Since the identi cation of proto-planetary nebulae (PPNe) as transition objects between the asymptotic giant branch stars and planetary nebulae more than two decades ago, astronomers have attempted to characterise these exciting objects. Today many questions still elude a conclusive answer, partly due to the sheer diversity observed within this small subset of stellar objects, and partly due to the low numbers detected. Fortunately, many of these objects display a rich spectrum of emission/absorption lines that can be used as diagnostics for these nebulae. This dissertation presents a study of six PPNe using the relatively new (at NIR wavelengths) integral eld spectroscopy technique. This method has allowed the investigation of distinct regions of these nebulae, and in certain cases the application of magneto-hydrodynamic shock models to the data. The goal of this research has been to investigate the evolution of PPNe by detailed examination of a small sample of objects consisting of a full range of evolutionary types. Near-IR ro-vibrational lines were employed as the primary tool to tackle this problem. In all six sources the 1!0S(1) line is used to map the spatial extent of the H2. In three of these objects the maps represent the rst images of their H2 emission nebulae. In the case of the earliest-type object (IRAS 14331-6435) in this sample, the line map gives the rst image of its nebula at any wavelength. In the only M-type object (OH 231.8+4.2) in the sample, high-velocity H2 is detected in discrete clumps along the edges of the bipolar out ow, while a possible ring of slower moving H2 is found around the equatorial region. This is the rst detection of H2 in such a late-type object but due its peculiarities, it is possibly not representative of what is expected of M-type objects. In IRAS 19500-1709, an intermediate-type object, the line map shows the H2 emission to originate in clumpy structures along the edges of a bipolar shell/out ow. The remaining three objects have all been the subject of previous studies but in each case new H2 lines are detected in this work along with other emission lines (Mg ii, Na i & CO). In the case of IRAS 16594-4656, MHD shock models have been used to determine the gas density and shock velocity. Two new python modules/classes have been written. The rst one to deal with the data cubes, extract ux measurements, rebin regions of interest, and produce line maps. The second class allows the easy calculation of many important parameters, for example, excitation temperatures, column density ratio values, extinction estimates from several line-pairs, column density values, and total mass of the H2. The class also allows the production of input les for the shock tting procedure, and simulated shocks for testing this tting process. A new framework to t NIR shock models to data has been developed, employing Monte Carlo techniques and the extensive computing cluster at the University of Hertfordshire (UH). This method builds on the approach used by many other authors, with the added advantages that this framework provides a method of correctly sampling the shock model parameter space, and providing error estimates on the model t. Using this approach, data from IRAS 16594-4656 have been successfully modelled using the shock models. A full description of this class of stellar objects from such a small sample is not possible due to their diverse nature. Although H2 was detected across the full spectral vi range of post-AGB objects, the phase at which H2 emission begins is still not clear. The only M-type object in this work is a peculiar object and may not be representative of a typical post-AGB star. The H2 PPNe appear to be located at lower Galactic latitudes (b 20 ) than the total PPNe population, possibly pointing to an above average mass and hence younger age of these objects.

Indiana University-Purdue University Indianapolis (IUPUI) / Recent years have seen a growth in research on system reliability and maintenance. Various studies in the scientific fields of reliability engineering, quality and productivity analyses, risk assessment, software reliability, and probabilistic machine learning are being undertaken in the present era. The dependency of human life on technology has made it more important to maintain such systems and maximize their potential. In this dissertation, some methodologies are presented that maximize certain measures of system reliability, explain the underlying stochastic behavior of certain systems, and prevent the risk of system failure. An overview of the dissertation is provided in Chapter 1, where we briefly discuss some useful definitions and concepts in probability theory and stochastic processes and present some mathematical results required in later chapters. Thereafter, we present the motivation and outline of each subsequent chapter. In Chapter 2, we compute the limiting average availability of a one-unit repairable system subject to repair facilities and spare units. Formulas for finding the limiting average availability of a repairable system exist only for some special cases: (1) either the lifetime or the repair-time is exponential; or (2) there is one spare unit and one repair facility. In contrast, we consider a more general setting involving several spare units and several repair facilities; and we allow arbitrary life- and repair-time distributions. Under periodic monitoring, which essentially discretizes the time variable, we compute the limiting average availability. The discretization approach closely approximates the existing results in the special cases; and demonstrates as anticipated that the limiting average availability increases with additional spare unit and/or repair facility. In Chapter 3, the system experiences two types of sporadic impact: valid shocks that cause damage instantaneously and positive interventions that induce partial healing. Whereas each shock inflicts a fixed magnitude of damage, the accumulated effect of k positive interventions nullifies the damaging effect of one shock. The system is said to be in Stage 1, when it can possibly heal, until the net count of impacts (valid shocks registered minus valid shocks nullified) reaches a threshold $m_1$. The system then enters Stage 2, where no further healing is possible. The system fails when the net count of valid shocks reaches another threshold $m_2 (> m_1)$. The inter-arrival times between successive valid shocks and those between successive positive interventions are independent and follow arbitrary distributions. Thus, we remove the restrictive assumption of an exponential distribution, often found in the literature. We find the distributions of the sojourn time in Stage 1 and the failure time of the system. Finally, we find the optimal values of the choice variables that minimize the expected maintenance cost per unit time for three different maintenance policies. In Chapter 4, the above defined Stage 1 is further subdivided into two parts: In the early part, called Stage 1A, healing happens faster than in the later stage, called Stage 1B. The system stays in Stage 1A until the net count of impacts reaches a predetermined threshold $m_A$; then the system enters Stage 1B and stays there until the net count reaches another predetermined threshold $m_1 (>m_A)$. Subsequently, the system enters Stage 2 where it can no longer heal. The system fails when the net count of valid shocks reaches another predetermined higher threshold $m_2 (> m_1)$. All other assumptions are the same as those in Chapter 3. We calculate the percentage improvement in the lifetime of the system due to the subdivision of Stage 1. Finally, we make optimal choices to minimize the expected maintenance cost per unit time for two maintenance policies. Next, we eliminate the restrictive assumption that all valid shocks and all positive interventions have equal magnitude, and the boundary threshold is a preset constant value. In Chapter 5, we study a system that experiences damaging external shocks of random magnitude at stochastic intervals, continuous degradation, and self-healing. The system fails if cumulative damage exceeds a time-dependent threshold. We develop a preventive maintenance policy to replace the system such that its lifetime is utilized prudently. Further, we consider three variations on the healing pattern: (1) shocks heal for a fixed finite duration $\tau$; (2) a fixed proportion of shocks are non-healable (that is, $\tau=0$); (3) there are two types of shocks---self healable shocks heal for a finite duration, and non-healable shocks. We implement a proposed preventive maintenance policy and compare the optimal replacement times in these new cases with those in the original case, where all shocks heal indefinitely. Finally, in Chapter 6, we present a summary of the dissertation with conclusions and future research potential.

