This thesis develops ways of localizing the Ritz values of non-Hermitian matrices. The restarted Arnoldi method with exact shifts, useful for determining a few desired eigenvalues of a matrix, employs Ritz values to refine eigenvalue estimates. In the Hermitian case, using selected Ritz values produces convergence due to interlacing. No generalization of interlacing exists for non-Hermitian matrices, and as a consequence no satisfactory general convergence theory exists. To study Ritz values, I propose the inverse field of values problem for k Ritz values, which asks if a set of k complex numbers can be Ritz values of a matrix. This problem is always solvable for k = 1 for any complex number in the field of values; I provide an improved algorithm for finding a Ritz vector in this case. I show that majorization can be used to characterize, as well as localize, Ritz values. To illustrate the difficulties of characterizing Ritz values, this work provides a complete analysis of the Ritz values of two 3 × 3 matrices: a Jordan block and a normal matrix. By constructing conditions for localizing the Ritz values of a matrix with one simple, normal, sought-after eigenvalue, this work develops sufficient conditions that guarantee convergence of the restarted Arnoldi method with exact shifts. For general matrices, the conditions provide insight into the subspace dimensions that ensure that shifts do not cluster near the wanted eigenvalue. As Ritz values form the basis for many iterative methods for determining eigenvalues and solving linear systems, an understanding of Ritz value behavior for non-Hermitian matrices has the potential to inform a broad range of analysis.

In this thesis, the feasibility of employing organometallic single-source precursors in the preparation of advanced transition metal pnictide materials such as colloidal nanoparticles and films has been investigated. In particular, the ternary FeMnP phase was targeted as a model for preparing advanced heterobimetallic phosphide materials, and the iron-rich Fe 3 P phase was targeted due to its favorable ferromagnetic properties as well as the fact that the preparation of advanced Fe 3 P materials has been elusive by commonly used methods. Progress towards the synthesis of advanced Fe 2-x Mn x P nanomaterials and films was facilitated by the synthesis of the novel heterobimetallic complexes FeMn(CO) 8 (μ-PR 1 R 2 ) (R 1 = H, R 2 = H or R 1 = H, R 2 = Ph), which contain the relatively rare μ-PH2 and μ-PPhH functionalities. Iron rich Fe 2-x Mn x P nanoparticles were obtained by thermal decomposition of FeMn(CO) 8 (μ-PH 2 ) using solution-based synthetic methods, and empirical evidence suggested that oleic acid was responsible for manganese depletion. Films containing Fe, Mn, and P with the desired stoichiometric ratio of 1:1:1 were prepared using FeMn(CO) 8 (μ-PH 2 ) in a simple low-pressure metal-organic chemical vapor deposition (MOCVD) apparatus. Although the elemental composition of the precursor was conserved in the deposited film material, spectroscopic evidence indicated that the films were not composed of pure-phase FeMnP, but were actually mixtures of crystalline FeMnP and amorphous FeP and Mn x O y . A new method for the preparation of phase-pure ferromagnetic Fe 3 P films on quartz substrates has also been developed. This approach involved the thermal decomposition of the single-source precursors H 2 Fe 3 (CO) 9 PR (R = t Bu or Ph) at 400 °C. The films were deposited using a simple home-built MOCVD apparatus and were characterized using a variety of analytical methods. The films exhibited excellent phase purity, as evidenced by X-ray diffraction, X-ray photoelectron spectroscopy, and field-dependent magnetization measurements, the results of which were all in good agreement with measurements obtained from bulk Fe 3 P. As-deposited Fe 3 P films were found to be amorphous, and little or no magnetic hysteresis was observed in plots of magnetization versus applied field. Annealing the Fe 3 P films at 550 °C resulted in improved crystallinity as well as the observation of magnetic hysteresis.

