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Abstract. The complete elliptic integrals are generalized by using the generalized trigonometric functions with two parameters. As an application of the integrals, an alternative proof of a result for a family of means by Bhatia and Li, which involves the logarithmic mean and the arithmetic-geometric mean, is given. Moreover, it is shown that a particular relation holds for the generalized integrals.Mathematics subject classification (2010): 33E05, 33C75, 34L10.

Ambient conditions may change rapidly and notably over time in urban areas. Conventional indices, such as the heat index and wet bulb globe temperature, are useful only in stationary ambient conditions. To estimate the risks of heat-related illness, human thermophysiological responses should be followed for ambient conditions in the time domain. We develop a computational method for estimating the time course of core temperature and water loss by combining micrometeorology and human thermal response. We firstly utilize an urban micrometeorology prediction to reproduce the environment surrounding walkers. The temperature elevations and sweating in a standard adult and child are then estimated for meteorological conditions. With the integrated computational method, we estimate the body temperature and thermophysiological responses for an adult and child walking along a street with two routes (sunny and shaded) in Tokyo on 7 August 2015. The difference in the core temperature elevation in the adult between the two routes was 0.11 • C, suggesting the necessity for a micrometeorology simulation. The differences in the computed body core temperatures and water loss of the adult and child were notable, and were mainly characterized by the surface area-to-mass ratio. The computational techniques will be useful for the selection of actions to manage the risk of heat-related illness and for thermal comfort.

Performance and completion rates in marathons attract considerable attention in race planning and risk management. Previous studies have explored the relationship between several parameters, including athlete age and weather, and marathon performance. In this study, for the first time, we correlate the computational estimation of core body temperature and perspiration of athletes with speed loss and completion rate in marathon races. A feature of this method is that nonlinear thermo-physiological responses, that is, core temperature change and sweating, are followed in the time domain for ambient conditions. Our computational estimation was in good agreement with group-level core temperature rise and perspiration in typical marathon races. When the ambient conditions in previous World Athletics Championships and Olympic Games were replicated, the estimated perspiration was better correlated with the marathon speed (R 2 = 0.50, p < 0.05 in men) than with the ambient temperature (R 2 = 0.37, not statistically significant), which was used in conventional studies. The correlation for female athletes was better than that for male athletes. A weak correlation was observed between the completion rate and ambient conditions, as well as the thermo-physiological response in male athletes (R 2 = 0.3), whereas a strong correlation was observed in female athletes (R 2 = 0.7). This method was applied to estimate pacing in the upcoming Olympic Games in 2021 to discuss the effect of location change on marathon performance. The findings of this study may be useful for race pacing and risk management of heat-induced illness.

Power consumption is an important parameter for the design of mixing equipment. e aim of this study is to develop a new correlation of the power consumption of a double impeller. e effect of impeller spacing on the double-impeller flow pattern and power consumption was investigated in the laminar region. As a result, the effect of impeller spacing on the flow pattern was described based on the ratio of impeller spacing to the impeller blade height. Moreover, the power consumption of a double impeller could be correlated with the same ratio.