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2 results

  • AN INCIDENCE RESULT FOR WELL-SPACED ATOMS IN ALL DIMENSIONS

  • Part of
  • PETER J. BRADSHAW
  • Journal:
    Journal of the Australian Mathematical Society / Volume 115 / Issue 1 / August 2023
    Published online by Cambridge University Press:
    20 February 2023, pp. 58-72
    Print publication:
    August 2023
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  • We prove an incidence result counting the k-rich $\delta $-tubes induced by a well-spaced set of $\delta $-atoms. Our result coincides with the bound that would be heuristically predicted by the Szemerédi–Trotter theorem and holds in all dimensions $d \geq 2$. We also prove an analogue of Beck’s theorem for $\delta $-atoms and $\delta $-tubes as an application of our result.

  • Loops as Invariant Sections in Groups, and their Geometry

  • Péter T. Nagy, Karl Strambach
  • Journal:
    Canadian Journal of Mathematics / Volume 46 / Issue 5 / 01 October 1994
    Published online by Cambridge University Press:
    20 November 2018, pp. 1027-1056
    Print publication:
    01 October 1994
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  • We investigate left conjugacy closed loops which can be given by invariant sections in the group generated by their left translations. These loops are generalizations of the conjugacy closed loops introduced in [13] just as Bol loops generalize Moufang loops. The relations of these loops to common classes of loops are studied. For instance on a connected manifold we construct proper topological left conjugacy closed loops satisfying the left Bol condition but show that any differentiable such loop must be a group. We show that the configurational condition in the 3-net corresponding to an isotopy class of left conjugacy closed loops has the same importance in the geometry of 3-nets as the Reidemeister or the Bol condition.