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University of Alabama

1. Green, Michael Timothy. Graphs of groups.

Degree: 2012, University of Alabama

URL: http://purl.lib.ua.edu/55036

Graphs of groups were first introduced by Jean-Pierre Serre in his book entitled Arbres, Amalgames, SL2 (1977), whose first English translation was Trees in 1980. In 1993, Hyman Bass wrote a paper called Covering theory for graphs of groups which discussed such concepts in the category of Graphs of Groups as morphisms, fundamental groups, and infinite covers. Hence, this area of geometric group theory is typically referred to as Bass-Serre Theory. The contents of this dissertation lie within this broad area of study. The main focus of the research is to try to apply to the category of Graphs of Groups what John Stallings did in the category of Graphs in his paper Topology of finite graphs. In that paper, he explored in graphs a vast number of topics such as pullbacks, paths, stars, coverings, and foldings. The goal of this dissertation is to apply many of those concepts to the category of Graphs of Groups. In this work, we develop our notion of paths, links, maps of graphs of groups, and coverings. We then explore the resultant path-lifting properties. (Published By University of Alabama Libraries)
*Advisors/Committee Members: Corson, Jon M., Dixon, Martyn, Evans, Martin, Trace, Bruce, Liem, Vo, Wright, Vivian, University of Alabama. Dept. of Mathematics.*

Subjects/Keywords: Electronic Thesis or Dissertation; – thesis; Mathematics; covering; graph; graphs of groups; group; path; path-lifting

Record Details Similar Records

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

APA (6^{th} Edition):

Green, M. T. (2012). Graphs of groups. (Thesis). University of Alabama. Retrieved from http://purl.lib.ua.edu/55036

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):

Green, Michael Timothy. “Graphs of groups.” 2012. Thesis, University of Alabama. Accessed September 26, 2020. http://purl.lib.ua.edu/55036.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Green, Michael Timothy. “Graphs of groups.” 2012. Web. 26 Sep 2020.

Vancouver:

Green MT. Graphs of groups. [Internet] [Thesis]. University of Alabama; 2012. [cited 2020 Sep 26]. Available from: http://purl.lib.ua.edu/55036.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Green MT. Graphs of groups. [Thesis]. University of Alabama; 2012. Available from: http://purl.lib.ua.edu/55036

Not specified: Masters Thesis or Doctoral Dissertation

Queens University

2. Amiss, David Scott Cameron. Obstructions to Motion Planning by the Continuation Method .

Degree: Chemical Engineering, 2013, Queens University

URL: http://hdl.handle.net/1974/7703

The subject of this thesis is the motion planning algorithm known as the continuation method. To solve motion planning problems, the continuation method proceeds by lifting curves in state space to curves in control space; the lifted curves are the solutions of special initial value problems called
path-lifting equations. To validate this procedure, three distinct obstructions
must be overcome. The first obstruction is that the endpoint maps of the control system
under study must be twice continuously differentiable. By extending a result
of A. Margheri, we show that this differentiability property is satisfied by an
inclusive class of time-varying fully nonlinear control systems. The second obstruction is the existence of singular controls, which are simply the singular points of a fixed endpoint map. Rather than attempting to completely characterize such controls, we demonstrate how to isolate control systems for which no controls are singular. To this end, we build on the
work of S. A. Vakhrameev to obtain a necessary and sufficient condition. In particular, this result accommodates time-varying fully nonlinear control
systems. The final obstruction is that the solutions of path-lifting equations may not
exist globally. To study this problem, we work under the standing assumption
that the control system under study is control-affine. By extending a result of Y. Chitour, we show that the question of global existence can be resolved by examining Lie bracket configurations and momentum functions. Finally, we show that if the control system under study is completely
unobstructed with respect to a fixed motion planning problem, then its corresponding endpoint map is a fiber bundle. In this sense, we obtain a necessary condition for unobstructed motion planning by the continuation method.

