This thesis records an attempt to prove the two conjecture: Conjecture A: Every finite non-regular primitive permutation group of degree n contains permutations fixing one point but fixing at most $n^{1/2}$ points. Conjecture C: Every finite irreducible linear group of degree m > 1 contains an element whose fixed-point space has dimension at most m/2. Variants of these conjectures are formulated, and C is reduced to a special case of A. The main results of the investigation are: Theorem 2: Every finite non-regular primitive permutation group of degree n contains permutations which fix one point but fix fewer than (n+3)/4 points. Theorem 3: Every finite non-regular primitive soluble permutation group of degree n contains permutations which fix one point but fix fewer than $n^{7/18}$ points. Theorem 4: If H is a finite group, F is a field whose characteristic is 0 or does not divide the order of H, and M is a non-trivial irreducible H-module of dimension m over F, then there is an element h in H whose fixed-point space in M has dimension less than m/2. Theorem 5: If H is a finite soluble group, F is any field, and M is a non-trivial irreducible H-module of dimension m over F, then there is an element h in H whose fixed-point space in M has dimension less than 7m/18. Proofs of these assertions are to be found in Chapter II; examples which show the limitations on possible strenghtenings of the conjectures and results are marshalled in Chapter III. A detailed formulation of the problems and results is contained in section 1.

Moral encroachment theses (MET) operate like pragmatic encroachment theses. When the stakes of belief are high, so are the standards for evidence. This means that evidence which is sufficient in a low stakes-of-belief scenario may be insufficient when the stakes are raised. Simply, METs aim to appeal to the varying moral intuitions that one may have in cases with different moral stakes and build an epistemological difference out of that moral distinction. For example, one might think that in cases of racial profiling, because the moral stakes of belief are high, what would otherwise constitute good evidence for belief is insufficient. However, most METs assume that the probabilistic evidence on which one relies to form their belief is good evidence. Instead of examining the reliability of statistical generalizations, like those used in cases of racial profiling, the moral encroacher focuses on the moral facts of the circumstance of belief formation to explain why the subsequent belief is wrong epistemically. I will focus on Sarah Moss's account because she focuses on cases in which one forms an opinion on the basis of probabilistic evidence. I use Moss's version of the MET as a target to illustrate the challenges METs face in general. Broadly, Moss holds that a judgment's moral risk bears on its epistemic status. In Section 1, I briefly outline Sarah Moss's MET and explain why it fails to identify which cases produce epistemically problematic judgments and fails to explain why those judgments are epistemically problematic. In Section 2, I offer an alternative account, which explains why statistical generalizations about marginalized social groups are likely unreliable as evidence. Thus, use of this kind of evidence leads to epistemically problematic beliefs. I conclude by introducing epistemic risk as an explanation for why the inference made in Shopper is epistemically problematic while the inference made in Fraternity Member is not. / Master of Arts / Imagine a shopkeeper who has just realized something was stolen from his shop. There are two possible suspects: a young white man and a young Black man. He did not see the shoplifting occur, and the only evidence he has is the statistical evidence that young Black men are 70% more likely to shoplift than young white men. By all accounts, he is not racially biased, this is simply a statistical fact that he is aware of. Based on this evidence, he forms the judgment that the young Black man is the likely culprit. Let's call this case Shopper. Now imagine a student on a college campus whose friend has been assaulted. There are two possible suspects: a young man who is not a fraternity member and a young man who is in a fraternity. The only evidence that the student has is the statistical evidence that men involved in fraternities are 70% more likely to have committed sexual violence than average. By all accounts she is not anti-fraternity, she is simply aware of this statistical evidence. Based on this evidence, she forms the judgment that the fraternity member is the likely assailant. Let's call this case Fraternity Member. I think there's a difference between these two cases. Specifically, I think it's okay to make the inference in the latter case, but not in the former. Even if you don't quite share my intuition, you might still think that however 'icky' it feels to draw the above sort of inference in Fraternity Member, it feels ickier still to draw it in Shopper. Either way, I don't think these intuitions are merely responsive to the moral facts of the cases: I think there's something different about the evidence relied upon in these cases. Specifically, we have reason to thinks that the processes with which we produce the evidence relied upon in Shopper are biased.

The focus of this thesis is finitely generated subgroups of the automorphism group of an infinite spherically homogeneous rooted tree (regular or irregular). The first chapter introduces the topic and outlines the main results. The second chapter provides definitions of the terminology used, and also some preliminary results. The third chapter introduces a group that appears to be a promising candidate for a finitely generated group of infinite upper rank with finite upper $p$-rank for all primes $p$. It goes on to demonstrate that in fact this group has infinite upper $p$-rank for all primes $p$. As a by-product of this construction, we obtain a finitely generated branch group with quotients that are virtually-(free abelian of rank $n$) for arbitrarily large $n$. The fourth chapter gives a complete classification of ternary automata with $C_2$-action at the root, and a partial classification of ternary automata with $C_3$-action at the root. The concept of a `windmill automaton' is introduced in this chapter, and a complete classification of binary windmill automata is given. The fifth chapter contains a detailed study of the non-abelian ternary automata with $C_3$-action at the root. It also contains some conjectures about possible isomorphisms between these groups.

