We explore the cardinality of generalised inverse limits. Among other things, we show that, for any $n\in \{ℵ_{0},c,1,2,3,\dots \}$, there is an upper semicontinuous function with the inverse limit having exactly $n$ points. We also prove that if $f$ is an upper semicontinuous function whose graph is a continuum, then the cardinality of the corresponding inverse limit is either 1, $ℵ_{0}$ or $c$. This generalises the recent result of I. Banič and J. Kennedy, which claims that the same is true in the case where the graph is an arc.

An example is given of a map $f$ defined between arcwise connected continua such that $C(f)$ is light and ${{2}^{f}}$ is not light, giving a negative answer to a question of Charatonik and Charatonik. Furthermore, given a positive integer $n$, we study when the lightness of the induced map ${{2}^{f}}$ or ${{C}_{n}}(f)$ implies that $f$ is a homeomorphism. Finally, we show a result in relation with the lightness of $C(C(f))$.

This paper proposes and evaluates an artificial neural network based method of modeling the dynamic behavior of spatially distributed continuous engineering processes. The technique is applicable to situations where the differential equations governing the behavior of a system are nonlinear and poorly understood, such as is the case for frost-heave and thaw-settlement processes in soils. A description is first provided of a means of modeling the unknown component of governing differential equations. A range of levels of understanding of these equations is considered. A method of discretizing the resultant neural models of these equations is then illustrated, and the way in which these can be used to simulate the behavior of a process is described. The performance of the proposed neural network approach is then assessed in a series of experiments simulating the nonlinear thermal behavior of translucent solid materials. The system is proven capable of providing highly accurate simulations of system behavior sustained over many thousands of simulation time steps. The paper concludes with an identification of several ongoing areas of further development and application of the proposed tool.