Subjects/Keywords: geometric control theory ; motion planning ; continuation method ; path-lifting equations ; singular controls ; nonlinear control theory

Record Details Similar Records

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

APA (6^{th} Edition):

Amiss, D. S. C. (2013). Obstructions to Motion Planning by the Continuation Method . (Thesis). Queens University. Retrieved from http://hdl.handle.net/1974/7703

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Amiss, David Scott Cameron. “Obstructions to Motion Planning by the Continuation Method .” 2013. Thesis, Queens University. Accessed September 26, 2020. http://hdl.handle.net/1974/7703.

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Amiss, David Scott Cameron. “Obstructions to Motion Planning by the Continuation Method .” 2013. Web. 26 Sep 2020.

Vancouver:

Amiss DSC. Obstructions to Motion Planning by the Continuation Method . [Internet] [Thesis]. Queens University; 2013. [cited 2020 Sep 26]. Available from: http://hdl.handle.net/1974/7703.

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Amiss DSC. Obstructions to Motion Planning by the Continuation Method . [Thesis]. Queens University; 2013. Available from: http://hdl.handle.net/1974/7703

Not specified: Masters Thesis or Doctoral Dissertation

Georgia Tech

3. Shim, Sangho. Large scale group network optimization.

Degree: PhD, Industrial and Systems Engineering, 2009, Georgia Tech

URL: http://hdl.handle.net/1853/31737

Every knapsack problem may be relaxed to a cyclic group problem. In 1969, Gomory found the subadditive characterization of facets of the master cyclic group problem. We simplify the subadditive relations by the substitution of complementarities and discover a minimal representation of the subadditive polytope for the master cyclic group problem. By using the minimal representation, we characterize the vertices of cardinality length 3 and implement the shooting experiment from the natural interior point.
The shooting from the natural interior point
is a shooting from the inside of the plus level set of the subadditive polytope. It induces the shooting for the knapsack problem. From the shooting experiment for the knapsack problem
we conclude that the most hit facet is the knapsack mixed integer cut which is the 2-fold lifting of a mixed integer cut.
We develop a cutting plane algorithm augmenting cutting planes generated by shooting, and implement it on Wong-Coppersmith digraphs observing that only small number of cutting planes
are enough to produce the optimal solution. We discuss a relaxation of shooting as a clue to quick shooting. A max flow model on covering space
is shown to be equivalent to the dual of shooting linear programming problem.
*Advisors/Committee Members: Ellis L. Johnson (Committee Chair), Brady Hunsaker (Committee Member), George Nemhauser (Committee Member), Jozef Siran (Committee Member), Shabbir Ahmed (Committee Member), William Cook (Committee Member).*

Subjects/Keywords: Shortest path; Column generation; Large scale optimization; Minimal word problem; Rubik's cube; Lifting; Network flow; Cayley digraph; Geometric group theory; Integer programming; Knapsack problem (Mathematics); Mathematical optimization

Record Details Similar Records

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

APA (6^{th} Edition):

Shim, S. (2009). Large scale group network optimization. (Doctoral Dissertation). Georgia Tech. Retrieved from http://hdl.handle.net/1853/31737

Chicago Manual of Style (16^{th} Edition):

Shim, Sangho. “Large scale group network optimization.” 2009. Doctoral Dissertation, Georgia Tech. Accessed September 26, 2020. http://hdl.handle.net/1853/31737.

MLA Handbook (7^{th} Edition):

Shim, Sangho. “Large scale group network optimization.” 2009. Web. 26 Sep 2020.

Vancouver:

Shim S. Large scale group network optimization. [Internet] [Doctoral dissertation]. Georgia Tech; 2009. [cited 2020 Sep 26]. Available from: http://hdl.handle.net/1853/31737.

Council of Science Editors:

Shim S. Large scale group network optimization. [Doctoral Dissertation]. Georgia Tech; 2009. Available from: http://hdl.handle.net/1853/31737