# Doctoral Dissertation Defense: David Trott

Location

Mathematics/Psychology : 401

Date & Time

April 25, 2014, 4:30 pm – 6:30 pm

Description

Title: Top heavy and special Bishop-Phelps cones, Lyapunov rank, and related topics.

Location: MP 401

Presentation: 4:30-5:30 (Open to everyone)

Questioning/Deliberations: 5:30-6:30 (Restricted to committee members)

Abstract:

Motivated by optimization considerations, this dissertation investigates

cones of the form C = cl(cone({1} x S)), where S is a non-empty arbitrary

subset of R^{n-1}, as well as the corresponding completely positive cone

K_C generated by C.

First, we investigate the interrelations between S and C. Specifically,

we show that many of the properties of the cone C are inherited from the

underlying set S. With this realization, we determine which traits of S

result in desirable properties for the cone C, such as pointedness,

connected interior, etc.

Further restricting our attention, we consider a special type of the cone

C that reduces to the form {(t,x): t >= ||x|| }, where ||*|| is a norm on

R^{n-1}. We show that cones of this type, which are called topheavy cones

or special Bishop-Phelps (BP) cones, are always irreducible when n is

greater than or equal to three.

Defining the Lyapunov rank of a proper cone as the dimension of the Lie

algebra of the automorphism group of the cone, in this dissertation, we

show that the Lyapunov rank of any special Bishop-Phelps polyhedral cone

in R^n, when n is greater than or equal to three, is one. We build on an

earlier known result for the l_{1,+}^n cone (which is a special

Bishop-Phelps cone with 1-norm) and show that any l_{p,+}^n cone, where p

is between one and infinity but not equal to two, has Lyapunov rank one.

In the last part of the dissertation, we study automorphisms of special

Bishop-Phelps cones. In particular, we give a complete description of the

automorphism group of the l_{1,+}^n cone as well as properties for the

automorphisms of l_{p,+}^n cone. We also introduce the concept of

conjugate pair preserving automorphisms on special Bishop-Phelps cones.

Location: MP 401

Presentation: 4:30-5:30 (Open to everyone)

Questioning/Deliberations: 5:30-6:30 (Restricted to committee members)

Abstract:

Motivated by optimization considerations, this dissertation investigates

cones of the form C = cl(cone({1} x S)), where S is a non-empty arbitrary

subset of R^{n-1}, as well as the corresponding completely positive cone

K_C generated by C.

First, we investigate the interrelations between S and C. Specifically,

we show that many of the properties of the cone C are inherited from the

underlying set S. With this realization, we determine which traits of S

result in desirable properties for the cone C, such as pointedness,

connected interior, etc.

Further restricting our attention, we consider a special type of the cone

C that reduces to the form {(t,x): t >= ||x|| }, where ||*|| is a norm on

R^{n-1}. We show that cones of this type, which are called topheavy cones

or special Bishop-Phelps (BP) cones, are always irreducible when n is

greater than or equal to three.

Defining the Lyapunov rank of a proper cone as the dimension of the Lie

algebra of the automorphism group of the cone, in this dissertation, we

show that the Lyapunov rank of any special Bishop-Phelps polyhedral cone

in R^n, when n is greater than or equal to three, is one. We build on an

earlier known result for the l_{1,+}^n cone (which is a special

Bishop-Phelps cone with 1-norm) and show that any l_{p,+}^n cone, where p

is between one and infinity but not equal to two, has Lyapunov rank one.

In the last part of the dissertation, we study automorphisms of special

Bishop-Phelps cones. In particular, we give a complete description of the

automorphism group of the l_{1,+}^n cone as well as properties for the

automorphisms of l_{p,+}^n cone. We also introduce the concept of

conjugate pair preserving automorphisms on special Bishop-Phelps cones.

**Tags:**