The modern theory of condition measures for convex optimization problems was initially developed for convex problems in conic format, and several aspects of the theory have now been extended to handle non-conic formats as well. In this theory, the (Renegar-) condition measure C(d) for a problem instance with data d=(A,b,c) has been shown to be connected to bounds on a wide variety of behavioral and computational characteristics of the problem instance, from sizes of optimal solutions to the complexity of algorithms. Herein we test the practical relevance of the condition measure theory, as applied to linear optimization problems that one might typically encounter in practice. Using the NETLIB suite of linear optimization problems as a test bed, we found that 71% of the NETLIB suite problem instances have infinite condition measure. In order to examine condition measures of the problems that are the actual input to a modern IPM solver, we also computed condition measures for the NETLIB suite problems after pre-preprocessing by CPLEX 7.1. Here we found that 19% of the post-processed problem instances in the NETLIB suite have infinite condition measure, and that log C(d) of the post-processed problems is fairly nicely distributed. Furthermore, there is a positive linear relationship between IPM iterations and log C(d) of the post-processed problem instances (significant at the 95% confidence level), and 42% of the variation in IPM iterations among the NETLIB suite problem instances is accounted for by log C(d) of the post-processed problem instances. / Singapore-MIT Alliance (SMA)

The vector-valued coarea formula provides a relationship between the integral of the Jacobian of a map from high dimensions down to low dimensions with the integral over the measure of the fibers of this map. We explore minimizers of this functional, proving existence using both a variational approach and an approach with currents. Additionally, we consider what properties these minimizers will have and provide examples. Finally, this problem is considered in metric spaces, where a third existence proof is given.

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Continuous weak measurement provide a convenient way to gather information about a quantum system without the need to prepare huge ensembles of identical systems as required by standard quantum measurement theory. Even though weak measurement alter the dynamics of the wave function slightly, they nevertheless are a good tool to monitor the dynamics of the wave function in real time in the presence of certain perturbations, for example, sudden momentum kicks due to collisions with particles of a surrounding gas. With weak measurement it is possible to monitor the dynamics of the wave function without knowing it initially. The continuous monitoring can be employed to influence the dynamics by means of feedback. This thesis focuses on the numeric simulation of the continuous monitoring of the position of a free massive particle as well as a particle bound in the following one-dimensional potentials: harmonic and double well. The monitoring scheme involves estimating the wave function of the hydrogen atom initially and then applying the results of the weak measurement its position to update the estimate through a numerically simulated stochastic evolution. We also simulate evolution of the true wave function. The key highlights of this thesis include: discussion of an alternative way to derive the stochastic differential equations that govern the evolution of the true and estimated wave functions of the system, as well as the explanation of the second order numerical scheme. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2012.

Diss. Västerås : Mälardalens högskola, 2007. / Härtill 5 uppsatser. S. 20-23: Bibliografi.

Thesis (Ph. D.)--University of Washington, 2004. / Vita. Includes bibliographical references (leaves 108-111).

Thesis (Ph. D.)--University of California, Riverside, 2009. / Includes abstract. Includes bibliographical references (leaves 202-204) and index. Issued in print and online. Available via ProQuest Digital Dissertations.