Abstract Methods for improving upper and lower bounds various coverings of planar sets are proposed. New numbers partition constituents presented, suggestions the generalization presented methods offered.

It is known that a definably compact group G is an extension of a compact Lie group L by a divisible torsion-free normal subgroup. We show that the o-minimal higher homotopy groups of G are isomorphic to the corresponding higher homotopy groups of L. As a consequence, we obtain that all abelian definably compact groups of a given dimension are definably homotopy equivalent, and that their unive...

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This paper gives an upper bound for the first eigenvalue of the universal cover of a complete, stable minimal surface in hyperbolic space, and a sharper one for least area disks.