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Rice University

1.
McGaffey, Tom.
* Regularity* and Nearness Theorems for Families of Local Lie Groups.

Degree: PhD, Natural Sciences, 2011, Rice University

URL: http://hdl.handle.net/1911/70349

In this work, we prove three types of results with the strategy that, together, the author believes these should imply the local version of Hilbert's Fifth problem. In a separate development, we construct a nontrivial topology for rings of map germs on Euclidean spaces. First, we develop a framework for the theory of (local) nonstandard Lie groups and within that framework prove a nonstandard result that implies that a family of local Lie groups that converge in a pointwise sense must then differentiability converge, up to coordinate change, to an analytic local Lie group, see corollary 6.3.1. The second result essentially says that a pair of mappings that almost satisfy the properties defining a local Lie group must have a local Lie group nearby, see proposition 7.2.1. Pairing the above two results, we get the principal standard consequence of the above work which can be roughly described as follows. If we have pointwise equicontinuous family of mapping pairs (potential local Euclidean topological group structures), pointwise approximating a (possibly differentiably unbounded) family of differentiable (sufficiently approximate) almost groups, then the original family has, after appropriate coordinate change, a local Lie group as a limit point. (See corollary 7.2.1 for the exact statement.) The third set of results give nonstandard renditions of equicontinuity criteria for families of differentiable functions, see theorem 9.1.1. These results are critical in the proofs of the principal results of this paper as well as the standard interpretations of the main results here. Following this material, we have a long chapter constructing a Hausdorff topology on the ring of real valued map germs on Euclidean space. This topology has good properties with respect to convergence and composition. See the detailed introduction to this chapter for the motivation and description of this topology.
*Advisors/Committee Members: Hardt, Robert M. (advisor).*

Subjects/Keywords: Pure sciences; Lie groups; Regularity theorems; Hilbert's Fifth problem; Theoretical mathematics

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

APA (6^{th} Edition):

McGaffey, T. (2011). Regularity and Nearness Theorems for Families of Local Lie Groups. (Doctoral Dissertation). Rice University. Retrieved from http://hdl.handle.net/1911/70349

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

McGaffey, Tom. “Regularity and Nearness Theorems for Families of Local Lie Groups.” 2011. Doctoral Dissertation, Rice University. Accessed December 18, 2018. http://hdl.handle.net/1911/70349.

MLA Handbook (7^{th} Edition):

McGaffey, Tom. “Regularity and Nearness Theorems for Families of Local Lie Groups.” 2011. Web. 18 Dec 2018.

Vancouver:

McGaffey T. Regularity and Nearness Theorems for Families of Local Lie Groups. [Internet] [Doctoral dissertation]. Rice University; 2011. [cited 2018 Dec 18]. Available from: http://hdl.handle.net/1911/70349.

Council of Science Editors:

McGaffey T. Regularity and Nearness Theorems for Families of Local Lie Groups. [Doctoral Dissertation]. Rice University; 2011. Available from: http://hdl.handle.net/1911/70349

University of Kansas

2. Kanyama, Isaac Kalonda. Shifting Preferences and Time-Varying Parameters in Demand Analysis: A Monte Carlo Study.

Degree: PhD, Economics, 2011, University of Kansas

URL: http://hdl.handle.net/1808/10337

Using Monte Carlo experiments, I address two issues in demand analysis. The first relates to the performance of local flexible functional forms in recovering the time-varying elasticities of a true model, and in correctly identifying goods as complements, substitutes, normal or inferior. The problem is illustrated with the nonlinear almost ideal demand system (NLAI) and the Rotterdam model (RM). For the AIDS, I also consider two versions of its linear approximation: one with simple formulas (LAISF) and the other with corrected formulas (LAICF). The second issue concerns the ability of the flexible functional structures to satisfy theoretical regularity in terms of the Slutsky matrix being negative semi-definite at each time period of time. I tackle these issues in the framework of structural time series models, computing the relevant time-varying elasticities by means of Kalman filtered and smoothed coefficients. The estimated time-varying coefficients are obtained under the pure random walk and the local trend hypotheses. I find that both the NLAI and the RM qualitatively perform well in approximating the signs of the time-varying income and substitution elasticities. Quantitatively, the RM tends to produce values of the time-varying elasticity of substitution close to the true ones within separable utility branches while the NLAI tends to produce overestimating values. On the other hand, the RM produces time-varying income elasticities with values close to the true ones while the NLAI tends to produce constant values over time. The LAISF model qualitatively performs similarly to the NLAI, but the LAICF does not. Finally, the NLAI achieves higher levels of the regularity index under the local trend specification while the RM achieves higher regularity levels under the random walk specification. In contrast, the LAISF and the LAICF models achieve lower levels of regularity under both specifications of the time-varying coefficients. Globally, the LAICF which widely adopted in applied work performs poorly compared to the RM and the NLAI. These findings are robust to different values of the time-varying parameters in the utility function. Two implications emerge from this research. First, the LAICF model should be considered as a model on its own rather than as an approximation of the NLAI. Second, the choice between an AIDS-type model and the RM should be motivated by their performance with respect to the properties a hypothesized true model for the data at hand, especially when working with real data.
*Advisors/Committee Members: Barnett, William A. (advisor), Comolli, Paul (cmtemember), Hillmer, Steve (cmtemember), Keating, John (cmtemember), Zhang, Jianbo (cmtemember).*

Subjects/Keywords: Economics; Almost ideal demand system; Monte Carlo study; Rotterdam model; State-space model; Theoretical regularity

Record Details Similar Records

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

APA (6^{th} Edition):

Kanyama, I. K. (2011). Shifting Preferences and Time-Varying Parameters in Demand Analysis: A Monte Carlo Study. (Doctoral Dissertation). University of Kansas. Retrieved from http://hdl.handle.net/1808/10337

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

Kanyama, Isaac Kalonda. “Shifting Preferences and Time-Varying Parameters in Demand Analysis: A Monte Carlo Study.” 2011. Doctoral Dissertation, University of Kansas. Accessed December 18, 2018. http://hdl.handle.net/1808/10337.

MLA Handbook (7^{th} Edition):

Kanyama, Isaac Kalonda. “Shifting Preferences and Time-Varying Parameters in Demand Analysis: A Monte Carlo Study.” 2011. Web. 18 Dec 2018.

Vancouver:

Kanyama IK. Shifting Preferences and Time-Varying Parameters in Demand Analysis: A Monte Carlo Study. [Internet] [Doctoral dissertation]. University of Kansas; 2011. [cited 2018 Dec 18]. Available from: http://hdl.handle.net/1808/10337.

Council of Science Editors:

Kanyama IK. Shifting Preferences and Time-Varying Parameters in Demand Analysis: A Monte Carlo Study. [Doctoral Dissertation]. University of Kansas; 2011. Available from: http://hdl.handle.net/1808/10337