting Sun

Atmospheric refraction is responsible for the bending of light-rays in the atmosphere. It is a result of the continuous decrease in the refractive index of the air as a function of altitude. A well-known consequence of this phenomenon is the apparently elliptic shape of the setting or rising Sun (or full-Moon). In the present paper we systematically investigate this phenomenon in a standard atmosphere. Theoretical and numerical calculations are compared with experimental data. The asymmetric rim of the Sun is computed as a function of its inclination angle, observational height and meteorological conditions characterized by pressure, temperature and lapse-rate. We reveal and illustrate some extreme and highly unusual situations.Comment: RevTex, 10 pages, 14 Figures. A web-page is accompanying this study: http://www.fi.uib.no/~neda/sunset/index.htm

Risk assessment in high- and low-MELD liver transplantation

Allocation of liver grafts triggers emotional debates, as those patients, not receiving an organ, are prone to death. We analyzed a high-Model of End-stage Liver Disease (MELD) cohort (laboratory MELD score ≥30, n = 100, median laboratory MELD score of 35; interquartile range 31-37) of liver transplant recipients at our center during the past 10 years and compared results with a low-MELD group, matched by propensity scoring for donor age, recipient age, and cold ischemia time. End points of our study were cumulative posttransplantation morbidity, cost, and survival. Six different prediction models, including donor age x recipient MELD (D-MELD), Difference between listing MELD and MELD at transplant (Delta MELD), donor-risk index (DRI), Survival Outcomes Following Liver Transplant (SOFT), balance-of-risk (BAR), and University of California Los Angeles-Futility Risk Score (UCLA-FRS), were applied in both cohorts to identify risk for poor outcome and high cost. All score models were compared with a clinical-oriented decision, based on the combination of hemofiltration plus ventilation. Median intensive care unit and hospital stays were 8 and 26 days, respectively, after liver transplantation of high-MELD patients, with a significantly increased morbidity compared with low-MELD patients (median comprehensive complication index 56 vs. 36 points [maximum points 100] and double cost [median US$179 631 vs. US$80 229]). Five-year survival, however, was only 8% less than that of low-MELD patients (70% vs. 78%). Most prediction scores showed disappointing low positive predictive values for posttransplantation mortality, such as mortality above thresholds, despite good specificity. The clinical observation of hemofiltration plus ventilation in high-MELD patients was even superior in this respect compared with D-MELD, DRI, Delta MELD, and UCLA-FRS but inferior to SOFT and BAR models. Of all models tested, only the BAR score was linearly associated with complications. In conclusion, the BAR score was most useful for risk classification in liver transplantation, based on expected posttransplantation mortality and morbidity. Difficult decisions to accept liver grafts in high-risk recipients may thus be guided by additional BAR score calculation, to increase the safe use of scarce organs

An asymptotic theorem for a class of nonlinear neutral differential equations

summary:The neutral differential equation (1.1) $$\frac{{\mathrm{d}}^n}{{\mathrm{d}} t^n} [x(t)+x(t-\tau)] + \sigma F(t,x(g(t))) = 0,$$ is considered under the following conditions: $n\ge 2$, $\tau >0$, $\sigma = \pm 1$, $F(t,u)$ is nonnegative on $[t_0, \infty) \times (0,\infty)$ and is nondecreasing in $u\in (0,\infty)$, and $\lim g(t) = \infty$ as $t\rightarrow \infty$. It is shown that equation (1.1) has a solution $x(t)$ such that (1.2) $$\lim_{t\rightarrow \infty} \frac{x(t)}{t^k}\ \text{exists and is a positive finite value if and only if} \int^{\infty}_{t_0} t^{n-k-1} F(t,c[g(t)]^k){\mathrm{d}} t 0.$$ Here, $k$ is an integer with $0\le k \le n-1$. To prove the existence of a solution $x(t)$ satisfying (1.2), the Schauder-Tychonoff fixed point theorem is used