Heat fluxes through roofs and their relevance to estimates of urban heat storage

The storage heat flux constitutes a large term in the heat balance of cities. This flux is difficult to measure but one approach is to parameterize it using relations between the net all-wave radiation and the heat flux conducted into and out of the typical materials that form the surface of cities and combine them into a weighted average to give the bulk storage. Urban heat storage parameterization could be improved if there were more and better estimates of the net radiation vs. storage relation for typical urban and suburban roofs. This paper presents the results of a study of the heat storage characteristics of 6 different roof assemblies (typical of many North American commercial/industrial and residential buildings) in Vancouver, Canada, observed across a range of wind and moisture conditions. Further, these observations are used to verify the Simplified Transient Analysis of Roofs (STAR) model, which is then used to estimate the heat storage parameterization for other roof types, thereby extending the usefulness of the scheme to a wider range of urban areas.     

The zero divisor problem of multivariable stochastic adaptive control

Stochastic adaptive minimum variance control algorithms require a division by a function of a recursively computed parameter estimate at each instant of time. In order that the analysis of these algorithms is valid, zero divisions must be events of probability zero. This property is established for the stochastic gradient adaptive control algorithm under the condition that the initial state of the system and all finite segments of its random disturbance process have a joint distribution which is absolutely continuous with respect to Lebesgue measure. This result is deduced from the following general result established in this paper: a non-constant rational function of a finite set of random variables {x₁},x_n} is absolutely continuous with respect to Lebesgue measure if the joint distribution function of {x₁,…,x_n} has this property.