[For the entire collection see Zbl 0742.00067.]This paper is devoted to a method permitting to determine explicitly all multilinear natural operators between vector-valued differential forms and between sections of several other natural vector bundles.

For an abelian lattice ordered group $G$ let $G$ be the system of all compatible convergences on $G$; this system is a meet semilattice but in general it fails to be a lattice. Let ${\alpha }_{nd}$ be the convergence on $G$ which is generated by the set of all nearly disjoint sequences in $G$, and let $\alpha$ be any element of $G$. In the present paper we prove that the join ${\alpha }_{nd}\vee \alpha$ does exist in $G$.

In 1985, V. G. Pestov described a neighborhood base at the identity of free topological groups on a Tychonoff space in terms of the elements of the fine uniformity on the Tychonoff space. In this paper, we extend Postev’s description to the free paratopological groups where we introduce a neighborhood base at the identity of free paratopological groups on any topological space in terms of the elements of the fine quasiuniformity on the space.

For Tychonoff $X$ and $\alpha$ an infinite cardinal, let $\alpha defX:=$ the minimum number of $\alpha$ cozero-sets of the Čech-Stone compactification which intersect to $X$ (generalizing $ℝ$-defect), and let $rtX:={min}_{\alpha }max(\alpha ,\alpha defX)$. Give $C\left(X\right)$ the compact-open topology. It is shown that $\tau C\left(X\right)\le n\chi C\left(X\right)\le rtX=max\left(L\right(X),L(X)defX)$, where: $\tau$ is tightness; $n\chi$ is the network character; $L\left(X\right)$ is the Lindel"of number. For example, it follows that, for $X$ Čech-complete, $\tau C\left(X\right)=L\left(X\right)$. The (apparently new) cardinal functions $n\chi C$ and $rt$ are compared with several others.

Utilizing the theory of the Poisson transform, we develop some new concrete models for the Hecke theory in a space $M_{\lambda }\left(N\right)$ of Maass forms with eigenvalue $1/4-{\lambda }^2$ on a congruence subgroup ${\Gamma }_1\left(N\right)$. We introduce the field $F_{\lambda }=\mathbb{Q}(\lambda ,\sqrt{n},n^{\lambda /2}\mid n\in \mathbb{N})$ so that $F_{\lambda }$ consists entirely of algebraic numbers if $\lambda =0$.The main result of the paper is the following. For a packet $\Phi =({\nu }_p\mid p\nmid N)$ of Hecke eigenvalues occurring in $M_{\lambda }\left(N\right)$ we then have that either every ${\nu }_p$ is algebraic over $F_{\lambda }$, or else $\Phi$ will – for some $m\in \mathbb{N}$ – occur in the first cohomology of a certain space $W_{\lambda ,m}$ which is a...