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  • Arithmetic dynamical questions with local rationality conditions

    Creator:
    Noytaptim, Chatchai
    Abstract:
    We introduce a numerical criterion which allows us to bound the degree of any algebraic integer having all of Galois conjugates in an interval of length less than 4. Using this criterion, we study two arithmetic dynamical questions with local rationality conditions. First, we classify all unicritical polynomials defined over...
    Resource Type:
    Dissertation
    Full Text:
    : Arithmetic Dynamical Questions with Local Rationality Conditions. Abstract approved: Clayton Petsche Let d

  • On Small Heights of Totally p-adic Numbers

    Creator:
    Stacy, Emerald T.
    Abstract:
    The height of an algebraic number A is a measure of how arithmetically complicated A is. We say A is totally p-adic if the minimal polynomial of A splits completely over the field of p-adic numbers. In this paper, we investigate what can be said about the smallest nonzero height...
    Resource Type:
    Dissertation
    Full Text:
    Heights of Totally p-adic Numbers. Abstract approved: Clayton J. Petsche The height of an algebraic

  • An inner product on adelic measures

    Creator:
    Oberly, Peter J.
    Abstract:
    We define an inner product on a vector space of adelic measures over a number field $K$. We find that the norm induced by this inner product governs weak convergence at each place of $K$. The canonical adelic measure associated to a rational map is in this vector space, and...
    Resource Type:
    Dissertation
    Full Text:
    Product on Adelic Measures. Abstract Approved: Clayton J. Petsche We define an inner product on a vector

  • Fourier Analysis and Equidistribution on the p-adic Integers

    Creator:
    Somasunderam, Naveen
    Abstract:
    In this dissertation, we use Fourier-analytic methods to study questions of equidistribution on the compact abelian group Zp of p-adic integers. In particu- lar, we prove a LeVeque-type Fourier analytic upper bound on the discrepancy of sequences. We establish p-adic analogues of the classical Dirichlet and Fejér kernels on R/Z,...
    Resource Type:
    Dissertation
    Full Text:
    Analysis and Equidistribution on the p-adic Integers Abstract approved: Clayton J. Petsche In this

  • Critically Separable Rational Maps in Families

    Creator:
    Petsche, Clayton
    Abstract:
    Given a number field K, we consider families of critically separable rational maps of degree d over K possessing a certain fixed-point and multiplier structure. With suitable notions of isomorphism and good reduction between rational maps in these families, we prove a finiteness theorem which is analogous to Shafarevich’s theorem...
    Resource Type:
    Article

  • HARD PASS: An Arts-Based Autoethnography Regarding the Experience of a Graduate Student of Mathematics

    Creator:
    Ohsiek, Katy
    Abstract:
    This arts-based autoethnography explores the experience of a graduate student of mathematics at a mid-sized research university through a collection of collage, songwriting, and personal essays. This research identifies issues in the mathematics academic pipeline associated with gender, burnout culture, perfectionism, mental health, qualifying exams, and isolation. The present research...
    Resource Type:
    Capstone Project
    Full Text:
    . • To Clayton Petsche and Kathryn McIntosh for taking the time to support this inquiry. • To my math

  • The Axiom of Choice and Homotopy Type Theory

    Creator:
    Schneider, Eric
    Abstract:
    In this work we introduce some basic concepts within homotopy type theory (HoTT), a proposed alternative mathematical foundation to classical set theory. In particular, our discussion revolves around the Axiom of Choice (AC). In Part I, we introduce the classical AC and some of its most important equivalents. In Part...
    Resource Type:
    Honors College Thesis
    Full Text:
    . APPROVED: David Wing, Mentor, representing Mathematics Clayton Petsche, Committee Member, representing

  • Subalgebras of the Split Octonions

    Creator:
    Bentz, Lida
    Abstract:
    We classify the subalgebras of the split octonions, paying particular attention to the null subalgebras and their extensions.
    Resource Type:
    Capstone Project

  • Classifying Octonionic-Linear Operators

    Creator:
    Putnam, Alexander
    Abstract:
    The goal of this paper is to classify linear operators with octonionic coefficients and octonionic variables. While building up to the octonions we also classify linear operators over the quaternions and show how to relate the linear operators over the quaternions and octonions to matrices. We also construct a basis...
    Resource Type:
    Capstone Project

  • A dynamical pairing between two rational maps

    Creator:
    Petsche, Clayton, Szpiro, Lucien, and Tucker, Thomas J.
    Abstract:
    Given two rational maps φ and ψ on Ρ¹ of degree at least two, we study a symmetric, nonnegative real-valued pairing〈φ, ψ〉which is closely related to the canonical height functions hφ and hψ associated to these maps. Our main results show a strong connection between the value of〈φ, ψ〉and the...
    Resource Type:
    Article
    Full Text:
    on November 10, 2011 A DYNAMICAL PAIRING BETWEEN TWO RATIONAL MAPS CLAYTON PETSCHE, LUCIEN SZPIRO