Geometric Origin of the Tennis Racket Effect.

The tennis racket effect is a geometric phenomenon which occurs in a free rotation of a three-dimensional rigid body. In a complex phase space, we show that this effect originates from a pole of a Riemann surface and can be viewed as a result of the Picard-Lefschetz formula. We prove that a perfect twist of the racket is achieved in the limit of an ideal asymmetric object. We give upper and lower bounds to the twist defect for any rigid body, which reveals the robustness of the effect. A similar approach describes the Dzhanibekov effect in which a wing nut, spinning around its central axis, suddenly makes a half-turn flip around a perpendicular axis and the monster flip, an almost impossible skateboard trick.

Relative heart rate, heart rate reserve, and VO2 during submaximal exercise in the elderly.

BACKGROUND: The purpose of this study was to examine the relationships among relative maximal heart rate (%HRmax), maximal heart rate reserve (%HRmax reserve), and maximal oxygen uptake (%VO2max) during submaximal exercise by elderly subjects. METHODS: VO2max and HRmax were determined on 36 women and 19 men, 60 to 80 yrs of age, by a maximal treadmill test to volitional exhaustion. On a separate day, subjects underwent a submaximal treadmill protocol consisting of three 6-min exercise stages at treadmill speeds and grades estimated to elicit 40%, 60%, and 80% of HRmax reserve. Cardiorespiratory responses were determined during mins 4-5 and 5-6 of each stage. RESULTS: Measured exercise intensities expressed by the three methods were: %HRmax reserve = 36, 55, and 79%; %HRmax = 65, 75, and 88%; %VO2max = 53, 69, and 88%. %HRmax was greater (p < .05) than %VO2max at 53 and 69% of VO2max. %HRmax reserve was less (p < .05) than %VO2max for all three intensities. Slopes and intercepts for the linear regression equations relating %VO2max with %HRmax and with %HRmax reserve differed between men and women (p < .05). The regression equation relating %VO2max and %HRmax was y = -22.8 + 1.2 (%HRmax) -13.0 (Gender) + 0.2 (%HRmax x Gender): standard error of the estimate (SEE) = 9.7% and R2 = .71. The regression equation relating %VO2max and %HRmax reserve was y = 32.4 + 0.7 (%HRmax reserve) -10.9 (Gender) + 0.2 (%HRmax reserve x Gender): SEE = 9.8% and R2 = .70 (Gender: F = 0; M = 1). CONCLUSIONS: The data indicate that there is considerable variability among methods of expressing exercise intensity and that %HRmax more closely represents %VO2max than does %HRmax reserve (p < .05) in older adults. These results are in contrast to what has been shown with younger subjects and with American College of Sports Medicine guidelines for exercise prescription.

The lack of effect of aerobic exercise training on propranolol pharmacokinetics in young and elderly adults.

The effects of exercise training on the pharmacokinetics of orally administered propranolol were studied in young and elderly healthy volunteers. Twenty-three young (30 +/- 5 years of age) and 20 elderly (67 +/- 5 years of age) adults were randomly assigned to endurance training (N = 12 young subjects, 10 elderly subjects) or nonexercising control (N = 11 young subjects, 10 elderly subjects) groups. Training consisted of treadmill walking, stair climbing, or both three times per week for 40 minutes at 70-85% of maximal heart rate reserve for 16 weeks. Resting plasma propranolol concentrations after a single dose of 80 mg of oral propranolol were measured by high performance liquid chromatography with fluorescence detection, and estimated hepatic blood flow measured was measured using indocyanine green during supine test. Aerobic training increased maximal oxygen uptake (VO2 max) by 13% and 14% in the exercising young and elderly groups, respectively. There was no change in VO2 max in either control group. Adjusted mean estimated hepatic blood flow after exercise corrected for body weight for the young subjects who did not exercise (15.6 mL/min/Kg) and those who did (18.2 mL/min/Kg) groups were of borderline significance. No statistical differences were detected in the experimental propranolol pharmacokinetic parameters (maximal concentration, time of maximal concentration, terminal half-life, area under the curve, and protein binding) or derived pharmacokinetic parameters (intrinsic clearance, bioavailability, clearance, and volume of distribution). These results provide evidence that changes in aerobic fitness do not produce corresponding changes in propranolol pharmacokinetics in young or elderly adults.