Triangulations and Heegaard splittings

Recently William Jaco, J. Hyam Rubinstein and Stephan Tillmann together proved that the generalized quaternion spaces S3 /Q4k, k ≥ 2, which are small Seifert fibered spaces Mk= (S2 : (2, 1),(2, 1),(k, -k + 1)), have complexity k, which is the minimal number of tetrahedra in a triangulation of Mk. The techniques used can be expanded to show that the layered chain pair triangulations of Seifert fibered spaces (S2: (2, -1),(r + 1, 1),(s + 1, 1)), r, s ≥ 1 are minimal.My thesis is to closely study the minimal, 0-efficient triangulations of the above two infinite families of Seifert fiberd spaces. One family is called the twisted layered loop triangulation, and the other family is called layered chain pair triangulations. They were named by Ben Burton. We classify all normal and almost normal surfaces by identifying one-sided incompressible surfaces, orientable incompressible surfaces and Heegaard splitting surfaces. We also use combinatorial methods to study and classify irreducible Heegaard splitting surfaces, up to isotopy, in these two infinite families of Seifert fibered manifolds.In the twisted layered loop triangulations of the Seifert fibered space Mk, k ≥ 2. We prove that a properly embedded surface S is a Heegaard splitting surface if and only if it is an almost normal tubed surface with the almost normal tube at the same level of a thin edge-linking tube. Furthermore, any genus two Heegaard splitting surface is vertical. A combinatorial proof is given that there is a unique irreducible genus 2 Heegaard splitting surface, up to isotopy, in Mk, k ≥ 2.In the layered chain pair triangulation of Seifert fibered spaces Mr,s=( S2: (2, -1),(r+ 1, 1),(s+1, 1)), r, s ≥ 1, we prove that an almost normal tubed surface with the almost normal tube at the same level of a thin edge-linking tube is a Heegaard splitting surface. Moreover, if the genus of it is 2, then it is not only an irreducible Heegaard splitting but also a vertical one. We give a combinatorial proof that there is a unique irreducible vertical Heegaard splitting surface, up to isotopy, in Mr,s, r, s ≥ 1.Our work follows the methods used by Jaco and Rubinstein in studying layered triangulations of the solid torus and their classification of normal surfaces in these triangulations

Dynamic characterization of vocal fold virbrations

An emerging trend among voice specialists is the use of quantitative protocols for the diagnosis and treatment of voice disorders. Vocal fold vibrations are directly related to voice quality. This research is devoted to providing an objective means of characterizing these vibrations. Our goal is to develop a dynamic model of vocal fold vibration, and map the parameter space of the model to a class of voice disorders; thus, furthering the assessment and diagnosis of voice disorder in clinical settings. To this end, this dissertation introduces a new seven-mass biomechanical model for the vibration of vocal folds. The model is based on the body-cover layer concept of the vocal fold biomechanics, and segments the cover layer into three masses along the longitudinal direction of the vocal fold. This segmentation facilitates the model comparison with the motion of the vocal glottis contour derived from modern high-speed digital imaging systems. The model simulation is compared to 14 sets of experimental data from human subjects with healthy vocal folds and pathological vocal folds including nodule, polyp, and unilateral paralysis. We also propose a semi-empirical two-stage procedure for tuning the parameters so that the model response matches as closely as possible the experimental data in the time and frequency domains. The first stage involves the manual coarse tuning of parameters based on limited data to expedite the process. The second stage is an automatic (or manual) fine tuning process on a subset of the parameters tuned in the first stage based on a larger amount of data. Once an ‘optimal’ set of model parameters has been identified, two model-based factors, quantifying the asymmetry between left and right vocal folds and anterior and posterior segments of the vocal folds, are introduced and calculated for each of the 14 cases. The two factors form an asymmetry plane. Based on the value of the asymmetry factors for the 14 cases, the plane is subdivided into four regions corresponding to healthy vocal folds, nodule, polyp, and unilateral paralysis. This yields a clear visual aid for clinicians, correlating the model parameters to voice quality

Different Estimation Methods for the Basic Independent Component Analysis Model

Inspired by classic cocktail-party problem, the basic Independent Component Analysis (ICA) model is created. What differs Independent Component Analysis (ICA) from other kinds of analysis is the intrinsic non-Gaussian assumption of the data. Several approaches are proposed based on maximizing the non-Gaussianity of the data, which is measured by kurtosis, mutual information, and others. With each estimation, we need to optimize the functions of expectations of non-quadratic functions since it can help us to access the higher-order statistics of non-Gaussian part of the data. In this thesis, our goal is to review the one of the most efficient estimation methods, that is, the Fast Fixed-Point Independent Component Analysis (FastICA) algorithm, illustrate it with some examples using an R package

Advanced Ignition Strategies for Future Internal Combustion Engines with Lean and Diluted Fuel-Air Mixtures

The main objective of this research was to study the mechanisms of the spark ignition process of lean or diluted fuel-air mixtures under enhanced gas flow conditions for applications in future internal combustion engines. Various spark ignition strategies were deployed by controlling the spark discharge process via different spark ignition hardware configurations. Modulated spark discharge parameters, such as enhanced discharge power, prolonged discharge duration, and boosted discharge current were facilitated in the research. The impact of gas flow on the spark discharge process in air was investigated under varying air flow conditions with a range of flow velocities from 0 m/s to 60 m/s. The ignition performance of the spark strategies was investigated with lean or diluted fuel-air mixtures under controlled gas flow conditions in an optical constant volume combustion chamber test platform. The mixture flow velocity across the spark gap ranged from 0 m/s to 35 m/s during the combustion tests.Experiments were carried out with air as the background media. Short circuits and restrikes were observed under air flow conditions. The frequency of these occurrences increased with increased air flow velocity. The length of the spark plasma increased, due to the stretch of the plasma channel by the air flow. The plasma was stretched at a speed similar to the air flow velocity across the spark gap. The maximum length of the spark plasma was affected by the air flow velocity and the spark gap size. The spark discharge duration reduced with increased air flow velocity. To enhance the ignition of a lean or diluted fuel-air mixture under quiescent conditions, high spark discharge power or high spark discharge current were applied. With equivalent spark discharge energy, a larger flame kernel was achieved by the high-power spark whereas the impacts of spark discharge current level and discharge duration during the arc and glow phases were insignificant on the flame kernel growth. A transient high-current spark also generated a larger flame kernel, although with much higher spark energy as compared with that from a conventional spark. Under gas flow conditions, both the spark discharge current magnitude and discharge duration were critical for the flame kernel growth. It is postulated that this kernel growth was the result of a prolonged spark discharge duration effectively increasing the interaction volume between the plasma channel and the combustible gas engulfed by the mixture flow. Consequently, a longer spark discharge duration proved beneficial in establishing a larger flame kernel, probably because the spark discharge current was sufficient to support the flame kernel growth. Indeed, it was observed that boosted spark current was advantageous for the flame kernel growth, especially at higher flow velocities. However, the high-power spark and transient high-current spark proved to be less effective with higher flow velocities, probably because of the short discharge duration