On the $k$-ary ‎M‎oment Map

On the Moment Map on Symplectic Manifolds

We consider a connected symplectic manifold M acted on by a connected Lie group G in a Hamiltonian fashion. If G is compact, we prove give an Equivalence Theorem for the symplectic manifolds whose squared moment map ‖ μ ‖ is constant. This result works also in the almost-Kähler setting. Then we study the case when G is a non compact Lie group acting properly on M and we prove a splitting result...

The Moment Map Revisited

In this paper, we show that the notion of moment map for the Hamiltonian action of a Lie group on a symplectic manifold is a special case of a much more general notion. In particular, we show that one can associate a moment map to a family of Hamiltonian symplectomorphisms, and we prove that its image is characterized, as in the classical case, by a generalized “energy-period” relation.

The Moment Map

Branches in random recursive k-ary trees

In this paper, using generalized {polya} urn models we find the expected value of the size of a branch in recursive $k$-ary trees. We also find the expectation of the number of nodes of a given outdegree in a branch of such trees.

On k-ary spanning trees of tournaments

It is well known that every tournament contains a Hamiltonian path, which can be restated as that every tournament contains a unary spanning tree. The purpose of this article is to study the general problem of whether a tournament contains a k-ary spanning tree. In particular, we prove that, for any fixed positive integer k, there exists a minimum number h(k) such that every tournament of order...

Percolation on a k-Ary Tree

Starting from the root, extend k branches and append k children with probability p, or terminate with probability q = 1− p. Then, we have a finite k-ary tree with probability one if 0 ≤ p ≤ 1/k. Moreover, we give the expectation and variance of the length of ideal codewords for representing the finite trees. Furthermore, we establish the probability of obtaining infinite tree, that is, of penet...