ICTP Mathematics Seminar - Wednesday, 24 June at 14:30

[Margherita Di Giovannantonio]

M A T H E M A T I C S   S E M I N A R S   2015


Wednesday, 24 June, at 14:30 hrs.


Mansoor Saburov (IIUM, Malaysia)


Title: Quadratic Stochastic Operators on Simplex


Abstract:In this talk, we first review some basic results of square 
stochastic matrices concerning the convergence of power of square 
stochastic matrix. We also discuss a linear operator associated with a 
square stochastic matrix. This gives an advantage to interpret all 
results by means of the language of dynamical system. In the second part 
of this talk, we discuss on a cubic stochastic matrix. Unlike square 
matrices, there is no proper multiplication of two cubic matrices in 
which the dimension of matrices would remain invariant. That’s why we 
can’t speak about the inverse of cubic matrix (which is the important 
concept) and the power of cubic matrices. However, for each cubic 
stochastic matrix we can associate a quadratic operator acting on the 
finite dimensional simplex. By this way, we can consider the similar 
problem which was studied in the linear case. For example, due to the 
Perron-Frobenius theorem, the trajectory of linear operator associated 
with the positive square stochastic matrix converges to a unique fixed 
point. It is natural to ask the similar question for quadratic 
operators, i.e., does the trajectory of quadratic operator associated 
with the positive cubic stochastic matrix converge to a unique fixed 
point? However, this is wrong in general. The main reason is that a 
quadratic operator associated with positive cubic stochastic matrix may 
have many fixed points. The first attempt to provide an example for 
quadratic stochastic operators with positive coefficients having three 
fixed points was done by A.A. Krapivin (Lyubich’s former graduate 
student) and Y.I. Lyubich. However, it turns out that their examples 
were wrong. We showed that Krapivin-Lyubich examples have a unique fixed 
point. Moreover, we provide an example for a family of quadratic 
stochastic operators with positive coefficients which have three fixed 
points. Consequently, the question mentioned above does not have an 
affirmative answer in general. If it is a case, can we find an 
affirmative answer for some class of quadratic operators? The last but 
not least, we also provide a class of quadratic stochastic operators 
(which are not contraction) in which their trajectory converges a unique 
fixed point.


VENUE:  Luigi Stasi Seminar Room (ICTP Leonardo da Vinci Building, first 
floor)