1. CMB Online first
||Relative equilibria in curved restricted $4$-body problems|
We consider the curved $4$-body problems on spheres and hyperbolic
spheres. After obtaining a criterion for the existence of quadrilateral
configurations on the equator of the sphere, we study two restricted
$4$-body problems, one in which two masses are negligible, and
another in which only one mass is negligible.
In the former we prove the evidence square-like relative equilibria,
whereas in the latter we discuss the existence of kite-shaped
Keywords:$N$-body problem, relative equilibria, celestial mechanics
2. CMB 1998 (vol 41 pp. 151)
||Equivalent presentations of modules over PrÃ¼fer domains |
If $F$ and $F^\prime$ are free
$R$-modules, then $M \cong F/H$ and $M \cong F^\prime/H^\prime$ are
viewed as equivalent presentations of the $R$-module $M$ if there is an
isomorphism $F \to F^\prime$ which carries the submodule $H$ onto $H^\prime$.
We study when presentations of modules of projective dimension $1$ over
Pr\"ufer domains of finite character are necessarily equivalent.