AbstractStochastic matrices play an important role in the study of probability theory
and statistics, and are often used in a variety of modeling problems in
economics, biology and operation research. Recently, the study
of tensors and
their applications
became a hot topic in numerical analysis and
optimization.
In this paper, we focus on studying stochastic tensors and, in
particular, we study the extreme points of a set of multi-stochastic tensors. Two
necessary and sufficient conditions for a multi-stochastic tensor to be an
extreme point are established.
These conditions characterize the “generators” of multi-stochastic tensors.
An algorithm to search the convex combination of extreme points for an arbitrary given
multi-stochastic tensor is developed. Based on our obtained results, some expression
properties for third-order and n-dimensional multi-stochastic tensors
(${n=3}$ and 4) are derived, and all extreme points of
3-dimensional and 4-dimensional triply-stochastic tensors can be produced in a simple
way. As an application, a new approach for the
partially filled square problem under the framework of multi-stochastic
tensors is given.
On the 3-dimensional Hopf bifurcation via averaging theory of third order