This is a further study of the work published in [2]. A generalized number system is established. The generalized functions denote the functions with generalized number system as their domain and range spaces which are different essentially from the Schwartz distributions. For such functions we define $(GNL)$ and $(G)$ integrals which can be considered as the generalization of the Lebesgue integral for real functions. Dirac $\delta$ function can be naturally represented by our generalized functions. This representation is more straightforward than the Schwartz distribution theory. Moreover, each distribution can be described by a generalized function in a natural way.

Let $A$ be a finite-dimensional algebra over an algebraically closed field $k$. A representation of the $A$-$A$ bimodule $DA=\mathrm{Hom}_{k}(A, k)$ is a module over the matrix algebra: \[\overline{A}= \begin{bmatrix} A & 0 \\ DA & A \end{bmatrix}\] We first prove that $\overline{A}$ is representation-finite (and in fact simply connected) whenever $A$ is an iterated tilted algebra of Dynbin type. We then assume that $A$ is a tilted algebra of Dynkin type, and characterise $\overline{A}$ by its Auslander-Reiten quiver.

In this paper we develop a fiber shape theory for maps between metric spaces. Our approach is based on the Mardešič-Segal method and, instead of ANR's, their fiber preserving analogues are used. A fiber preserving version of Chapman's complement theorem is proved.

This paper contains some results concerning completely movable maps and shape fibrations. It is shown that if $f:X\to Y$ is a completely movable map and $X$ is a finite dimensional ANR then $Y$ is an ANR iff $Y$ is finite dimensional. It is also shown that if $X$ is an ANR and $f:X\to Y$ is a shape fibration, then $Y$ is an ANR iff $Y$ is semi-locally contractible.