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.

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.

The aquatic biota of the now extinct lacustrine complex of the Mexico basin

The commonest organisms of the original Mexico lake complex are listed, including those that exist today in the Lago Viejo. In addition, a brief hydraulic history of this endorheic basin is given.

Models of the Oxygen-Evolving Complex of Photosystem II

In the five chapters that follow, I delineate my efforts over the last five years to synthesize structurally and chemically relevant models of the Oxygen Evolving Complex (OEC) of Photosystem II. The OEC is nature’s only water oxidation catalyst, in that it forms the dioxygen in our atmosphere necessary for oxygenic life. Therefore understanding its structure and function is of deep fundamental interest and could provide design elements for artificial photosynthesis and manmade water oxidation catalysts. Synthetic endeavors towards OEC mimics have been an active area of research since the mid 1970s and have mutually evolved alongside biochemical and spectroscopic studies, affording ever-refined proposals for the structure of the OEC and the mechanism of water oxidation. This research has culminated in the most recent proposal: a low symmetry Mn₄CaO₅ cluster with a distorted Mn₃CaO₄ cubane bridged to a fourth, dangling Mn. To give context for how my graduate work fits into this rich history of OEC research, Chapter 1 provides a historical timeline of proposals for OEC structure, emphasizing the role that synthetic Mn and MnCa clusters have played, and ending with our Mn₃CaO₄ heterometallic cubane complexes.

In Chapter 2, the triarylbenzene ligand framework used throughout my work is introduced, and trinuclear clusters of Mn, Co, and Ni are discussed. The ligand scaffold consistently coordinates three metals in close proximity while leaving coordination sites open for further modification through ancillary ligand binding. The ligands coordinated could be varied, with a range of carboxylates and some less coordinating anions studied. These complexes’ structures, magnetic behavior, and redox properties are discussed.

Chapter 3 explores the redox chemistry of the trimanganese system more thoroughly in the presence of a fourth Mn equivalent, finding a range of oxidation states and oxide incorporation dependent on oxidant, solvent, and Mn salt. Oxidation states from Mn^II₄ to Mn^IIIMn^IV₃ were observed, with 1-4 O^2– ligands incorporated, modeling the photoactivation of the OEC. These complexes were studied by X-ray diffraction, EPR, XAS, magnetometry, and CV.

As Ca²⁺ is a necessary component of the OEC, Chapter 4 discusses synthetic strategies for making highly structurally accurate models of the OEC containing both Mn and Ca in the Mn₃CaO₄ cubane + dangling Mn geometry. Structural and electrochemical characterization of the first Mn₃CaO₄ heterometallic cubane complex— and comparison to an all-Mn Mn₄O₄ analog—suggests a role for Ca²⁺ in the OEC. Modification of the Mn₃CaO₄ system by ligand substitution affords low symmetry Mn₃CaO₄ complexes that are the most accurate models of the OEC to date.

Finally, in Chapter 5 the reactivity of the Mn₃CaO₄ cubane complexes toward O- atom transfer is discussed. The metal M strongly affects the reactivity. The mechanisms of O-atom transfer and water incorporation from and into Mn₄O₄ and Mn₄O₃ clusters, respectively, are studied through computation and ¹⁸O-labeling studies. The μ3-oxos of the Mn₄O₄ system prove fluxional, lending support for proposals of O^2– fluxionality within the OEC.

MODIFIED DIRECT TWOS-COMPLEMENT PARALLEL ARRAY MULTIPLICATION ALGORITHM FOR COMPLEX MATRIX OPERATION

A direct twos-complement parallel array multiplication algorithm is introduced and modified for digital optical numerical computation. The modified version overcomes the problems encountered in the conventional optical twos-complement algorithm. In the array, all the summands are generated in parallel, and the relevant summands having the same weights are added simultaneously without carries, resulting in the product expressed in a mixed twos-complement system. In a two-stage array, complex multiplication is possible with using four real subarrays. Furthermore, with a three-stage array architecture, complex matrix operation is straightforwardly accomplished. In the experiment, parallel two-stage array complex multiplication with liquid-crystal panels is demonstrated.

Studies on the complex formation of iron with humic substances [Translation from: Vom Wasser 53, 243-247, 1979]

Most of the humic substances which occur in natural waters have an iron content of a few percent, indicated by the mg/1 content of organically-bonded carbon. This iron is apparently bound in a complex with the humic substances, for it quite plainly differs in its chemical and physico-chemical properties from what one would expect from the purely inorganic iron-water system. The deviations range from the solubility to the redox behaviour, and thus are frequently the basis of analytical and technical difficulties. The key to the solution of most of this problem lies in a better understanding of the aforementioned bonds between the iron and the humic substances. This paper studies the iron content of the humic substance concentration from a bog lake sample and the complexing of iron by humic substances from the surface of the bog lake.

Three-dimensional superresolution by three-zone complex pupil filters

Complex pupil filters are introduced to improve the three-dimensional resolving power of an optical imaging system. Through the design of the essential parameters of such filters, the transmittance and radius of the first zone, three-dimensional superresolution is realized. The Strehl ratio and the transverse and axial gains of such filters are analyzed in detail. A series of simulation examples of such filters are also presented that prove that three-dimensional superresolution can be realized. The advantage of such filters is that it is easy to realize three-dimensional superresolution, and the disadvantage is that the sidelobes of the axial intensity distribution are too high. But this can be overcome by the application of a confocal system. (C) 2005 Optical Society of America.

The complex angular momentum theory of the production of three particles in collisions of two strongly interacting particles at high energy

The problem of the continuation to complex values of the angular momentum of the partial wave amplitude is examined for the simplest production process, that of two particles → three particles. The presence of so-called "anomalous singularities" complicates the procedure followed relative to that used for quasi two-body scattering amplitudes. The anomalous singularities are shown to lead to exchange degenerate amplitudes with possible poles in much the same way as "normal" singularities lead to the usual signatured amplitudes. The resulting exchange-degenerate trajectories would also be expected to occur in two-body amplitudes.

The representation of the production amplitude in terms of the singularities of the partial wave amplitude is then developed and applied to the high energy region, with attention being paid to the emergence of "double Regge" terms. Certain new results are obtained for the behavior of the amplitude at zero momentum transfer, and some predictions of polarization and minima in momentum transfer distributions are made. A calculation of the polarization of the ρ^o meson in the reaction π ^- p → π ^- ρ^op at high energy with small momentum transfer to the proton is compared with data taken at 25 Gev by W. D. Walker and collaborators. The result is favorable, although limited by the statistics of the available data.

Ceratium hirundinella - ecology of a complex, mobile, and successful plant

Recent work carried out in the English Lake District (Esthwaite Water and Blelham Tarn) is reported. The seasonal growth cycle, diel growth cycle, photosynthesis, vertical distribution and migrations, horizontal distribution, and the interaction of environmental factors, were investigated.