University of Macau / Faculty of Science and Technology. / Department of Mathematics

Although the concept of fractional calculus was known for centuries, it was not considered in engineering due to the lack of implementation tools and acceptable performance of integer order models and control. However, recently, engineers and researchers started to investigate the potentially high performance of fractional calculus in various fields among which are acoustics, conservation of mass, diffusion equation and specifically in this thesis control theory. The intention of this thesis is to analyze the relative performance of fractional versus integer order PID controller for a quadcopter. Initially, the dynamics of the quadcopter is presented with additional consideration of the ground effect and torque saturation. Then, are introduced the concept of fractional calculus and the mathematical tools to be used for modeling fractional order controller. Finally, the performance of the fractional order controller is evaluated by comparing it to an integer order controller.

Lam, Wai Kit. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 55-56). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese.

acase@tulane.edu / In this thesis, we tackle the statistical problem of demixing a multivariate stochastic process made up of independent, fractional process entries. We consider both Gaussian and non-Gaussian frameworks. The observable, mixed process is then a multivariate fractional stochastic process. In particular, when the components of the unmixed process are self-similar, the mixed process is operator self-similar. Multivariate mixed fractional processes are parameterized by a vector of Hurst parameters and a mixing matrix. We propose a 2-step wavelet-based estimation method to produce estimators of both the demixing matrix and the Hurst parameters. In the first step, an estimator of the demixing matrix is obtained by applying a classical joint diagonalization algorithm to two wavelet variance matrices of the mixed process. In the second step, a univariate-like wavelet regression method is applied to each entry of the demixed process to provide estimators of each individual Hurst parameter. The limiting distribution of the estimators is established for both Gaussian and nonGaussian (Rosenblatt-like) instances. Monte Carlo experiments show that the finite sample estimation performance is very satisfactory. As an application, we model bivariate series of annual tree ring measurements from bristlecone pine trees in White Mountains, California. / 1 / Hui Li

The purpose of this research is to investigate the power of various often-used cointegration tests against the fractionally cointegrated alternaive from Monte Carlo simulation. According to the simulation results, the Saikkonen¡¦s Infinite VAR model is most adequate cointegration test.

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The fractional quantum Hall effect (FQHE), now entering it's fourth decade, continues to draw attention from the condensed matter community. New experiments in recent years are raising hopes that it will be possible to observe quasi-particles with non-abelian anyonic statistics. These particles could form the building blocks of a quantum computer. The quantum Hall states have topologically protected energy gaps to the low-lying set of excitations. This topological order is not a locally measurable quantity but rather a non-local object, and it is one of the keys to it's stability. From an early stage understanding of the FQHE has been facilitate by constructing trial wave functions. The topological classification of these wave functions have given further insight to the nature of the FQHE. An early, and successful, wave function construction for filling fractions ν=p/(2p+1) was that of composite fermions on planar and spherical geometries. Recently, new developments using conformal field theory have made it possible to also construct the full Haldane-Halperin hierarchy wave functions on planar and spherical geometries. In this thesis we extend this construction to a toroidal geometry, i.e. a flat surface with periodic boundary conditions. One of the defining features of topological states of matter in two dimensions is that the ground state is not unique on surfaces with non trivial topology, such as a torus. The archetypical example is the fractional quantum Hall effect, where a state at filling fraction ν=p/q, has at least a q-fold degeneracy on a torus. This has been shown explicitly for a few cases, such as the Laughlin states and the the Moore-Read states, by explicit construction of candidate electron wave functions with good overlap with numerically found states. In this thesis, we construct explicit torus wave functions for a large class of experimentally important quantum liquids, namely the chiral hierarchy states in the lowest Landau level. These states, which includes the prominently observed positive Jain sequence at filling fractions ν=p/(2p+1), are characterized by having boundary modes with only one chirality. Our construction relies heavily on previous work that expressed the hierarchy wave functions on a plane or a sphere in terms of correlation functions in a conformal field theory. This construction can be taken over to the torus when care is taken to ensure correct behaviour under the modular transformations that leave the geometry of the torus unchanged. Our construction solves the long standing problem of engineering torus wave functions for multi-component many-body states. Since the resulting expressions are rather complicated, we have carefully compared the simplest example, that of ν=2/5, with numerically found wave functions. We have found an extremely good overlap for arbitrary values of the modular parameter τ, that describes the geometry of the torus. Having explicit torus wave functions allows us to use the methods developed by Read and Read \&amp; Rezayi to numerically compute the quantum Hall viscosity. Hall viscosity is conjectured to be a topologically protected macroscopic transport coefficient characterizing the quantum Hall state. It is related to the shift of the same QH-fluid when it is put on a sphere. The good agreement with the theoretical prediction for the 2/5 state strongly suggests that our wave functions encodes all relevant topologically information. We also consider the Hall viscosity in the limit of a very thin torus. There we find that the viscosity changes as we approach the thin torus limit. Because of this we study the Laughlin state in that limit and see how the change in viscosity arises from a change in the Hamiltonian hopping elements. Finally we conclude that there are both qualitative and quantitative difference between the thin and the square torus. Thus, one has to be careful when interpreting results in the thin torus limit. / <p>At the time of the doctoral defense, the following paper was unpublished and had a status as follows: Paper 4: Manuscript.</p>

