Abstract. { In this paper, we develop some stochastic dominance theorems for the location and scale family and linear combinations of random variables and for risk lovers as well as risk averters that extend results in Hadar and Russell (1971) and Tesfatsion (1976). The results are discussed and applied to decision-making.

In this paper, we extend Fishburn's convex stochastic dominance theorem to include any distribution function. This paper also considers risk takers as well as risk averters, and discusses third order stochastic dominance. We apply separation and representation theorems to obtain a concise alternative proof of the theorem. Our results are used to extend a theorem of Bawa et.al. on comparison between a convex combinations of several contiuous distributions and a single continuous distribution.

We show that the linear group of automorphism of Hermitian matrices which preserves the set of separable states is generated by natural automorphisms: change of an orthonormal basis in each tensor factor, partial transpose in each tensor factor, and interchanging two tensor factors of the same dimension. We apply our results to preservers of the product numerical range.