Random propagation in complex systems : nonlinear matrix recursions and epidemic spread

This dissertation studies long-term behavior of random Riccati recursions and mathematical epidemic model. Riccati recursions are derived from Kalman filtering. The error covariance matrix of Kalman filtering satisfies Riccati recursions. Convergence condition of time-invariant Riccati recursions are well-studied by researchers. We focus on time-varying case, and assume that regressor matrix is random and identical and independently distributed according to given distribution whose probability distribution function is continuous, supported on whole space, and decaying faster than any polynomial. We study the geometric convergence of the probability distribution. We also study the global dynamics of the epidemic spread over complex networks for various models. For instance, in the discrete-time Markov chain model, each node is either healthy or infected at any given time. In this setting, the number of the state increases exponentially as the size of the network increases. The Markov chain has a unique stationary distribution where all the nodes are healthy with probability 1. Since the probability distribution of Markov chain defined on finite state converges to the stationary distribution, this Markov chain model concludes that epidemic disease dies out after long enough time. To analyze the Markov chain model, we study nonlinear epidemic model whose state at any given time is the vector obtained from the marginal probability of infection of each node in the network at that time. Convergence to the origin in the epidemic map implies the extinction of epidemics. The nonlinear model is upper-bounded by linearizing the model at the origin. As a result, the origin is the globally stable unique fixed point of the nonlinear model if the linear upper bound is stable. The nonlinear model has a second fixed point when the linear upper bound is unstable. We work on stability analysis of the second fixed point for both discrete-time and continuous-time models. Returning back to the Markov chain model, we claim that the stability of linear upper bound for nonlinear model is strongly related with the extinction time of the Markov chain. We show that stable linear upper bound is sufficient condition of fast extinction and the probability of survival is bounded by nonlinear epidemic map.

New insights into the formation and modification of carbonate-bearing minerals and methane gas in geological systems using multiply substituted isotopologues

This thesis describes the use of multiply-substituted stable isotopologues of carbonate minerals and methane gas to better understand how these environmentally significant minerals and gases form and are modified throughout their geological histories. Stable isotopes have a long tradition in earth science as a tool for providing quantitative constraints on how molecules, in or on the earth, formed in both the present and past. Nearly all studies, until recently, have only measured the bulk concentrations of stable isotopes in a phase or species. However, the abundance of various isotopologues within a phase, for example the concentration of isotopologues with multiple rare isotopes (multiply substituted or 'clumped' isotopologues) also carries potentially useful information. Specifically, the abundances of clumped isotopologues in an equilibrated system are a function of temperature and thus knowledge of their abundances can be used to calculate a sample’s formation temperature. In this thesis, measurements of clumped isotopologues are made on both carbonate-bearing minerals and methane gas in order to better constrain the environmental and geological histories of various samples.

Clumped-isotope-based measurements of ancient carbonate-bearing minerals, including apatites, have opened up paleotemperature reconstructions to a variety of systems and time periods. However, a critical issue when using clumped-isotope based measurements to reconstruct ancient mineral formation temperatures is whether the samples being measured have faithfully recorded their original internal isotopic distributions. These original distributions can be altered, for example, by diffusion of atoms in the mineral lattice or through diagenetic reactions. Understanding these processes quantitatively is critical for the use of clumped isotopes to reconstruct past temperatures, quantify diagenesis, and calculate time-temperature burial histories of carbonate minerals. In order to help orient this part of the thesis, Chapter 2 provides a broad overview and history of clumped-isotope based measurements in carbonate minerals.

In Chapter 3, the effects of elevated temperatures on a sample’s clumped-isotope composition are probed in both natural and experimental apatites (which contain structural carbonate groups) and calcites. A quantitative model is created that is calibrated by the experiments and consistent with the natural samples. The model allows for calculations of the change in a sample’s clumped isotope abundances as a function of any time-temperature history.

In Chapter 4, the effects of diagenesis on the stable isotopic compositions of apatites are explored on samples from a variety of sedimentary phosphorite deposits. Clumped isotope temperatures and bulk isotopic measurements from carbonate and phosphate groups are compared for all samples. These results demonstrate that samples have experienced isotopic exchange of oxygen atoms in both the carbonate and phosphate groups. A kinetic model is developed that allows for the calculation of the amount of diagenesis each sample has experienced and yields insight into the physical and chemical processes of diagenesis.