Mathematical modeling has made great strides since the Lotka-Volterra predator-prey models. Newer models attempt to describe sub-cellular signal transduction pathways, such as the JAK-STAT and NF-κB pathways. However, the tools to accurately determine reaction and translocation rates in these pathways still have a number of drawbacks, including the effects of concentration scale on determining reaction rates and the effects of bulky additions to translocation rates. One method of overcoming these problems in signal transduction rate determination is to sample and stain cells from a full population at specific time points. However, fixed cell methods can only generate an average population rate. This could become an issue if the rate depends on the genotype of one of the proteins in the pathway. Another method of overcoming these problems in signal transduction rates is to use unmarked nuclei in live-cell imaging techniques. However, live cell imaging methods poses different problems, primarily how to find and track nuclei and cytoplasm when cells are actively moving and the nuclear and cytoplasmic intensities are by necessity fluctuating. To date, there is only one software package designed for tracking cells under these conditions - Cell Tracker (Shen et al., 2006). Cell Tracker is designed to handle the tracking of live cell images for protein translocation studies. They recommend using a separate color channel to mark the nucleus, although results can be obtained using unmarked nuclei. The results from Cell Tracker with unmarked nuclei are often less than optimal. We have developed a novel segmentation scheme and variation of the particle filter algorithm to allow more accurate tracking in time series with unmarked nuclei. The proposed segmentation scheme uses a non-parametric level set algorithm to refine a fast initial thresholding step. The tracking scheme uses a dense optical flow calculation to assist the particle filter algorithm in continuing to follow the true positions of the nuclei. To test the proposed algorithm, a novel mimicry of cell movement has been developed using random perturbations of a triangular mesh structure through the use of the finite element method.

Existing growth methods produce single-walled carbon nanotubes (SWCNTs) with a range of structures and electronic properties, but many potential applications require pure nanotube samples. Density gradient ultracentrifugation (DGU) has recently emerged as a technique for sorting as-grown mixtures of single-walled nanotubes into their distinct ( n,m ) structural forms, but this approach has been limited to samples containing only a small number of nanotube structures, and has often required repeated DGU processing. For the first time, it has been shown that the use of tailored nonlinear density gradient ultracentrifugation (NDGU) can significantly improve DGU separations. This new sorting process readily separated highly polydisperse samples of SWCNTs grown by the HiPco method in a single step to give fractions enriched in any of ten different ( n,m ) species. In addition, minor variants of the method allowed separation of the minor-image isomers (enantiomers) of seven ( n,m ) species. Optimization of this new approach was aided by the development of instrumentation that spectroscopically mapped nanotube contents inside undisturbed centrifuge tubes. Besides, sorted nanotube samples enabled the discovery of novel oxygen-doped SWCNTs with remarkable photophysical properties. Modified nanotube samples were produced using mild oxidation of SWCNTs with ozone followed by a photochemical conversion step that induced well-defined changes in emissive properties. As demonstrated for a set of ten separated SWCNT ( n,m ) structures, chemically altered nanotubes possess slightly lower band gap energies with correspondingly longer photoluminescence wavelengths. Treated samples showed distinct, structure-specific near-infrared fluorescence at wavelengths 10 to 15% longer than the pristine semiconducting SWCNTs. Quantum chemical modeling suggests that dopant sites harvest light energy absorbed in undoped nanotube regions by trapping mobile excitons. The oxygen-doped SWCNTs are much easier to detect and image in biological specimen than pristine SWCNTs because they give stronger near-IR emission and do not absorb at the shifted emission wavelength. This novel modification of SWCNT properties may lead to new optical and electronic applications, as it provides a way to change optical band gaps in whole nanotubes or in selected sections.

