Geometric estimates for complex Monge–Ampère equations

We prove uniform gradient and diameter estimates for a family of geometric complex Monge–Ampère equations. Such estimates can be applied to study geometric regularity of singular solutions of complex Monge–Ampère equations. We also prove a uniform diameter estimate for collapsing families of twisted Kähler–Einstein metrics on Kähler manifolds of nonnegative Kodaira dimensions.

The impact of baseline artery diameter on flow-mediated vasodilation: a comparison of brachial and radial artery responses to matched levels of shear stress

An inverse relationship between baseline artery diameter (BAD) and flow-mediated vasodilation (FMD) has been identified using reactive hyperemia (RH) to create a shear stress (SS) stimulus in human conduit arteries. However, RH creates a SS stimulus that is inversely related to BAD. The purpose of this study was to compare FMD in response to matched levels of SS in two differently sized upper limb arteries [brachial (BA) and radial (RA) artery]. With the use of exercise, three distinct, shear rate (SR) stimuli were created (SR = blood velocity/vessel diameter; estimate of SS) in the RA and BA. Artery diameter and mean blood velocity were assessed with echo and Doppler ultrasound in 15 healthy male subjects (19–25 yr). Data are means ± SE. Subjects performed 6 min of adductor pollicis and handgrip exercise to increase SR in the RA and BA, respectively. Exercise intensity was modulated to achieve uniformity in SR between arteries. The three distinct SR levels were as follows: steady-state exercise 39.8 ± 0.6, 57.3 ± 0.7, and 72.4 ± 1.2 s−1 ( P < 0.001). %FMD and AbsFMD (mm) at the end of exercise were greater in the RA vs. the BA at each shear level [at the highest level: RA = 15.7 ± 1.5%, BA = 5.4 ± 0.8% ( P < 0.001)]. The mean slope of the within-subject SR-%FMD regression line was greater in the RA (RA = 0.33 ± 0.04, BA = 0.13 ± 0.02, P < 0.001), and a strong within-subjects relationship between %FMD and SR was observed in both arteries (RA: r2 = 0.92 ± 0.02; BA: r2 = 0.90 ± 0.03). Within the RA, there was a significant relationship between baseline diameter and %FMD; however, this relationship was not present in the BA (RA: r2 = 0.76, P < 0.001; BA: r2 = 0.03, P = 0.541). These findings suggest that the response to SS is not uniform across differently sized vessels, which is in agreement with previous studies.