Hyperfocused Arcs

t-Pancyclic Arcs in Tournaments

Let $T$ be a non-trivial tournament. An arc is emph{$t$-pancyclic} in $T$, if it is contained in a cycle of length $ell$ for every $tleq ell leq |V(T)|$. Let $p^t(T)$ denote the number of $t$-pancyclic arcs in $T$ and $h^t(T)$ the maximum number of $t$-pancyclic arcs contained in the same Hamiltonian cycle of $T$. Moon ({em J. Combin. Inform. System Sci.}, {bf 19} (1994), 207-214) showed that $...

The statistical difference between bending arcs and regular polar arcs

In this work, the Polar UVI data set by Kullen et al. (2002) of 74 polar arcs is reinvestigated, focusing on bending arcs. Bending arcs are typically faint and form (depending on interplanetary magnetic field (IMF) By direction) on the dawnside or duskside oval with the tip of the arc splitting off the dayside oval. The tip subsequently moves into the polar cap in the antisunward direction, whi...

Existence of Quasi-arcs

We show that doubling, linearly connected metric spaces are quasiarc connected. This gives a new and short proof of a theorem of Tukia.

An Exaltation of Arcs

Arcs in Desarguesian nets

A trivial upper bound on the size k of an arc in an r-net is k ≤ r + 1. It has been known for about 20 years that if the r-net is Desarguesian and has odd order, then the case k = r + 1 cannot occur, and k ≥ r−1 implies that the arc is contained in a conic. In this paper, we show that actually the same must hold provided that the difference r − k does not exceed p k/18. Moreover, it is proved t...