Carbon nanotubes have been of great interest given their unique electronic and mechanical properties. Scholars have focused on the addition of carbon nanotubes to various matrices in order to develop novel materials. These new hybrid materials would combine the properties of both the nanotubes and the matrix of choice, which can both enhance the mechanical and electronic properties of the matrix material, and allow for the matrix to be used for other applications. In order to take advantage of the properties of the nanotubes, it is vital for them to be well dispersed in a solution or matrix as individual tubes, rather than as bundles. Additionally, it is cost effective to have individually dispersed tubes in a matrix. In order to individually disperse the tubes in the matrix, they can be pre-treated or functionalized via both covalent and non covalent processes. Subsequent to functionalization, the nanotubes can be coated with the matrix material or other metal compounds. This can help with the dispersion and interface interaction with the matrix material, or create materials with novel properties. This thesis focuses on conditions of growing various metal compounds or metal oxides on nanotubes using chemical bath deposition (CBD) and liquid phase deposition (LPD) methods. CBD and LPD use aqueous mediums for growth and deposition of compounds, which makes it both environmentally friendly and cost effective. Different pre-treatments are first employed on the nanotubes in order for them to be both well dispersed in solution and provide nucleation sites for the deposition and growth of various metal and metal oxides on the surface of the nanotubes. Once an ideal deposition is achieved, applications of the coated tubes are studied, tested and discussed.

The cohesin complex is responsible for the fidelity of chromosomal segregation during mitosis. It consists of four core subunits namely Rad21/Mcd1/Sccl, Smc1, Smc3 and one of the yeast Scc3 orthologs SA1 or SA2. Sister chromatid cohesion is formed by the cohesin complex during DNA replication and maintained until the onset of anaphase. Among the many proposed models of how cohesin holds sister chromatids together, the 'core' cohesin subunits Smc1, Smc3 and Rad21/Mcd1/Scc1 are almost universally displayed as forming a contiguous ring. However, other than its supportive role in the cohesin ring, little is known about the fourth core protein SA1/SA2 - despite its physical association to the cohesin ring. To gain deeper insight into how physically and physiologically SA2 interacts with the cohesin complex, we performed structural characterization of SA2 and Rad21 and mapped the interaction region of the two proteins in vitro and ex vivo . We found SA2 interacts with Rad21 at multiple domains while Rad21 only interacts with SA2 through a 10 amino acid α-helical motif from 383-392aa. Deletion of these 10 amino acids or mutation of three conserved amino acids (L385, F389, and T390) in this α-helical motif prevents Rad21 from physically interacting with SA2/SA1 and causes premature sister chromatid separation in mitotic cells that often leads to aneuploidy. Our studies provide a model for how SA2 structurally strengthens the cohesin ring through its interaction with Rad21. Results from our structural characterization of these two proteins also provided directions for further investigation of the structural basis of protein-protein interaction in the cohesin complex.

The quantum Spin Hall Insulator (QSHI) is a two-dimensional variant of a novel class of materials characterized by topological order, whose unique properties have recently triggered much interest and excitement in the condensed matter community. Most notably, the topological properties of these systems hold great promise in mitigating the difficult problem of decoherence in implementations of quantum computers. Although QSHI has been theoretically predicted in a few different materials, prior to the work presented in this thesis, only the HgTe/CdTe semiconductor system has shown direct evidence for the existence of this phase. Ideally insulating in the bulk, QSHI is characterized by one-dimensional channels at the sample perimeter, which have a helical property, with carrier spin tied to the carrier direction of motion, and protected from elastic back-scattering by time-reversal symmetry. In this thesis we present low temperature transport measurements, showing strong evidence for the existence of proposed helical edge channels in InAs/CaSb quantum wells, which thus emerge as an important alternate to HgTe/CdTe quantum wells in studies of two-dimensional topological insulators and superconductors. Surprisingly, edge modes persist in spite of comparable bulk conduction of non-trivial origin and show only weak dependence on magnetic field in mesoscopic devices. We elucidate that the seeming independence of edge on bulk transport comes due to the disparity in Fermi wave-vectors between the bulk and the edge, leading to a total internal reflection of the edge modes. Furthermore, low Schottky barrier of this material system and good interface to superconductors allows us to probe topological properties of helical channels in Andreev reflection measurements, opening a promising route towards the realization of topologically superconducting phases hosting exotic Majorana modes.