<p>Recent years have seen a growth in research on system reliability and maintenance. Various studies in the scientific fields of reliability engineering, quality and productivity analyses, risk assessment, software reliability, and probabilistic machine learning are being undertaken in the present era. The dependency of human life on technology has made it more important to maintain such systems and maximize their potential. In this dissertation, some methodologies are presented that maximize certain measures of system reliability, explain the underlying stochastic behavior of certain systems, and prevent the risk of system failure.</p> <p><br></p> <p>An overview of the dissertation is provided in Chapter 1, where we briefly discuss some useful definitions and concepts in probability theory and stochastic processes and present some mathematical results required in later chapters. Thereafter, we present the motivation and outline of each subsequent chapter.</p> <p><br></p> <p>In Chapter 2, we compute the limiting average availability of a one-unit repairable system subject to repair facilities and spare units. Formulas for finding the limiting average availability of a repairable system exist only for some special cases: (1) either the lifetime or the repair-time is exponential; or (2) there is one spare unit and one repair facility. In contrast, we consider a more general setting involving several spare units and several repair facilities; and we allow arbitrary life- and repair-time distributions. Under periodic monitoring, which essentially discretizes the time variable, we compute the limiting average availability. The discretization approach closely approximates the existing results in the special cases; and demonstrates as anticipated that the limiting average availability increases with additional spare unit and/or repair facility.</p> <p><br></p> <p>In Chapter 3, the system experiences two types of sporadic impact: valid shocks that cause damage instantaneously and positive interventions that induce partial healing. Whereas each shock inflicts a fixed magnitude of damage, the accumulated effect of k positive interventions nullifies the damaging effect of one shock. The system is said to be in Stage 1, when it can possibly heal, until the net count of impacts (valid shocks registered minus valid shocks nullified) reaches a threshold $m_1$. The system then enters Stage 2, where no further healing is possible. The system fails when the net count of valid shocks reaches another threshold $m_2  (> m_1)$. The inter-arrival times between successive valid shocks and those between successive positive interventions are independent and follow arbitrary distributions. Thus, we remove the restrictive assumption of an exponential distribution, often found in the literature. We find the distributions of the sojourn time in Stage 1 and the failure time of the system. Finally, we find the optimal values of the choice variables that minimize the expected maintenance cost per unit time for three different maintenance policies.</p> <p><br></p> <p>In Chapter 4, the above defined Stage 1 is further subdivided into two parts: In the early part, called Stage 1A, healing happens faster than in the later stage, called Stage 1B. The system stays in Stage 1A until the net count of impacts reaches a predetermined threshold $m_A$; then the system enters Stage 1B and stays there until the net count reaches another predetermined threshold $m_1 (>m_A)$. Subsequently, the system enters Stage 2 where it can no longer heal. The system fails when the net count of valid shocks reaches another predetermined higher threshold $m_2 (> m_1)$. All other assumptions are the same as those in Chapter 3. We calculate the percentage improvement in the lifetime of the system due to the subdivision of Stage 1. Finally, we make optimal choices to minimize the expected maintenance cost per unit time for two maintenance policies.</p> <p><br></p> <p>Next, we eliminate the restrictive assumption that all valid shocks and all positive interventions have equal magnitude, and the boundary threshold is a preset constant value. In Chapter 5, we study a system that experiences damaging external shocks of random magnitude at stochastic intervals, continuous degradation, and self-healing. The system fails if cumulative damage exceeds a time-dependent threshold. We develop a preventive maintenance policy to replace the system such that its lifetime is utilized prudently. Further, we consider three variations on the healing pattern: (1) shocks heal for a fixed finite duration $\tau$; (2) a fixed proportion of shocks are non-healable (that is, $\tau=0$); (3) there are two types of shocks---self healable shocks heal for a finite duration, and non-healable shocks. We implement a proposed preventive maintenance policy and compare the optimal replacement times in these new cases with those in the original case, where all shocks heal indefinitely.</p> <p><br></p> <p>Finally, in Chapter 6, we present a summary of the dissertation with conclusions and future research potential.</p>