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You searched for +publisher:"University of Notre Dame" +contributor:("Julia F. Knight, Committee Chair"). Showing records 1 – 2 of 2 total matches.

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University of Notre Dame

1. Steven M. VanDenDriessche. Embedding Computable Infinitary Equivalence nto P-Groups</h1>.

Degree: Mathematics, 2013, University of Notre Dame

We examine the relation between the uniformity of a collection of operators witnessingTuring computable embeddings, and the existence of an operator witnessingthe universality of a class. The primary equivalence relation studied here is computableinfinitary Σα equivalence. This project of exploiting uniformity of Turingcomputable embeddings to construct a limit embedding is carried out entirely in thecontext of countable reduced abelian p-groups. One may look at this program as eithera project in the computable structure theory of abelian p-groups, or as a projectin the construction of limits of sequences of uniform Turing computable operators. In an attempt to explore the boundary between computable infinitary Σα equivalenceand isomorphism, we show that for any computable , certain classes of countablereduced abelian p-groups are universal for ∼cα under Turing computable embedding.Further, the operators witnessing these embeddings are extremely uniform. Exploiting the uniformity of the embeddings, we produce operators which are,in some sense, limits of the embeddings witnessing the universality of the classesof countable reduced abelian p-groups. This is approached in three dierent ways:transnite recursion on ordinal notation, Barwise-Kreisel Compactness, and hyperarithemeticalsaturation. Finally, we work in admissible set theory, and use BarwiseCompactness and ΣA-saturation to generalize selected results. Advisors/Committee Members: Russell G. Miller, Committee Member, Julia F. Knight, Committee Chair, Cameron Hill, Committee Member, Sergei Starchenko, Committee Member.

Subjects/Keywords: computable structure theory; abelian groups

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

APA (6th Edition):

VanDenDriessche, S. M. (2013). Embedding Computable Infinitary Equivalence nto P-Groups</h1>. (Thesis). University of Notre Dame. Retrieved from https://curate.nd.edu/show/9z902z12x9k

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

VanDenDriessche, Steven M.. “Embedding Computable Infinitary Equivalence nto P-Groups</h1>.” 2013. Thesis, University of Notre Dame. Accessed December 04, 2020. https://curate.nd.edu/show/9z902z12x9k.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

VanDenDriessche, Steven M.. “Embedding Computable Infinitary Equivalence nto P-Groups</h1>.” 2013. Web. 04 Dec 2020.

Vancouver:

VanDenDriessche SM. Embedding Computable Infinitary Equivalence nto P-Groups</h1>. [Internet] [Thesis]. University of Notre Dame; 2013. [cited 2020 Dec 04]. Available from: https://curate.nd.edu/show/9z902z12x9k.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

VanDenDriessche SM. Embedding Computable Infinitary Equivalence nto P-Groups</h1>. [Thesis]. University of Notre Dame; 2013. Available from: https://curate.nd.edu/show/9z902z12x9k

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation


University of Notre Dame

2. Victor A Ocasio. Computability in the class of Real Closed Fields</h1>.

Degree: Mathematics, 2014, University of Notre Dame

The class of Real Closed Fields (RCF) has nice model theoretic properties, among them O-minimality and quantifier elimination. We examine RCF and the non-elementary subclass of RCF composed of archimedean real closed fields, ARCF, from a computable structure theory perspective. Also, we explore connections with the class of linear orders (LO). We focus on two main notions: Turing computable embeddings and relative categoricity.The notion of Turing computable embedding (TCE) is an effective version of the Borel embedding defined by Friedman and Stanley in 1989 [1]. In fact, many Borel embeddings are computable. The relation < is a preordering on classes of structures. In [1], the authors provide a computable Borel reduction showing that LO is a maximal element of the preordering defined by TCE’s. An embedding, due to Marker and discussed by Levin in his thesis, shows that LO < RCF, hence RCF is also maximal.We consider the class DG, of daisy graphs, a non-elementary subclass of the class of undirected graphs. Each structure A in DG codes a family S of subsets of the natural numbers N. We show that each of DG and ARCF is TC embeddable into the other. We use = to denote this.Theorem 1: DG = ARCF.To prove Theorem 1 we construct a computable perfect tree T whose paths represent algebraically independent reals. We pass from an element of DG to a family of paths through T, and from there to a real closed field which is the real closure of this family of reals.We generalize Theorem 1 and obtain the following.Theorem 2: Let K be a class of structures such that any A, B satisfying the same quantifier-free types are isomorphic. Then K < ARCF.A computable structure A is relatively Delta-alpha categorical if for any copy B, isomorphicto A there is a Delta-alpha relative to B isomorphism between A and B, here “alpha” stands for a computable ordinal. Ash, Knight, Manasse, and Slaman, and independently Chisholm, showed that a structure is relatively Delta-alpha categorical if it has a formally Sigma-alpha Scott family of formulas. Results by Nurtazin imply that a real closed field is relatively computably categorical if it has finite transcendence degree. Later work by Calvert shows that if a real closed field is archimedean then it is relatively Delta-2 categorical.Marker’s embedding takes a linear order L to a real closed field we denote by RL. Essentially RL is the real closure of L, where the elements of L are positive and infinite in RL, and if l is less than s in L, then all positive powers of l are less than s in RL.Theorem 3: If a computable linear order L is relatively Delta-alpha categorical, then RL isrelatively Delta-(1+alpha) categorical.The converse of Theorem 3 holds for alpha = 1. We show it also holds for alpha a natural number, assuming further computability properties on one copy of L. We are able then to show the following.Theorem 4: For arbitrarily large computable ordinals, there is a real closed field that is relatively Delta-alpha categorical and not relatively Delta-beta categorical for any… Advisors/Committee Members: Greg Igusa, Committee Member, Julia F. Knight, Committee Chair, Peter Cholak, Committee Member, Russell Miller, Committee Member.

Subjects/Keywords: Linear Orders; Turing computable embeddings; Logic; Computability; Computable structure theory; Real Closed Fields

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

APA (6th Edition):

Ocasio, V. A. (2014). Computability in the class of Real Closed Fields</h1>. (Thesis). University of Notre Dame. Retrieved from https://curate.nd.edu/show/v979v12176z

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16th Edition):

Ocasio, Victor A. “Computability in the class of Real Closed Fields</h1>.” 2014. Thesis, University of Notre Dame. Accessed December 04, 2020. https://curate.nd.edu/show/v979v12176z.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7th Edition):

Ocasio, Victor A. “Computability in the class of Real Closed Fields</h1>.” 2014. Web. 04 Dec 2020.

Vancouver:

Ocasio VA. Computability in the class of Real Closed Fields</h1>. [Internet] [Thesis]. University of Notre Dame; 2014. [cited 2020 Dec 04]. Available from: https://curate.nd.edu/show/v979v12176z.

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Ocasio VA. Computability in the class of Real Closed Fields</h1>. [Thesis]. University of Notre Dame; 2014. Available from: https://curate.nd.edu/show/v979v12176z

Note: this citation may be lacking information needed for this citation format:
Not specified: Masters Thesis or Doctoral Dissertation

.