This thesis discusses the efficient and accurate generation of self-similar traffic for ATM networks. ATM networks have been developed to carry multiple service categories. Since the traffic on a number of existing networks is bursty, much research focuses on how to capture the characteristics of traffic to reduce the impact of burstiness. Conventional traffic models do not represent the characteristics of burstiness well, but self-similar traffic models provide a closer approximation. Self-similar traffic models have two fundamental properties, long-range dependence and infinite variance, which have been found in a large number of measurements of real traffic. Therefore, generation of self-similar traffic is vital for the accurate simulation of ATM networks. The main starting point for self-similar traffic generation is the production of fractional Brownian motion (FBM) or fractional Gaussian noise (FGN). In this thesis six algorithms are brought together so that their efficiency and accuracy can be assessed. It is shown that the discrete FGN (dPGN) algorithm and the Weierstrass-Mandelbrot (WM) function are the best in terms of accuracy while the random midpoint displacement (RMD) algorithm, successive random addition (SRA) algorithm, and the WM function are superior in terms of efficiency. Three hybrid approaches are suggested to overcome the inefficiency or inaccuracy of the six algorithms. The combination of the dFGN and RMD algorithm was found to be the best in that it can generate accurate samples efficiently and on-the-fly. After generating FBM sample traces, a further transformation needs to be conducted with either the marginal distribution model or the storage model to produce self-similar traffic. The storage model is a better transformation because it provides a more rigorous mathematical derivation and interpretation of physical meaning. The suitability of using selected Hurst estimators, the rescaled adjusted range (R/S) statistic, the variance-time (VT) plot, and Whittle's approximate maximum likelihood estimator (MLE), is also covered. Whittle's MLE is the better estimator, the R/S statistic can only be used as a reference, and the VT plot might misrepresent the actual Hurst value. An improved method for the generation of self-similar traces and their conversion to traffic has been proposed. This, combined with the identification of reliable methods for the estimators of the Hurst parameter, significantly advances the use of self-similar traffic models in ATM network simulation.

Thesis (M.S.)--Virginia Polytechnic Institute, 1957. / Vita. Abstract. Includes bibliographical references (leaves 136-140). Also available via the Internet.

Thesis (Sc. D.)--University of Michigan, 1934. / "This paper was presented at the 26th semi-annual meeting of the American institute of chemical engineers, New York, N.Y., May, 1934, and is reprinted from volume no. 30 of the Transactions." "References dealing with distillation and vapor-liquid equilibria": p. 26-27.