This paper studies the construction of the exact solution for parabolic coupled systems of the type , , , , , and , where , , , and are arbitrary matrices for which the block matrix is nonsingular, and is a positive stable matrix.

The Birkhoff-Kakutani Theorem asserts that a topological group is metrizable if and only if it has countable character. We develop and apply tools for the estimation of the character for a wide class of nonmetrizable topological groups.We consider abelian groups whose topology is determined by a countable cofinal family of compact sets. These are the closed subgroups of Pontryagin-van Kampen duals of metrizable abelian groups, or equivalently, complete abelian groups whose dual is metrizable. By investigating these connections, we show that also in these cases, the character can be estimated, and that it is determined by the weights of the compact subsets of the group, or of quotients of the group by compact subgroups. It follows, for example, that the density and the local density of an abelian metrizable group determine the character of its dual group. Our main result applies to the more general case of closed subgroups of Pontryagin-van Kampen duals of abelianČech-complete groups.In the special case of free abelian topological groups, our results extend a number of results of Nickolas and Tkachenko, which were proved using combinatorial methods.In order to obtain concrete estimations, we establish a natural bridge between the studied concepts and pcf theory, that allows the direct application of several major results from that theory. We include an introduction to these results and their use.

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Abstract. Let I be an infinite set, {Gi : i ∈ I} be a family of (topological) groups and G = i∈I Gi be its direct product. For J ⊆ I, pJ : G → j∈J Gj denotes the projection. We say that a subgroup H of G is: (i) uniformly controllable in G provided that for every finite set J ⊆ I there exists a finite set K ⊆ I such that pJ (H) = pJ (H ∩ i∈K Gi); (ii) controllable in G provided that pJ (H) = pJ (H ∩ i∈I Gi) for every finite set J ⊆ I; (iii) weakly controllable in G if H ∩ i∈I Gi is dense in H, when G is equipped with the Tychonoff product topology. One easily proves that (i)→(ii)→(iii). We thoroughly investigate the question as to when these two arrows can be reversed. We prove that the first arrow can be reversed when H is compact, but the second arrow cannot be reversed even when H is compact. Both arrows can be reversed if all groups Gi are finite. When Gi = A for all i ∈ I, where A is an abelian group, we show that the first arrow can be reversed for all subgroups H of G if and only if A is finitely generated. Connections with coding theory are highlighted.

Abstract. Combining ideas of Troallic [20] and Cascales, Namioka, and Vera [3], we prove several characterizations of almost equicontinuity and hereditarily almost equicontinuity for subsets of metric-valued continuous functions when they are defined on aČech-complete space. We also obtain some applications of these results to topological groups and dynamical systems.