For various values of n and m we investigate homomorphisms from Out(F_n) to Out(F_m) and from Out(F_n) to GL_m(K), i.e. the free and linear representations of Out(F_n) respectively. By means of a series of arguments revolving around the representation theory of finite symmetric subgroups of Out(F_n) we prove that each homomorphism from Out(F_n) to GL_m(K) factors through the natural map p_n from Out(F_n) to GL(H_1(F_n,Z)) = GL_n(Z) whenever n=3, m < 7 and char(K) is not an element of {2,3}, and whenever n>5, m< n(n+1)/2 and char(K) is not an element of {2,3,...,n+1}. We also construct a new infinite family of linear representations of Out(F_n) (where n > 2), which do not factor through p_n. When n is odd these have the smallest dimension among all known representations of Out(F_n) with this property. Using the above results we establish that the image of every homomorphism from Out(F_n) to Out(F_m) is finite whenever n=3 and n < m < 6, and of cardinality at most 2 whenever n > 5 and n < m < n(n-1)/2. We further show that the image is finite when n(n-1)/2 -1 < m < n(n+1)/2. We also consider the structure of normal finite index subgroups of Out(F_n). If N is such then we prove that if the derived subgroup of the intersection of N with the Torelli subgroup T_n < Out(F_n) contains some term of the lower central series of T_n then the abelianisation of N is finite.

Dado um espaço localmente compacto Hausdorff K e um espaço de Banach X, Co(K,X) representa o espaço de Banach das funções contínuas em K com valores em X que se anulam no infinito com a norma do supremo. No presente trabalho, unificaremos e melhoraremos várias generalizações do teorema clássico de Banach-Stone para os espaços Co(K,X) devidas a Cambern, Amir, Behrends e Jarosz. No caso em que X=lp com $ 2 p, nossos resultados são maximais. / Let K be a locally compact Hausdor space and X a Banach space. By Co(K,X) we denote the Banach space of all X-valued continuous functions dened on K which vanish at innity, provided with the supremum norm. In the present work, we unify and strengthen several generalizations obtained in recent years of the classical Banach-Stone theorem for Co(K,X) spaces. In the case where X = lp such that 2 p < 1, our results are optimal.

In this thesis, we provide constructive proofs of serveral generalizations of Sperner's Lemma, a combinatorial result which is equivalent to the Brouwer Fixed Point Theorem. This lemma makes a statement about the number of a certain type of simplices in the triangulation of a simplex with a special labeling. We prove generalizations for polytopes with simplicial facets, for arbitrary 3-polytopes, and for polygons. We introduce a labeled graph which we call a nerve graph to prove these results. We also suggest a possible non-constructive proof for a polytopal generalization.

While the traditional form of continued fractions is well-documented, a new form, designed to approximate real numbers between 1 and 2, is less well-studied. This report first describes prior research into the new form, describing the form and giving an algorithm for generating approximations for a given real number. It then describes a rational function giving the rational number represented by the continued fraction made from a given tuple of integers and shows that no real number has a unique continued fraction. Next, it describes the set of real numbers that are hardest to approximate; that is, given a positive integer $n$, it describes the real number $\alpha$ that maximizes the value $|\alpha - T_n|$, where $T_n$ is the closest continued fraction to $\alpha$ generated from a tuple of length $n$. Finally, it lays out plans for future work.

Dado um espaço localmente compacto Hausdorff K e um espaço de Banach X, Co(K,X) representa o espaço de Banach das funções contínuas em K com valores em X que se anulam no infinito com a norma do supremo. No presente trabalho, unificaremos e melhoraremos várias generalizações do teorema clássico de Banach-Stone para os espaços Co(K,X) devidas a Cambern, Amir, Behrends e Jarosz. No caso em que X=lp com $ 2 p, nossos resultados são maximais. / Let K be a locally compact Hausdor space and X a Banach space. By Co(K,X) we denote the Banach space of all X-valued continuous functions dened on K which vanish at innity, provided with the supremum norm. In the present work, we unify and strengthen several generalizations obtained in recent years of the classical Banach-Stone theorem for Co(K,X) spaces. In the case where X = lp such that 2 p < 1, our results are optimal.

This thesis covers the Cantor Ternary Set and generalizations of the Cantor Set, and gives a complete existential theory for three set properties: denumerability, exhaustibility, and zero measure.

This thesis studies some classes of non-periodic branch groups. In particular their growth, relations between elements and their Hausdorff dimensions.