Let P be a set of n points in R, and k ≤ n an integer. A sphere σ is k-rich with respect to P if |σ ∩ P | ≥ k, and is η-nondegenerate, for a fixed fraction 0 < η < 1, if no circle γ ⊂ σ contains more than η|σ ∩ P | points of P . We improve the previous bound given in [1] on the number of k-rich η-nondegenerate spheres in 3space with respect to any set of n points inR, from O(n/k+n/k), which hol...

In this paper we prove that for every bumpy Finsler metric F on every rationally homological n-dimensional sphere Sn with n ≥ 2, there exist always at least two distinct prime closed geodesics.

Connes and Cuntz showed in [Comm. Math. Phys. 114 (1988), 515–526] that suitable cyclic cocycles can be represented as Chern characters of finitely summable semifinite Fredholm modules. We show an analogous result in twisted cyclic cohomology using Chern characters of modular Fredholm modules. We present examples of modular Fredholm modules arising from Podleś spheres and from SUq(2).

We recognize the Gromoll-Meyer sphere Σ as the geodesic join of a simple closed geodesic and a minimal subsphere Σ ⊂ Σ, which can be equivariantly identified with the Brieskorn sphere W 5 3 . As applications we in particular determine the full isometry group of Σ, classify all closed subgroups which act freely, determine the homotopy types of the corresponding orbit spaces, identify the Hirsch-...

INTRODUCTION Along with the miniaturization of optical devices such as medical endoscopes and fiber-optic communication systems, micro-aspheric lenses (1 mm in diameter) have become very important parts [1]. Micro-aspheric lenses are necessary for downsizing and higher image quality. Form accuracy of micro-aspheric lenses is required to be better than 100 nm to make the most of those devises. H...

A Dehn sphere Σ [P] in a closed 3-manifold M is a 2-sphere immersed in M with only double curve and triple point singularities. The sphere Σ fills M [Mo2] if it defines a celldecomposition of M . The inverse image in S of the double curves of Σ is the Johansson diagram of Σ [J1] and if Σ fills M it is possible to reconstruct M from the diagram. A Johansson representation of M is the Johansson d...