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Denote by CH, CSH, CQH, and CSQH the planar cubic homogeneous, cubic semi-homogeneous, cubic quasi-homogeneous and cubic semi-quasi-homogeneous differential systems, respectively. The problems on limit cycles and global dynamics of these systems have been solved for CH, and partially for CSH. This paper studies the same problems for CQH and CSQH. We prove that CQH have no limit cycles and CSQH can have at most one limit cycle with the limit cycle realizable. Moreover, we classify all the global phase portraits of CSQH. [ABSTRACT FROM AUTHOR]

We establish some equivalent conditions to the Hilbert-Smith conjecture on connected n-manifolds M. For n = 3; we show that regularly almost periodic homeomorphisms of M are periodic; this result extends Theorem 5.34 of Gottschalk and Hedlund (Topological Dynamics. American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, ). For the special case of $${M = \mathbb{R}^3}$$ , we extend the result of Brechner (Pac J Math 59(2):367-374, ) saying that 'almost periodic homeomorphisms of the plane are periodic' to $${\mathbb{R}^3}$$ , and we show that any compact abelian group of homeomorphisms of $${\mathbb{R}^3}$$ is either finite or topologically equivalent to a subgroup of the orthogonal group O(3). [ABSTRACT FROM AUTHOR]

In this paper we present purely combinatorial conditions that allow us to recognize the topological equivalence (or non-equivalence) of two given dissections. Using a computer program based on this result, we are able to generate a set which contains all topologically non-equivalent dissections of a p0-gon into convex pi-gons, i = 1...n, where n, p0,...,pn are integers such that n ≥ 2, pi ≥ 3. By analyzing generated structures, we are able to find all (up to similarity) dissections of a given type. Since the number of topologically non-equivalent dissections is huge even when the number of parts is small, it is necessary to find additional combinatorial conditions depending on the type of sought dissections, which will allow us to exclude the majority of generated structures. We present such conditions for some special dissections of a triangle into triangles. Finally we prove two new results concerning perfect dissections of a triangle into similar triangles. [ABSTRACT FROM AUTHOR]