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An important problem in analysis on fractals is the existence and the determination of an eigenform on a given finitely ramified fractal. It is known that on every fractal either with three vertices or with connected interior, an eigenform exists for suitable weights on the cells. In this paper, we prove that if the fractal has three vertices and connected interior, the form having all coefficients equal to 1 is an eigenform for suitable weights on the cells. [ABSTRACT FROM AUTHOR]

Purpose: The three-parameter model of critical power (3-p) implies that in the severe exercise intensity domain time to exhaustion (Tlim) decreases hyperbolically with power output starting from the power asymptote (critical power, ẇcr) and reaching 0 s at a finite power limit (ẇ0) thanks to a negative time asymptote (k). We aimed to validate 3-p for short Tlim and to test the hypothesis that ẇ0 represents the maximal instantaneous muscular power.Methods: Ten subjects performed an incremental test and nine constant-power trials to exhaustion on an electronically braked cycle ergometer. All trials were fitted to 3-p by means of non-linear regression, and those with Tlim greater than 2 min also to the 2-parameter model (2-p), obtained constraining k to 0 s. Five vertical squat jumps on a force platform were also performed to determine the single-leg (i.e., halved) maximal instantaneous power.Results: Tlim ranged from 26 ± 4 s to 15.7 ± 4.9 min. In 3-p, with respect to 2-p, ẇcr was identical (177 ± 26 W), while curvature constant W' was higher (17.0 ± 4.3 vs 15.9 ± 4.2 kJ, p < 0.01). 3-p-derived ẇ0 was lower than single-leg maximal instantaneous power (1184 ± 265 vs 1554 ± 235 W, p < 0.01).Conclusions: 3-p is a good descriptor of the work capacity above ẇcr up to Tlim as short as 20 s. However, since there is a discrepancy between estimated ẇ0 and measured maximal instantaneous power, a modification of the model has been proposed. [ABSTRACT FROM AUTHOR]

If, as postulated, the end of the steady state phase (φ2) of cardiovascular responses to apnoea corresponds to the physiological breaking point, then we may hypothesize that φ2 should become visible if exercise apnoeas are performed in pure oxygen. We tested this hypothesis on 9 professional divers by means of continuous recording of blood pressure (BP), heart rate ( f H ), stroke volume ( Q S ), and arterial oxygen saturation (SpO 2 ) during dry maximal exercising apnoeas in ambient air and in oxygen. Apnoeas lasted 45.0 ± 16.9 s in air and 77.0 ± 28.9 s in oxygen (p < 0.05). In air, no φ2 was observed. Conversely, in oxygen, a φ2 of 28 ± 5 s duration appeared, during which systolic BP (185 ± 29 mmHg), f H (93 ± 16 bpm) and Q S (91 ± 16 ml) remained stable. End-apnoea SpO 2 was 95.5 ± 1.9% in air and 100% in oxygen. The results support the tested hypothesis. [ABSTRACT FROM AUTHOR]