21 
A Quotientlike Construction involving Elementary SubmodelsBurton, Peter 21 November 2012 (has links)
This article is an investigation of a recently developed method of deriving a topology from a space and
an elementary submodel containing it. We first define and give the basic properties of this construction,
known as X/M. In the next section, we construct some examples and analyse the topological relationship
between X and X/M. In the final section, we apply X/M to get novel results about Lindelof spaces,
giving partial answers to a question of F.D. Tall and another question of Tall and M. Scheepers.

22 
Nonchainable Continua and Lelek's ProblemHoehn, Logan Cedric 09 June 2011 (has links)
The set of compact connected metric spaces (continua) can be divided into classes according to the complexity of their descriptions as inverse limits of polyhedra. The simplest such class is the collection of chainable continua, i.e. those which are inverse limits of arcs.
In 1964, A. Lelek introduced a notion which is related to chainability, called span zero. A continuum X has span zero if any two continuous maps from any other continuum to X with identical ranges have a coincidence point. Lelek observed that every chainable continuum has span zero; he later asked whether span zero is in fact a characterization of chainability.
In this thesis, we construct a nonchainable continuum in the plane which has span zero, thus providing a counterexample for what is now known as Lelek's Problem in continuum theory. Moreover, we show that the plane contains an uncountable family of pairwise disjoint copies of this continuum. We discuss connections with the classical problem of determining up to homeomorphism all the homogeneous continua in the plane.

23 
Contributions towards a Fine Structure Theory of Aronszajn Orderings.MartinezRanero, Carlos Azarel 31 August 2011 (has links)
The purpose of this thesis is to add to the structure theory of Aronszajn orderings. We shall focus essentially in four topics. The first topic of discussion is about the relation between Lipschitz and coherent trees. I will demonstrate that the tree $T(\rho_0)$ is coherent without any extra set theoretic hypothesis. The second topic presents an application of Todorcevic's $\rho$ functions to provide some partial answers to an old question of Juhaz asking whether a standard weakening of Jensen's diamond principle implies the existence of Suslin trees. In the third topic we focus on providing a satisfactory rough classification result of the class of Aronszajn lines. Our main result is that, assuming PFA, the class of Aronszajn lines is wellquasiordered by embeddability. The last topic is an investigation of the gap structure of the class of coherent Aronszajn trees. I will show that, assuming PFA, the class of coherent Aronszajn trees quasiordered by embeddability is the unique saturated linear order of cardinality $\aleph_2$.

24 
The Left Regular Representation of a SemigroupRowe, Barry James 11 January 2012 (has links)
As with groups, one can study the left regular representation of a semigroup. If one considers such representations, then it is natural to ask similar questions to the group case.
We start by formulating several questions in the semigroup case and then work towards understanding the structure of the representations given. We present results describing what the elements of the image under the representation map can look like (the semigroup problem), whether or not two semigroups will give isomorphic representations (the isomorphism problem), and whether or not the representation of a semigroup is reflexive (the reflexivity problem).
This research has been funded in part by a scholarship from the Natural Sciences and Engineering Research Council of Canada.

25 
Lie 2algebras as Homotopy Algebras Over a Quadratic OperadSquires, Travis 11 January 2012 (has links)
We begin by discussing motivation for our consideration of a structure called a Lie 2algebra, in particular an important class of Lie 2algebras are the Courant Algebroids introduced in 1990 by Courant. We wish to attach some natural definitions from operad theory, mainly the notion of a module over an algebra, to Lie 2algebras and hence to Courant algebroids. To this end our goal is to show that Lie 2algebras can be described as what are called \emph{homotopy algebras over an operad}. Describing Lie 2algebras using operads also solves the problem of showing that the equations defining a Lie 2algebra are consistent.
Our technical discussion begins by introducing some notions from operad theory, which is a generalization of the theory of operations on a set and their compositions. We define the idea of a quadratic operad and a homotopy algebra over a quadratic operad. We then proceed to describe Lie 2algebras as homotopy algebras over a given quadratic operad using a theorem of Ginzburg and Kapranov.
Next we briefly discuss the structure of a braided monoidal category. Following this, motivated by our discussion of braided monoidal categories, a new structure is introduced, which we call a commutative 2algebra. As with the Lie 2algebra case we show how a commutative 2algebra can be seen as a homotopy algebra over a particular quadratic operad.
Finally some technical results used in previous theorems are mentioned. In discussing these technical results we apply some ideas about distributive laws and Koszul operads.

