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You searched for `+publisher:"University of Notre Dame" +contributor:("Julia F. Knight, Committee Chair")`

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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

URL: https://curate.nd.edu/show/9z902z12x9k

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

Record Details Similar Records

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

APA (6^{th} 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 (16^{th} 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 (7^{th} 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

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

URL: https://curate.nd.edu/show/v979v12176z

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

Record Details Similar Records

❌

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

APA (6^{th} 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

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} 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.

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} 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.

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

Not specified: Masters Thesis or Doctoral Dissertation