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University of Michigan

1. Walton, Chelsea M. On Degenerations and Deformations of Sklyanin Algebras.

Degree: PhD, Mathematics, 2011, University of Michigan

URL: http://hdl.handle.net/2027.42/86397

A subfield of noncommutative algebra, entitled noncommutative projective algebraic geometry, was launched in the 1980s through Michael Artin, John Tate, William Schelter, and Michel van den Bergh's classification of noncommutative projective spaces. The toughest challenge was the investigation of three-dimensional Sklyanin algebras, denoted Skly3. The ring-theoretic and homological behavior of Skly3 could not be determined using purely algebraic techniques and the geometry of smooth projective curves was surprisingly used to achieve the desired classification results. In this dissertation, we employ variants of Skly3 to broaden both the class of noncommutative algebras and the type of geometric data featured in the theory of this rapidly expanding field.
Traditional techniques of this area study noncommutative graded algebras with use of projective geometric data of finite type. Even so, our research objectives include:
(1) the analysis of degenerate three-dimensional Sklyanin algebras, S(deg), which are associated to geometric data not of finite type, and;
(2) the study of the representation theory of ungraded deformations, S(def), of Skly3.
We have completed problem (1), details of which are provided in Chapter 3 of this dissertation. Namely, we conclude that the geometry of S(deg) yields an analogue to a result of Artin-Tate-van den Bergh, namely that a generalized twisted homogeneous coordinate ring arises as a factor of S(deg). This has resulted in several open problems regarding noncommutative graded rings and corresponding noncommutative coordinate rings; such further directions are also discussed in Chapter 3.
We only give a partial answer to problem (2), yet this is still sufficient as the work has several consequences not only for the advancement of the representation theory of noncommutative algebras, but also for Berenstein et al.'s study of noncommutative vacua in string theory. In particular, we present results on the dimensions of (not necessarily graded) simple finite-dimensional modules of Skly3 in Chapter 4. We also analyze the representation theory of a central extension D of Skly3 in Chapter 5 by investigating the existence of fat point modules over D and the structure of the center of D. This is all with a view towards understanding the irreducible finite-dimensional representations of S(def).
*Advisors/Committee Members: Smith, Karen E. (committee member), Stafford, J. Tobias (committee member), Derksen, Hendrikus Gerardus (committee member), Hochster, Melvin (committee member), Pando Zayas, Leopoldo A. (committee member).*

Subjects/Keywords: Noncommutative Algebra; Noncommutative Algebraic Geometry; Representation Theory; Sklyanin Algebra; Degenerate Sklyanin Algebra; Deformed Sklyanin Algebra; Mathematics; Science

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

APA (6^{th} Edition):

Walton, C. M. (2011). On Degenerations and Deformations of Sklyanin Algebras. (Doctoral Dissertation). University of Michigan. Retrieved from http://hdl.handle.net/2027.42/86397

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

Walton, Chelsea M. “On Degenerations and Deformations of Sklyanin Algebras.” 2011. Doctoral Dissertation, University of Michigan. Accessed November 25, 2020. http://hdl.handle.net/2027.42/86397.

MLA Handbook (7^{th} Edition):

Walton, Chelsea M. “On Degenerations and Deformations of Sklyanin Algebras.” 2011. Web. 25 Nov 2020.

Vancouver:

Walton CM. On Degenerations and Deformations of Sklyanin Algebras. [Internet] [Doctoral dissertation]. University of Michigan; 2011. [cited 2020 Nov 25]. Available from: http://hdl.handle.net/2027.42/86397.

Council of Science Editors:

Walton CM. On Degenerations and Deformations of Sklyanin Algebras. [Doctoral Dissertation]. University of Michigan; 2011. Available from: http://hdl.handle.net/2027.42/86397

University of Michigan

2. Bae, Changhun. Energy-bandwidth Tradeoff in Wireless Networks.

Degree: PhD, Electrical Engineering: Systems, 2010, University of Michigan

URL: http://hdl.handle.net/2027.42/77816

To effectively forward data from source to destination in energy-constrained wireless multi-hop networks, this thesis investigates the end-to-end energy-bandwidth tradeoff, and illustrates how an optimal routing path can be selected based on this tradeoff in arbitrary 2-dimensional networks.
In this tradeoff, set of relays communicates with each other, trading off the costs in power and bandwidth to achieve the minimum total energy consumption for a given target rate.
By contrast, classical energy-bandwidth tradeoff analysis in multi-hop routing network only employs the transmission energy consumption in equi-spaced linear networks.
In this thesis we consider energy-constrained wireless multi-hop networks, where each node is powered by a small battery. Under this constraint, reducing energy consumption and utilizing available spectrum are the most important design considerations for such networks,
which motivates the analysis of the energy-bandwidth tradeoff. After summarizing a model for the wireless channel, we introduce two performance measures in energy and bandwidth:
(1) total energy consumption per information bit that incorporates both the energy radiated by transmitters and the energy consumed by receivers
to process the received signals, and (2) end-to-end rate that captures the number of information bits end-to-end transmitted per channel use by any node in the network that is forwarding the data. We analyze the tradeoff between the total energy consumption per information bit and the end-to-end rate for various multi-hop routing scenarios based upon different types of transmission energy and rate. Using convex optimization methods, we show that the best energy-bandwidth tradeoffs or lower bounds can be achieved for general channel capacity models. We show that the total energy consumption per information bit can be minimized by optimally choosing the end-to-end rate, determined from the location of relays and the end-to-end distance. Thus by comparing the total energy consumption per information bit for different routing paths, a routing path which achieves the best energy-bandwidth tradeoff can be obtained. We then extend this analysis to the case with spatial reuse to improve the bandwidth efficiency, and practical channel code with modulation schemes. Finally we investigate cooperation between nodes to further increase the bandwidth utilization for various relaying strategies.
*Advisors/Committee Members: Stark, Wayne E. (committee member), Anastasopoulos, Achilleas (committee member), Derksen, Hendrikus Gerardus (committee member), Sadanandarao, Sandeep P. (committee member).*

Subjects/Keywords: Energy Efficiency; Bandwidth Efficiency; Wireless Multi-hop Networks; Convex Optimization; Electrical Engineering; Engineering

Record Details Similar Records

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

APA (6^{th} Edition):

Bae, C. (2010). Energy-bandwidth Tradeoff in Wireless Networks. (Doctoral Dissertation). University of Michigan. Retrieved from http://hdl.handle.net/2027.42/77816

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

Bae, Changhun. “Energy-bandwidth Tradeoff in Wireless Networks.” 2010. Doctoral Dissertation, University of Michigan. Accessed November 25, 2020. http://hdl.handle.net/2027.42/77816.

MLA Handbook (7^{th} Edition):

Bae, Changhun. “Energy-bandwidth Tradeoff in Wireless Networks.” 2010. Web. 25 Nov 2020.

Vancouver:

Bae C. Energy-bandwidth Tradeoff in Wireless Networks. [Internet] [Doctoral dissertation]. University of Michigan; 2010. [cited 2020 Nov 25]. Available from: http://hdl.handle.net/2027.42/77816.

Council of Science Editors:

Bae C. Energy-bandwidth Tradeoff in Wireless Networks. [Doctoral Dissertation]. University of Michigan; 2010. Available from: http://hdl.handle.net/2027.42/77816