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You searched for +publisher:"University of Colorado" +contributor:("Robert Tubbs"). Showing records 1 – 2 of 2 total matches.

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University of Colorado

1. Wakefield, Nathan Paul. Primitive Divisors in Generalized Iterations of Chebyshev Polynomials.

Degree: PhD, Mathematics, 2013, University of Colorado

Let (<em>gi</em>)<em>i</em> ≥1 be a sequence of Chebyshev polynomials, each with degree at least two, and define (<em>fi</em>) <em>i</em> ≥1 by the following recursion: f1 = g1, <em>fn</em> = <em>gn</em> ∘ <em> fn</em>–1, for n ≥ 2. Choose α ∈ [special characters omitted] such that {[special characters omitted](α) : n ≥ 1} is an infinite set. The main result is as follows: Let γ ∈ {0, ±1}, if <em>f n</em>(α) = [special characters omitted] is written in lowest terms, then for all but finitely many n > 0, the numerator, <em>An</em>, has a primitive divisor; that is, there is a prime p which divides <em> An</em> but does not divide <em>Ai</em> for any i < n. In addition to the main result, several of the tools developed to prove the main result may be of interest. A key component of the main result was the development of a generalization of canonical height. Namely: If [f] is a set of rational maps, all commuting with a common function f, and f = [special characters omitted] is a generalized iteration of rational maps formed by <em>f n</em>(x) = <em>gn</em>(<em> fn</em>–1(x)) with <em> gi</em> coming from [f], then there is a unique canonical height funtion ĥf : K → [special characters omitted] which is identical to the canonical height function associated to f. Another key component of the main result was proving that under certain circumstances, being acted upon by a Chebyshev polynomial does not lead to significant differences between the size of the numerator and denominator of the result. Specifically, let γ ∈ {0, ±1, ±2} be fixed, and <em>gi</em> be a sequence of Chebyshev polynomials. Let f given by the following recurrence f 1(z) = g1(z), and <em>fi</em> = <em>gi</em>(<em> fi</em>–1(z)) for i ≥ 2. Pick any α ∈ [special characters omitted] with |α + γ| < 2, such that α + γ is not pre-periodic for one hence any Chebyshev polynomial. Write <em>f n</em>(α + γ) − γ = [special characters omitted] in lowest terms. Then limn→∞ logAn logBn =1. Finally, some areas of future research are discussed. Advisors/Committee Members: Su-Ion Ih, Katherine Stange, Robert Tubbs, Eric Stade, Juan Restrepo.

Subjects/Keywords: Arithmetic Dynamics; Chebyshev; Generalized Iteration; Primitive Divisors; Mathematics

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APA (6th Edition):

Wakefield, N. P. (2013). Primitive Divisors in Generalized Iterations of Chebyshev Polynomials. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/math_gradetds/25

Chicago Manual of Style (16th Edition):

Wakefield, Nathan Paul. “Primitive Divisors in Generalized Iterations of Chebyshev Polynomials.” 2013. Doctoral Dissertation, University of Colorado. Accessed January 18, 2021. https://scholar.colorado.edu/math_gradetds/25.

MLA Handbook (7th Edition):

Wakefield, Nathan Paul. “Primitive Divisors in Generalized Iterations of Chebyshev Polynomials.” 2013. Web. 18 Jan 2021.

Vancouver:

Wakefield NP. Primitive Divisors in Generalized Iterations of Chebyshev Polynomials. [Internet] [Doctoral dissertation]. University of Colorado; 2013. [cited 2021 Jan 18]. Available from: https://scholar.colorado.edu/math_gradetds/25.

Council of Science Editors:

Wakefield NP. Primitive Divisors in Generalized Iterations of Chebyshev Polynomials. [Doctoral Dissertation]. University of Colorado; 2013. Available from: https://scholar.colorado.edu/math_gradetds/25


University of Colorado

2. Feaver, A.F. Amy. Euclid's Algorithm in Multiquadratic Fields.

Degree: PhD, Mathematics, 2014, University of Colorado

In this thesis we find that all imaginary n-quadratic fields with n>3 have class number larger than 1 and therefore cannot be Euclidean. We also examine imaginary triquadratic fields, presenting a complete list of 17 imaginary triquadratic fields with class number 1, and classifing many of them according to whether or not they are norm-Euclidean. We find that at least three of these fields are norm-Euclidean, and at least five are not. Advisors/Committee Members: Katherine Stange, David Grant, Robert Tubbs, Eric Stade, Franck Vernerey.

Subjects/Keywords: Class Number; Euclidean Rings; Number Fields; Mathematics

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Feaver, A. F. A. (2014). Euclid's Algorithm in Multiquadratic Fields. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/math_gradetds/30

Chicago Manual of Style (16th Edition):

Feaver, A F Amy. “Euclid's Algorithm in Multiquadratic Fields.” 2014. Doctoral Dissertation, University of Colorado. Accessed January 18, 2021. https://scholar.colorado.edu/math_gradetds/30.

MLA Handbook (7th Edition):

Feaver, A F Amy. “Euclid's Algorithm in Multiquadratic Fields.” 2014. Web. 18 Jan 2021.

Vancouver:

Feaver AFA. Euclid's Algorithm in Multiquadratic Fields. [Internet] [Doctoral dissertation]. University of Colorado; 2014. [cited 2021 Jan 18]. Available from: https://scholar.colorado.edu/math_gradetds/30.

Council of Science Editors:

Feaver AFA. Euclid's Algorithm in Multiquadratic Fields. [Doctoral Dissertation]. University of Colorado; 2014. Available from: https://scholar.colorado.edu/math_gradetds/30

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