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Connections between connected topological spaces on the set of positive integers

Paulina Szczuka (2013)

Open Mathematics

In this paper we introduce a connected topology T on the set ℕ of positive integers whose base consists of all arithmetic progressions connected in Golomb’s topology. It turns out that all arithmetic progressions which are connected in the topology T form a basis for Golomb’s topology. Further we examine connectedness of arithmetic progressions in the division topology T′ on ℕ which was defined by Rizza in 1993. Immediate consequences of these studies are results concerning local connectedness of...

Construction of Šindel sequences

Michal Křížek, Alena Šolcová, Lawrence Somer (2007)

Commentationes Mathematicae Universitatis Carolinae

We found that there is a remarkable relationship between the triangular numbers T k and the astronomical clock (horologe) of Prague. We introduce Šindel sequences { a i } of natural numbers as those periodic sequences with period p that satisfy the following condition: for any k there exists n such that T k = a 1 + + a n . We shall see that this condition guarantees a functioning of the bellworks, which is controlled by the horologe. We give a necessary and sufficient condition for a periodic sequence to be a Šindel sequence....

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