Ground state of nonlinear Schrodinger systems with saturable nonlinearity

We prove the existence of ground state in a multidimensional nonlinear Schrodinger model of paraxial beam propagation in isotropic local media with saturable nonlinearity. Such ground states exist in the form of bright counterpropagating solitons. From the proof, a general threshold condition on the beam coupling constant for the existence of such fundamental solitons follows.

On the Existence of the MLE for a Directed Random Graph Network Model with Reciprocation

Holland and Leinhardt (1981) proposed a directed random graph model, the p1 model, to describe dyadic interactions in a social network. In previous work (Petrovic et al., 2010), we studied the algebraic properties of the p1 model and showed that it is a toric model specified by a multi-homogeneous ideal. We conducted an extensive study of the Markov bases for p1 that incorporate explicitly the constraint arising from multi-homogeneity. Here we consider the properties of the corresponding toric variety and relate them to the conditions for the existence of the maximum likelihood and extended maximum likelihood estimators or the model parameters. Our results are directly relevant to the estimation and conditional goodness-of-fit testing problems in p1 models.