Weak differentiability of product measures

In this paper, we study cost functions over a finite collection of random variables. For these types of models, a calculus of differentiation is developed that allows us to obtain a closed-form expression for derivatives where "differentiation" has to be understood in the weak sense. The technique for proving the results is new and establishes an interesting link between functional analysis and gradient estimation. The key contribution of this paper is a product rule of weak differentiation. In addition, a product rule of weak analyticity is presented that allows for Taylor series approximations of finite products measures. In particular, from characteristics of the individual probability measures, a lower bound (i.e., domain of convergence) can be established for the set of parameter values for which the Taylor series converges to the true value. Applications of our theory to the ruin problem from insurance mathematics and to stochastic activity networks arising in project evaluation review techniques are provided. © 2010 INFORMS

Oxygen uptake kinetics in trained adolescent females

Little evidence exists with regard to the effect that exercise training has upon oxygen uptake kinetics in adolescent females. PURPOSE: The aim of the study was to compare [Formula: see text] and muscle deoxygenation kinetics in a group of trained (Tr) and untrained (Utr) female adolescents. METHOD: Twelve trained (6.4 ± 0.9 years training, 10.3 ± 1.4 months per year training, 5.2 ± 2.0 h per week) adolescent female soccer players (age 14.6 ± 0.7 years) were compared to a group (n = 8) of recreationally active adolescent girls (age 15.1 ± 0.6 years) of similar maturity status. Subjects underwent two, 6-min exercise transitions at a workload equivalent to 80 % of lactate threshold from a 3-min baseline of 10 W. All subjects had a passive rest period of 1 h between each square-wave transition. Breath-by-breath oxygen uptake and muscle deoxygenation were measured throughout and were modelled via a mono-exponential decay with a delay relative to the start of exercise. RESULT: Peak [Formula: see text] was significantly (p < 0.05) greater in the Tr compared to the Utr (Tr: 43.2 ± 3.2 mL kg(-1 )min(-1) vs. Utr: 34.6 ± 4.0 mL kg(-1 )min(-1)). The [Formula: see text] time constant was significantly (p < 0.05) faster in the Tr compared to the Utr (Tr: 26.3 ± 6.9 s vs. Utr: 35.1 ± 11.5 s). There was no inter-group difference in the time constant for muscle deoxygenation kinetics (Tr: 8.5 ± 3.0 s vs. Utr: 12.4 ± 8.3 s); a large effect size, however, was demonstrated (-0.804). CONCLUSION: Exercise training and/or genetic self-selection results in faster kinetics in trained adolescent females. The faster [Formula: see text] kinetics seen in the trained group may result from enhanced muscle oxygen utilisation