Date of Award

12-2017

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematical Sciences

First Advisor

Munevver Mine Subasi

Second Advisor

Susan Earles

Third Advisor

Jewgeni Dshalalow

Fourth Advisor

Nezammoddin Nezammoddini-Kachouie

Abstract

The contribution of this dissertation to the literature is twofold. First, we use a geometric perspective to present all possible subdivisions of R³ into tetrahedra with disjoint interiors and adopt a combinatorial approach to obtain a special subdivision of Rⁿ into simplices with disjoint interiors, where two simplices are called neighbors if they share a common facet. We then use the neighborhood relationship of the simplices in each subdivision to fully describe the sufficient conditions for the strong unimodality/logconcavity of the trivariate discrete distributions and further extend these results to present a new sufcient condition for the strong unimodality/logconcavity of multivariate discrete distributions defined on Zⁿ. We show that the multivariate P´olya-Eggenberger distribution, multivariate Poisson distribution, and multivariate Ewens distribution are strongly unimodal, and hence logconcave.

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