SDSU presents MS Mathematics

THESIS DEFENSEThursday, August 18, 2016 11:00am GMCS 418

*The Inclusion-Exclusion and Mixing Ratio Conjectures From the Ingleton Inequality*

Abstract:

The Ingleton Inequality, which is based on a vector space theorem and adapted to entropy functionals by Matus, plays a role in describing the entropy region for four discrete random variables, depending on whether or not it is satisfied. The entropy regions that are defined by entropic vectors for discrete random variables are not known, in general. For , recent work has been accomplished by Mao and Boston in characterizing this space of -dimensional entropic vectors. They study how *group-characterizable* entropic vectors sometimes yield violations of the Ingleton Inequality/Ratio, thus expanding the understanding of this region. We discovered a decomposition of the Ingleton Ratio into what we dub the Mixing Ratio and the Inclusion-Exclusion Ratio, which resembles the inclusion-exclusion principle from set theory. We propose two conjectures based on the simpler Inclusion-Exclusion and Mixing Ratios, attempting to streamline the search for Ingleton Inequality-violators. Examples of these violators are reproduced in order to provide support for our conjectures. Characterizing the entropy region associated with discrete random variables is related to determining the maximum communication rate of a network coding system that uses discrete random variables. Characterizing this region for arbitrary remains an open problem.

Thesis Committee:

Michael O’Sullivan, Thesis Chair, Department of Mathematics & Statistics

Carmelo Interlando, Department of Mathematics & Statistics

Jean mark Gawron, Department of Linguistics