Topological Circle Geometries

“…A generalization of all these results and our results for n-unisolvent sets of continuous functions R 2 → R 2 seems to be possible to some extent. See [21], [22], [1], [26] for details about higher-dimensional affine planes and Laguerre planes.…”

Section: 12mentioning

confidence: 99%

“…For general information about the topics mentioned in this section the reader is referred to: [5], [6], [26] for flat Laguerre planes; [21], [22] for flat affine and projective planes; [1] for Möbius strip planes; [10], [23] for Chebyshev systems; [2], [15], [19], [20], [27] for unisolvent sets.…”

Section: Flat Laguerre Planes Flat Affine Planes and Unisolvent Setsmentioning

confidence: 99%

𝑛-unisolvent sets and flat incidence structures

Polster

1998

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. For the past forty years or so topological incidence geometers and mathematicians interested in interpolation have been studying very similar objects. Nevertheless no communication between these two groups of mathematicians seems to have taken place during that time. The main goal of this paper is to draw attention to this fact and to demonstrate that by combining results from both areas it is possible to gain many new insights about the fundamentals of both areas.In particular, we establish the existence of nested orthogonal arrays of strength n, for short nested n-OAs, that are natural generalizations of flat affine planes and flat Laguerre planes. These incidence structures have point sets that are "flat" topological spaces like the Möbius strip, the cylinder, and strips of the form I × R, where I is an interval of R. Their circles (or lines) are subsets of the point sets homeomorphic to the circle in the first two cases and homeomorphic to I in the last case. Our orthogonal arrays of strength n arise from n-unisolvent sets of half-periodic functions, n-unisolvent sets of periodic functions, and n-unisolvent sets of functions I → R, respectively.Associated with every point p of a nested n-OA, n > 1, is a nested (n − 1)-OA-the derived (n − 1)-OA at the point p. We discover that, in our examples that arise from n-unisolvent sets of n − 1 times differentiable functions that solve the Hermite interpolation problem, deriving in our geometrical sense coincides with deriving in the analytical sense.

show abstract

Invertible Sharplyn-Transitive Sets

Polster

1998

Journal of Combinatorial Theory, Series A

View full text Add to dashboard Cite

We prove various results about sharply n-transitive sets of homeomorphisms of`n ice'' topological spaces like the real line and the circle. Our main results concern sharply 3-transitive sets G of homeomorphisms of the circle to itself. If G=G &1 , then G contains a real hyperbolic part (a set of involutions of the circle having special properties). We show that every real hyperbolic part is the hyperbolic part of a real abstract oval in the sense of Buekenhout and that every real abstract oval arises from a topological oval in a flat projective plane. This establishes a new relationship between flat Minkowski planes that admit automorphisms which are circle-symmetries and flat projective planes containing topological ovals. We also consider sharply n-transitive sets of permutations acting on finite sets. We find that our results about flat geometries and sharply n-transitive sets of homeomorphisms have counterparts in the finite case.

show abstract

Topological Circle Geometries

Cited by 18 publications

References 90 publications

Semi-biplanes on the cylinder

Semi-biplanes on the cylinder

𝑛-unisolvent sets and flat incidence structures

Invertible Sharplyn-Transitive Sets

Contact Info

Product

Resources

About