26 
Real and Complex Dynamics of Unicritical MapsClark, Trevor Collin 06 August 2010 (has links)
In this thesis, we prove two results. The first concerns the dynamics of typical maps in families of higher degree unimodal maps, and the second concerns the Hausdorff dimension of the Julia sets of certain quadratic maps.
In the first part, we construct a lamination of the space of unimodal maps whose
critical points have fixed degree d greater than or equal to 2 by the hybrid classes. As in [ALM], we show that the hybrid classes laminate neighbourhoods of all but countably many maps in the families under consideration. The structure of the lamination yields a partition of the
parameter space for oneparameter real analytic families of unimodal maps of degree d and allows us to transfer a priori bounds from the phase space to the parameter space.
This result implies that the statistical description of typical unimodal maps obtained
in [ALM], [AM3] and [AM4] also holds in families of higher degree unimodal maps, in
particular, almost every map in such a family is either regular or stochastic.
In the second part, we prove the Poincare exponent for the Fibonacci map is less than
two, which implies that the Hausdor ff dimension of its Julia set is less than two.

27 
Nonchainable Continua and Lelek's ProblemHoehn, Logan Cedric 09 June 2011 (has links)
The set of compact connected metric spaces (continua) can be divided into classes according to the complexity of their descriptions as inverse limits of polyhedra. The simplest such class is the collection of chainable continua, i.e. those which are inverse limits of arcs.
In 1964, A. Lelek introduced a notion which is related to chainability, called span zero. A continuum X has span zero if any two continuous maps from any other continuum to X with identical ranges have a coincidence point. Lelek observed that every chainable continuum has span zero; he later asked whether span zero is in fact a characterization of chainability.
In this thesis, we construct a nonchainable continuum in the plane which has span zero, thus providing a counterexample for what is now known as Lelek's Problem in continuum theory. Moreover, we show that the plane contains an uncountable family of pairwise disjoint copies of this continuum. We discuss connections with the classical problem of determining up to homeomorphism all the homogeneous continua in the plane.

28 
Contributions towards a Fine Structure Theory of Aronszajn Orderings.MartinezRanero, Carlos Azarel 31 August 2011 (has links)
The purpose of this thesis is to add to the structure theory of Aronszajn orderings. We shall focus essentially in four topics. The first topic of discussion is about the relation between Lipschitz and coherent trees. I will demonstrate that the tree $T(\rho_0)$ is coherent without any extra set theoretic hypothesis. The second topic presents an application of Todorcevic's $\rho$ functions to provide some partial answers to an old question of Juhaz asking whether a standard weakening of Jensen's diamond principle implies the existence of Suslin trees. In the third topic we focus on providing a satisfactory rough classification result of the class of Aronszajn lines. Our main result is that, assuming PFA, the class of Aronszajn lines is wellquasiordered by embeddability. The last topic is an investigation of the gap structure of the class of coherent Aronszajn trees. I will show that, assuming PFA, the class of coherent Aronszajn trees quasiordered by embeddability is the unique saturated linear order of cardinality $\aleph_2$.

29 
The Left Regular Representation of a SemigroupRowe, Barry James 11 January 2012 (has links)
As with groups, one can study the left regular representation of a semigroup. If one considers such representations, then it is natural to ask similar questions to the group case.
We start by formulating several questions in the semigroup case and then work towards understanding the structure of the representations given. We present results describing what the elements of the image under the representation map can look like (the semigroup problem), whether or not two semigroups will give isomorphic representations (the isomorphism problem), and whether or not the representation of a semigroup is reflexive (the reflexivity problem).
This research has been funded in part by a scholarship from the Natural Sciences and Engineering Research Council of Canada.

30 
Should the Pythagorean Theorem Actually be Called the 'Pythagorean' TheoremMoledina, Amreen 05 December 2013 (has links)
This paper investigates whether it is reasonable to bestow credit to one person or group for the famed theorem that relates to the side lengths of any rightangled triangle, a theorem routinely referred to as the “Pythagorean Theorem”. The author investigates the firstdocumented occurrences of the theorem, along with its first proofs. In addition, proofs that stem from different branches of mathematics and science are analyzed in an effort to display that credit for the development of the theorem should be shared amongst its many contributors rather than crediting the whole of the theorem to one man and his supporters.

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