On random subsets of projective spaces

on projective spaces

We prove that a one-dimensional foliation with generic singularities on a projective space, exhibiting a Lie group transverse structure in the complement of some codimension one algebraic subset is logarithmic, i.e., it is the intersection of codimension one foliations given by closed one-forms with simple poles. If there is only one singularity in a suitable affine space, then the foliation is...

Topologies on Spaces of Subsets

Projective embedding of projective spaces

In this paper, embeddings φ : M → P from a linear space (M,M) in a projective space (P,L) are studied. We give examples for dimM > dimP and show under which conditions equality holds. More precisely, we introduce properties (G) (for a line L ∈ L and for a plane E ⊂ M it holds that |L ∩ φ(M)| 6 = 1) and (E) (φ(E) = φ(E) ∩ φ(M), whereby φ(E) denotes the by φ(E) generated subspace of P ). If (G) a...

On Random Shuffles & Subsets

Four simple algorithms are presented for producing a random rearrangement of the values in an array X[1], X[2], ..., X[n] using a pseudorandom number generator. These range from one that is naive, inefficient, and biased to one that is elegeant, efficient, and unbiased. By taking the first k entries in X one obtains a random (ordered) subset of size k.

Geodesics on Weighted Projective Spaces

We study the inverse spectral problem for weighted projective spaces using wave-trace methods. We show that in many cases one can “hear” the weights of a weighted projective space.