On Points Drawn from a Circle

On Points Drawn from a Circle

Choose n points from the unit circle — no two with the same x-coordinate or y-coordinate (we call such a set of points generic; another term is noncorectilinear) — label them 1–n by height, reading bottom-to-top, and record these labels reading left-to-right. This operation produces a permutation. For example, the set of points shown below on the left gives the permutation 45312, the plot of wh...

Predicting (0,l)-Functions on Randomly Drawn Points

We consider the problem of predicting (0 , l ) valued functions on R" and smaller domains, based on their values on randomly drawn points. Our model is related to Valiant's learnability model, but does not require the hypotheses used for prediction to be represented in any specified form. First we disregard computational complexity and show how to construct prediction strategies that are optima...

Covering Points on a Circle with Circular Arcs

Fixed Points of Circle Actions

Let X be a finitistic space with its rational cohomology isomorphic to that of the wedge sum P (n)∨S or S ∨S∨S. We study continuous S actions on X and determine the possible fixed point sets up to rational cohomology depending on whether or not X is totally non-homologous to zero in XS1 in the Borel fibration X →֒ XS1 −→ BS1 . We also give examples realizing the possible cases.

On Semifree Symplectic Circle Actions with Isolated Fixed Points

Let M be a symplectic manifold, equipped with a semifree symplectic circle action with a finite, nonempty fixed point set. We show that the circle action must be Hamiltonian, and M must have the equivariant cohomology and Chern classes of (P ).