The thesis then switches gear and turns its attention to clumped isotope measurements of methane. Methane is critical greenhouse gas, energy resource, and microbial metabolic product and substrate. Despite its importance both environmentally and economically, much about methane’s formational mechanisms and the relative sources of methane to various environments remains poorly constrained. In order to add new constraints to our understanding of the formation of methane in nature, I describe the development and application of methane clumped isotope measurements to environmental deposits of methane. To help orient the reader, a brief overview of the formation of methane in both high and low temperature settings is given in Chapter 5.

In Chapter 6, a method for the measurement of methane clumped isotopologues via mass spectrometry is described. This chapter demonstrates that the measurement is precise and accurate. Additionally, the measurement is calibrated experimentally such that measurements of methane clumped isotope abundances can be converted into equivalent formational temperatures. This study represents the first time that methane clumped isotope abundances have been measured at useful precisions.

In Chapter 7, the methane clumped isotope method is applied to natural samples from a variety of settings. These settings include thermogenic gases formed and reservoired in shales, migrated thermogenic gases, biogenic gases, mixed biogenic and thermogenic gas deposits, and experimentally generated gases. In all cases, calculated clumped isotope temperatures make geological sense as formation temperatures or mixtures of high and low temperature gases. Based on these observations, we propose that the clumped isotope temperature of an unmixed gas represents its formation temperature — this was neither an obvious nor expected result and has important implications for how methane forms in nature. Additionally, these results demonstrate that methane-clumped isotope compositions provided valuable additional constraints to studying natural methane deposits.

Direct simulation of quantum dynamics in complex systems

Accurate simulation of quantum dynamics in complex systems poses a fundamental theoretical challenge with immediate application to problems in biological catalysis, charge transfer, and solar energy conversion. The varied length- and timescales that characterize these kinds of processes necessitate development of novel simulation methodology that can both accurately evolve the coupled quantum and classical degrees of freedom and also be easily applicable to large, complex systems. In the following dissertation, the problems of quantum dynamics in complex systems are explored through direct simulation using path-integral methods as well as application of state-of-the-art analytical rate theories.

On the Role of Delays in Biological Systems : Analysis and Design

This work quantifies the nature of delays in genetic regulatory networks and their effect on system dynamics. It is known that a time lag can emerge from a sequence of biochemical reactions. Applying this modeling framework to the protein production processes, delay distributions are derived in a stochastic (probability density function) and deterministic setting (impulse function), whilst being shown to be equivalent under different assumptions. The dependence of the distribution properties on rate constants, gene length, and time-varying temperatures is investigated. Overall, the distribution of the delay in the context of protein production processes is shown to be highly dependent on the size of the genes and mRNA strands as well as the reaction rates. Results suggest longer genes have delay distributions with a smaller relative variance, and hence, less uncertainty in the completion times, however, they lead to larger delays. On the other hand large uncertainties may actually play a positive role, as broader distributions can lead to larger stability regions when this formalization of the protein production delays is incorporated into a feedback system.

Furthermore, evidence suggests that delays may play a role as an explicit design into existing controlling mechanisms. Accordingly, the reccurring dual-feedback motif is also investigated with delays incorporated into the feedback channels. The dual-delayed feedback is shown to have stabilizing effects through a control theoretic approach. Lastly, a distributed delay based controller design method is proposed as a potential design tool. In a preliminary study, the dual-delayed feedback system re-emerges as an effective controller design.

Bacterial production of nitrate in aquatic systems: progress report. Nitrate-loss processes in raw water

This short interim progress report builds on previous progress reports which have described the quantification of the process both within and between lakes of different degrees of eutrophication. These data indicated that slight changes in methodology, particularly when investigating sediment deposits, could grossly affect the measured activity. The aim of the present research was an attempt to rationalize these differences. If this could be achieved it would enable meaningful interpretation of published data obtained using different methods and therefore enlarge the available database. In addition some observations have been made on the production of nitrite by Grasmere profundal sediment slurries sampled during the circulation period.