We consider solutions $u^\varepsilon$ of Poisson problems with the Dirichlet condition on domains $\Omega_\varepsilon$ with holes concentrated at subsets of a domain $\Omega$ nonperiodically. We show $u^\varepsilon$ converges to a solution of a Poisson problem with a simple function potential. This is a generalized result of a sample model given by Cioranescu and Murat (1997). They showed a result for case that holes are distributed at $\Omega$ periodically.

This paper treats resolvent $L_q$ estimate and maximal $L_p-L_q$ regularity for the heat equation with various boundary conditions in an infinite layer. We need to consider two boundary conditions on upper boundary and lower boundary. We are able to choose any pair of Dirichlet, Neumann and Robin boundary conditions. We construct the solutions of Fourier multiplier operators and we use a theorem for an integral operator, which derives $L_q$-boundedness and $L_p-L_q$ boundedness. The key is that the holomorphic symbols can be properly estimated from above.

The model to represent mortality that Lee and Carter proposed is still widely used. They proposed a method to estimate the parameters in the model with mortality data using the singular value decomposition. In addition, in the forecasting of future mortality, the time-dependent parameter is linearly estimated using ARIMA (0,1,0). This method treats the parameters as nonstochastic for part of the model construction using observed data, while it is stochastic for the forecasting of future mortality. This results in inconsistencies throughout the whole model. Girosi and King interpreted the parameters in the Lee-Carter model as random variables and provided an integrated expression for the two parts, which derived a single stochastic model; however, the covariance matrix in the single stochastic model is not the one by an ordinal ARIMA(0,1,0) and an estimation method was not clearly discussed, and estimating the parameters in their model appears to be difficult owing to the complicated covariance matrix. Therefore, this study proposes a new integrated model of the Lee-Carter model based on Girosi and King’s interpretation. Our model is defined as a stochastic model that does not conflict with the concepts of Lee and Carter’s existing model with the covariance matrix deduced by Girosi and King. Furthermore, we provide an estimation method for the parameters in our model by applying the idea of conditional distributions used in classical AR and MA models.

We show that the principal 2-blocks of infinite series of 2- dimensional general linear groups $GL_2(q)$ with wreathed Sylow 2-subgroups are splendidly Morita equivalent. Consequently, Puig’s conjecture holds in this case. To construct the splendid Morita equivalences, we use relative stable equivalences of Morita type introduced by Wang and Zhang.