In this paper we show how to give an intrinsic definition of a constructible circle on the sphere. The classical definition of constructible circle in the plane, using straight edge and compass is there by translated in ters of so called Lenart tools. The process by which we achieve our goal involves concepts from the algebra of Hermitian matrices, complex variables, and Sterographic projection. However, the discussion is entirely elementary throughout and hopefully can serve as a guide for teachers in advanced geometry.

In this paper we will study the 1p spaces. We will begin with definitions and different examples of 1p spaces. In particular, we will prove Holder's and Minkowski's inequalities for 1p sequence.

Diese Arbeit beschäftigt sich mit der Speziestheorie aus Mochizukis Beweis(versuch) der abc-Vermutung. Es wird ein Standpunkt eingeführt, der Parallelen zwischen der Kategorientheorie und der Speziestheorie aufzeigt und es werden so die Besonderheiten der Speziestheorie herausgearbeitet. In der Speziestheorie möchte man Konstruktionen ausführen, welche von keinen Auswahlen abhängen. Dieses Problem wird in einem allgemeinen Kontext für universelle Morphismen gelöst. An Beispielen wird die in der Arbeit behandelte Theorie erklärt.

In the first chapter, we define Steinhaus set as a set that meets every isometric copy of another set at exactly one point. We show that there is no Steinhaus set for any four-point subset in a plane.In the second chapter, we define the orbit tree of a permutation group of natural numbers, and further introduce compressed orbit trees. We show that any rooted finite tree can be realized as a compressed orbit tree of some permutation group. In the third chapter, we investigate certain classes of closed permutation groups of natural numbers with respect to their universal and surjectively universal groups. We characterize two-sided invariant groups, and prove that there is no universal group for countable groups, nor universal group for two-sided invariant groups in permutation groups of natural numbers.

I study sets of attractors and non-attractors of finite iterated function systems. I provide examples of compact sets which are attractors of iterated function systems as well as compact sets which are not attractors of any iterated function system. I show that the set of all attractors is a dense Fs set and the space of all non-attractors is a dense Gd set it the space of all non-empty compact subsets of a space X. I also investigate the small trans-finite inductive dimension of the space of all attractors of iterated function systems generated by similarity maps on [0,1].

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Civil Engineering, 1980. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND ENGINEERING. / Bibliography: leaves 100-102. / by Steven B. Pitschke. / M.S.

We introduce the Partition Principle PP, an axiom introduced by Russell in the context of its similarities and differences with the Axiom of Choice AC. We start by proving some properties of PP, and AC, and show that AC, entails PP. To address the problem of whether the converse holds, we develop the Zermelo-Fraenkel ZF set theory and examine its consistency and build a model in which AC, fails. We follow this with a discussion of forcing, a technique introduced by Paul Cohen to build new models of set theory from existing ones, which have differing properties from the starting model. We conclude by examining candidate models called permutation models where AC, fails, which may be useful as candidate models for forcing a model in which PP, holds but AC, does not. We conjecture that such a model exists, and that PP, does not entail AC. / Thesis / Master of Science (MSc)

<div>We further the research in o-minimal topology by studying in full generality definable topological spaces in o-minimal structures. These are topological spaces $(X, \tau)$, where $X$ is a definable set in an o-minimal structure and the topology $\tau$ has a basis that is (uniformly) definable. Examples include the canonical o-minimal "euclidean" topology, “definable spaces” in the sense of van den Dries [17], definable metric spaces [49], as well as generalizations of classical non-metrizable topological spaces such as the Split Interval and the Alexandrov Double Circle.</div><div><br></div><div>We develop a usable topological framework in our setting by introducing definable analogues of classical topological properties such as separability, compactness and metrizability. We characterize these notions, showing in particular that, whenever the underlying o-minimal structure expands $(\mathbb{R},<)$, definable separability and compactness are equivalent to their classical counterparts, and a similar weaker result for definable metrizability. We prove the equivalence of definable compactness and various other properties in terms of definable curves and types. We show that definable topological spaces in o-minimal expansions of ordered groups and fields have properties akin to first countability. Along the way we study o-minimal definable directed sets and types. We prove a density result for o-minimal types, and provide an elementary proof within o-minimality of a statement related to the known connection between dividing and definable types in o-minimal theories.</div><div><br></div><div>We prove classification and universality results for one-dimensional definable topological spaces, showing that these can be largely described in terms of a few canonical examples. We derive in particular that the three element basis conjecture of Gruenhage [25] holds for all infinite Hausdorff definable topological spaces in o-minimal structures expanding $(\mathbb{R},<)$, i.e. any such space has a definable copy of an interval with the euclidean, discrete or lower limit topology.</div><div><br></div><div>A definable topological space is affine if it is definably homeomorphic to a euclidean space. We prove affineness results in o-minimal expansions of ordered fields. This includes a result for Hausdorff one-dimensional definable topological spaces. We give two new proofs of an affineness theorem of Walsberg [49] for definable metric spaces. We also prove an affineness result for definable topological spaces of any dimension that are Tychonoff in a definable</div><div>sense, and derive that a large class of locally affine definable topological spaces are affine.</div>

Prior work in the field of set theory has looked at the properties of union-free families. This thesis investigates families based on a different set operation, the symmetricc difference. It provides a complete characterization of maximal symmetric differencefree families of subsets of {1, . . . n}

<p>This work strengthens and extends an algorithm for computing Betti numbers of symmetric semialgebraic sets developed by Basu and Riener in, <em>Vandermonde Varieties, Mirrored Spaces, and the Cohomology of Symmetric Semi-Algebraic Sets</em>. We first adapt a construction of Gabrielov and Vorobjov in, <em>Approximation of Definable Sets by Compact Families, and Upper Bounds on Homotopy and Homology,</em> for replacing arbitrary definable sets by compact ones to the symmetric case. The original construction provided maps from the homotopy and homology groups of the replacement set to those of the original; we show that for sets symmetric relative to the action of some finite reflection group <em>G</em>, we may construct these maps to be equivariant. This modification to the construction for compact replacement allows us to extend Basu and Riener's theorem on which submodules appear in the isotypic decomposition of each cohomology space to sets not necessarily closed and bounded. Furthermore, by utilizing this equivariant compact approximation, we may obtain a precise description of the aforementioned decomposition of each cohomology space, and not merely the final dimension of the space, from Basu and Riener's algorithm.</p> <p><br></p> <p>    Though our equivariant compact replacement holds for <em>G</em> any finite reflection group, Basu and Riener's results only consider the case of the action the of symmetric group, sometimes termed type <em>A</em>. As a first step towards generalizing Basu and Riener's work, we examine the next major class of symmetry: the action of the group of signed permutations (known as type <em>B</em>). We focus our attention on Vandermonde varieties, a key object in Basu and Riener's proofs. We show that the intersection of a type <em>B</em> Vandermonde variety with a fundamental region of type <em>B</em> symmetry is topologically regular. We also prove a result about the intersection of a type <em>B</em> Vandermonde variety with the walls of this fundamental region, leading to the elimination of factors in a different decomposition of the homology spaces.</p>