# Majorization Polytopes

## Description

Given two vectors of real components, we say that one vector (say u) majorizes or dominates the other vector (say v) if the components of u are more spread-out than the components of v. This idea of comparing vectors is used to analyze probability distributions and to formulate optimization problems in real life; and it leads to Majorization theory, a branch of mathematics that has applications in various fields, including inequalities, geometry, combinatorics, optimization, and statistics. A majorization polytope is a convex polytope associated with majorization. The classical Birkhoff polytope is a majorization polytope that consists of doubly stochastic matrices. The Birkhoff theorem states that this polytope is generated by permutation matrices. The concept of Birkhoff polytope has been extended to multi-dimensional arrays (aka hypermatrices or tensors). This talk summarizes the studies of the polytope of line- and plane- stochastic tensors. This is a joint work with Xiao-Dong Zhang.

## Date of Event

Thursday, February 15, 2024

## Location

Parker Building Room 338

Majorization Polytopes

Parker Building Room 338

Given two vectors of real components, we say that one vector (say u) majorizes or dominates the other vector (say v) if the components of u are more spread-out than the components of v. This idea of comparing vectors is used to analyze probability distributions and to formulate optimization problems in real life; and it leads to Majorization theory, a branch of mathematics that has applications in various fields, including inequalities, geometry, combinatorics, optimization, and statistics. A majorization polytope is a convex polytope associated with majorization. The classical Birkhoff polytope is a majorization polytope that consists of doubly stochastic matrices. The Birkhoff theorem states that this polytope is generated by permutation matrices. The concept of Birkhoff polytope has been extended to multi-dimensional arrays (aka hypermatrices or tensors). This talk summarizes the studies of the polytope of line- and plane- stochastic tensors. This is a joint work with Xiao-Dong Zhang.

## Presenter Bio

Dr. Fuzhen Zhang serves as a professor of mathematics at NSU Florida. He earned his Ph.D. in mathematics from the University of California - Santa Barbara (UCSB) in 1993. Dr. Zhang joined NSU in 1993 and has served as mentor and professor to hundreds of NSU students. His research interests include matrix analysis, linear algebra, multilinear algebra, functional analysis, operator theory, and combinatorics.