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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]

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]