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.
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