Pure Mathematics
http://hdl.handle.net/10012/9932
Fri, 26 Apr 2019 00:06:24 GMT
20190426T00:06:24Z

The Logarithmic Derivative and ModelTheoretic Analysability in Differentially Closed Fields
http://hdl.handle.net/10012/14379
The Logarithmic Derivative and ModelTheoretic Analysability in Differentially Closed Fields
Jin, Ruizhang
This thesis deals with internal and analysable types, mainly in the context of the stable theory of differentially closed fields. Two main problems are dealt with: the construction of types analysable in the constants with specific properties, and a criterion for a given analysable type to be actually internal to the constants.
For analysable types, the notion of canonical analyses is introduced. A type has a canonical analysis if all its analyses of shortest length are interalgebraic. Given a finite sequence of ranks, it is constructed, in the theory of differentially closed field, a type analysable in the constants such that it admits a canonical analysis and each step of the analysis is of the given rank. The construction of such a type starts from the wellknown example of δ(logδx)=0, whose generic type is analysable in the constants in 2 steps but is not internal to the constants. Along the way, techniques for comparing analyses in stable theories are developed, including in particular the notions of analyses by reductions and by coreductions.
The property of the logδ function is further studied when the following question is raised: given a type internal to the constants, is its preimage under logδ, which is 2step analysable in the constants, ever internal to the constants? The question is answered positively, and a criterion for when the preimage is indeed internal is proposed. Partial results are proven for this conjectured criterion, namely the cases where the group of automorphisms (the binding group) of the given internal type is additive, multiplicative, or trivial. In particular, the conjecture is resolved for generic types of equations of the form δx=f(x) where f is a rational function over the constants. It is discovered that the related problem where logδ is replaced by δ is significantly different, and the analogue of the conjecture fails in this case.
Also included in this thesis are two examples asked for in the literature: internality of a particular twisted Dgroup, and a 2step analysable set with independent fibres.
Tue, 22 Jan 2019 00:00:00 GMT
http://hdl.handle.net/10012/14379
20190122T00:00:00Z

Compressions of Compact Tuples
http://hdl.handle.net/10012/14299
Compressions of Compact Tuples
Passer, Benjamin; Shalit, Orr
We study the matrix range of a tuple of compact operators on a Hilbert space and examine the notions of minimal, nonsingular, and fully compressed tuples. In this pursuit, we refine previous results by characterizing nonsingular compact tuples in terms of matrix extreme points of the matrix range. Further, we find that a compact tuple A is fully compressed if and only if it is multiplicityfree and the Shilov ideal is trivial, which occurs if and only if A is minimal and nonsingular. Fully compressed compact tuples are therefore uniquely determined up to unitary equivalence by their matrix ranges. We also produce a proof of this fact which does not depend on the concept of nonsingularity.
The final publication is available at Elsevier via https://doi.org/10.1016/j.laa.2018.12.002. © 2018. This manuscript version is made available under the CCBYNCND 4.0 license http://creativecommons.org/licenses/byncnd/4.0/
Fri, 01 Mar 2019 00:00:00 GMT
http://hdl.handle.net/10012/14299
20190301T00:00:00Z

A lower bound for the twovariable Artin conjecture and prime divisors of recurrence sequences
http://hdl.handle.net/10012/14034
A lower bound for the twovariable Artin conjecture and prime divisors of recurrence sequences
Murty, M. Ram; Séguin, François; Stewart, Cameron L.
In 1927, Artin conjectured that any integer other than −1 or a perfect square generates the multiplicative group (Z/pZ)× for infinitely many p. In 2000, Moree and Stevenhagen considered a twovariable version of this problem, and proved a positive density result conditionally to the generalized Riemann Hypothesis by adapting a proof by Hooley for the original conjecture. In this article, we prove an unconditional lower bound for this twovariable problem. In particular, we prove an estimate for the number of distinct primes which divide one of the first N terms of a nondegenerate binary recurrence sequence. We also prove a weaker version of the same theorem, and give three proofs that we consider to be of independent interest. The first proof uses a transcendence result of Stewart, the second uses a theorem of Bombieri and Schmidt on Thue equations and the third uses Mumford's gap principle for counting points on curves by their height. We finally prove a disjunction theorem, where we consider the set of primes satisfying either our twovariable condition or the original condition of Artin's conjecture. We give an unconditional lower bound for the number of such primes.
The final publication is available at Elsevier via https://dx.doi.org/10.1016/j.jnt.2018.06.017 © 2019. This manuscript version is made available under the CCBYNCND 4.0 license https://creativecommons.org/licenses/byncnd/4.0/
Tue, 01 Jan 2019 00:00:00 GMT
http://hdl.handle.net/10012/14034
20190101T00:00:00Z

On fixed points of self maps of the free ball
http://hdl.handle.net/10012/14017
On fixed points of self maps of the free ball
Shamovich, Eli
In this paper, we study the structure of the fixed point sets of noncommutative self maps of the free ball. We show that for such a map that fixes the origin the fixed point set on every level is the intersection of the ball with a linear subspace. We provide an application for the completely isometric isomorphism problem of multiplier algebras of noncommutative complete Pick spaces.
The final publication is available at Elsevier via https://dx.doi.org/10.1016/j.jfa.2018.03.004 © 2018. This manuscript version is made available under the CCBYNCND 4.0 license https://creativecommons.org/licenses/byncnd/4.0/
Sun, 01 Jul 2018 00:00:00 GMT
http://hdl.handle.net/10012/14017
20180701T00:00:00Z