On Nonlinear Filtering Problems: Structure Theorem and A New Suboptimal Filter

In this thesis, we introduce two methods to solve the nonlinear filtering problem. In chapter 2, we extend Yau and his coauthors' work of Mitter conjecture for low dimensional algebras in nonlinear filtering problem. We prove the Mitter conjecture when the dimension $n=6$. Using this result, we give the structure theorem of six-dimensional estimation algebra. We shall show the structure of six-dimensional estimation algebra is not unique while when $n\leqslant 5$, the structure of estimation algebra is unique. It is hard to solve nonlinear filtering problem when we want to find the optimal filter. In Chapter 3, we first introduce several widely used suboptimal filters, Extended Kalman filter, Unscented Kalman filter, Ensemble Kalman filter, Particle filter, and splitting up method. Then, we introduce a new suboptimal filter for polynomial filtering problems. With our special assumption, we can construct a closed form for conditional mean and conditional moments for the state process. Numerical results show that our new suboptimal filter works perfectly

Assembly of Polymer-Grafted Magnetic Nanoparticles in Polymer Melts

Hydrophobic iron oxide nanoparticles grafted with hydrophobic polymer chains of varying molecular weights and graft densities are synthesized to underpin the role of brush entanglement and dipolar forces on creating nanostructures. Grafting density on magnetic nanoparticles is controlled in grafting-to method by changing the concentration of functionalized polymer in solution. The grafting density and brush length have varied systemically to observe the changes in nanostructures. Bridging between grafted chains and dipolar forces become effective only at low grafting density and result in long chains of particles. We demonstrate experimentally that structural transition of magnetic nanoparticles is controlled with the balance between grafted chain entanglements and dipolar forces

The “critical” point at which the noninvasive tumor growing in the ECM with switches from a dormant state to a proliferative state as functions of <i>α</i> and (a).

<p>A schematic phase diagram that characterizes the growth dynamics of a noninvasive tumor growing in the ECM with under different <i>α</i> and (b).</p

Effect of Ionic Groups on Polymer-Grafted Magnetic Nanoparticle Assemblies

Conductivity in ionomer melts is governed by the density of conducting ions and ionic aggregation within low dielectric polymers. New material design strategies are needed to direct ion aggregation by utilizing low ion densities that will improve ion conductivity in polymer composite films. Here, we report the dispersion of ionomer-grafted magnetic nanoparticles (NPs) in polymers to explore their potential in energy applications. Iron oxide NPs coated with a uniform silane layer are grafted with polystyrene (PS) chains and are randomly sulfonated to various extents. We examine the interplay between ionic interactions and chain repulsion by varying the ion concentration and length of grafted chains. Transmission electron microscopy and small-angle X-ray scattering results show that ion-containing polymer-grafted NPs form highly ordered chain-like structures below 3 mol % sulfonation in bulk at two particle loadings (5 and 15 wt %). Moreover, increasing grafted chain length leads to long-range spacing correlations between sulfonated strings. This strategy to create discrete and connected highly ordered string nanostructures can be used as a means of controlling the ion aggregation and transport in polymer nanocomposites

Upper panel: statistics of a simulated noninvasive tumor growing in the ECM with and microenvironmental suppression factors, as predicted by the “CA dormancy model”.

<p>(a) Tumor area <i>A_T</i> normalized by the area <i>A</i>₀ of the growth permitting region. (b) Areas of different cell populations normalized by the area <i>A</i>₀ of the growth permitting region. Lower panel: statistics of a simulated noninvasive tumor growing in the ECM with without suppression. (c) Tumor area <i>A_T</i> normalized by the area <i>A</i>₀ of the growth permitting region. (d) Areas of different cell populations normalized by the area <i>A</i>₀ of the growth permitting region.</p

Snapshots of a simulated noninvasive tumor growing in the ECM with on different days given by the CA dormancy model.

<p>Upper panel: Dormancy period. Lower panel: Regrowth period.</p

Predicting Energy Conversion Efficiency of Dye Solar Cells from First Principles

In this work we target on accurately predicting energy conversion efficiency of dye-sensitized solar cells (DSC) using parameter-free first principles simulations. We present a set of algorithms, mostly based on solo first principles calculations within the framework of density functional theory, to accurately calculate key properties in energy conversion including sunlight absorption, electron injection, electron–hole recombination, open circuit voltages, and so on. We choose two series of donor-π-acceptor dyes with detailed experimental photovoltaic data as prototype examples to show how these algorithms work. Key parameters experimentally measured for DSC devices can be nicely reproduced by first-principles with as less empirical inputs as possible. For instance, short circuit current of model dyes can be well reproduced by precisely calculating their absorption spectra and charge separation/recombination rates. Open circuit voltages are evaluated through interface band offsets, namely, the difference between the Fermi level of electrons in TiO₂ and the redox potential of the electrolyte, after modification with empirical formulas. In these procedures the critical photoelectron injection and recombination dynamics are calculated by real-time excited state electronic dynamics simulations. Estimated solar cell efficiency reproduces corresponding experimental values, with errors usually below 1–2%. Device characteristics such as light harvesting efficiency, incident photon-to-electron conversion efficiency, and the current–voltage characteristics can also be well reproduced and compared with experiment. Thus, we develop a systematic ab initio approach to predict solar cell efficiency and photovoltaic performance of DSC, which enables large-scale efficient dye screening and optimization through high-throughput first principles calculations with only a few parameters taken from experimental settings for electrode and electrolyte toward a renewable energy based society

Parameters characterizing the interactions between tumor suppression factors and tumor cells in the CA dormancy model.

<p>Note that the two “critical threshold” parameters themselves do not incorporate any additional CA rules.</p><p>Parameters characterizing the interactions between tumor suppression factors and tumor cells in the CA dormancy model.</p

Design of Ion-Containing Polymer-Grafted Nanoparticles for Conductive Membranes

While sulfonated polymers are commonly used in membranes for fuel cells and water filtration applications, challenges of controlling ionic aggregation and understanding morphology effects on conductivity and transport still remain. In this work, we investigate the aggregation of copolymer-grafted nanoparticles that are designed to form conductive structures with low sulfonation amounts of chains. We demonstrate that long grafts of polystyrene chains with sulfonated end groups form side-by-side aggregated strings and retain their structures in ionic liquid, 1-hexyl-3-methylimidazolium bis­(trifluoro­methyl­sulfonyl)­imide, [HMIM]­[TFSI]. Transmission electron tomography results revealed that these aggregates are monolayers of particles at low sulfonations and planar-like networks at 3 mol % sulfonation in the ionic liquid. Organization of magnetic nanoparticles with the polymer grafting approach is shown, for the first time, to enhance conductivity upon incorporation of an ionic liquid

Tumor area <i>A_T</i> normalized by the area <i>A</i>₀ of the growth permitting region of a simulated noninvasive tumor growing in the ECM under different killing rates by microenvironmental suppression factors.

<p>The parameter <i>k</i>₀ is the fraction that the suppression factors from the microenvironment kill the actively dividing proliferative cells when the suppression factors counteract these cells.</p