Amyloid-proteins are among the most actively researched biological topics today, because they have been associated with many serious human diseases, such as Alzheimer's disease and type II diabetes. In particular the deposition of protein aggregates on cell membranes has been suspected as the causes of the diseases, although the proof is still elusive. Studying the interactions of amyloid-forming peptides with lipid-bilayer membranes may clarify the pathway of the β-aggregate formation and provide new insights into the amyloid hypothesis of diseases. In this thesis, I investigate how three peptides, penetratin, amylin, and LL-37, interact with lipid membranes by using several techniques well-developed in our lab. In the study of penetratin interacting with lipid membranes, we were able to clarify the energy pathway of amyloid formation mediated by membrane-binding. This provides the sole experimental proof for the Jarrett-Lansbury theory of β- amyloid formation. Our investigation on amylin-membrane interaction clarifies how amylin in different forms damage bilayer membranes. Between penetratin and amylin we have clarified the complicated pattern of interactions between amyloid-forming peptides and lipid bilayers. The third peptide LL-37 studied in my thesis turned out to a pore forming peptide. I found the mistake made by previous investigators in several different laboratories that made them erroneously conclude that LL-37 was not a pore forming peptide. The results of these three peptides show that methods we used are a comprehensive set of tools that can reveal a broad range of peptide properties. Both the formation of amyloid aggregates and formation of membrane pores can be explained by a two-state model proposed by Huang describing peptide-membrane interactions. For LL-37, the second state is a pore in membrane. But for penetratin and amylin the second state is an aggregation in the β form. We found that β-aggregates have low affinity within a lipid bilayer, and therefore exit from the bilayer structure. However, this exit process extracts lipid molecules from the bilayer and incorporates them in the peptide aggregates. We suggest that this is the molecular process of how amylin might damage of the membranes of β-cells.

This thesis presents the experimental study on the two-spin component, strongly interacting 6 Li Fermi gases in 3D and 1D traps. The interaction strength is tuned from the molecular BEC regime to the BCS regime using a Feshbach resonance. The trap dimension can be tuned from 3D to 1D with the implementation of optical lattice. The evaporation of imbalanced Fermi gases in 3D trap is studied. The anisotropic and fast evaporation is the cause of the deformation observed in the 2006 Rice experiment. In a balanced Fermi system, the fraction of correlated states is measured as a function of interaction and temperature. At unitarity, the fraction of correlated states is ∼85% and exists above T c . The one-body-like photoexcitation rate can be related to the contact quantity. Lastly, the spin-imbalance in a one-dimensional Fermi gas is studied. The 1D phase diagram is mapped out. The result agrees well with the 1D theory, in which the partially polarized regime is predicted to be a FFLO phase, an exotic superfluid with pairs carrying finite center-of-mass momentum proposed almost 50 years ago.

This dissertation describes the development of a rapid method for determining the acetylation and methylation sites of core histories (H2B, H2A, H3 and H4) that gives the relative abundance of the acetylation isoforms by a method superior to the traditional micro-chemical sequencing method. The method created in this dissertation emphasizes speed, sensitivity and general applicability to all sorts of histone acetylation (methylation) scenarios. In this study, mass spectrometry method was first used to identify histone acetylation and methylation sites. Although there are 15 possible acetylation sites, only four acetylated peptide sequences were observed. The tetra-acetylated form is acetylated at lysines 5, 8, 12 and 16, the tri-acetylated form is mostly acetylated at lysines 8, 12 and 16, and the diacetylated form is acetylated at lysine 12 and 16. The only significant amount of mono-acetylated form is modified at lysine 16. These results are consistent with the hypothesis of a “zip” model whereby acetylation of histone H4 proceeds in the direction from Lys16 to Lys5 and deacetylation goes in the reverse direction. This acetylation pattern is conserved in mammal species including HeLa cells, colon carcinoma cells and chicken erythrocyte. Core histones from chicken erythrocyte were isolated by the acid precipitation method. LC/MS simultaneously identified each subclass of histories and its potential acetylated or methylated isoforms. Acetylation and methylation sites of chicken core histories were identified. Histone H4 is acetylated at lysine residues 16, 12, 8 and 5 in the direction from lysine 16 to 5 as observed in butyrate treated HeLa cells. Lysine 20 in all isoforms of histone H4 is predominately di-methylated. About 17% of H2A is acetylated with a roughly even distribution of mono-acetylated at lysine 9 and di-acetylated at lysines 5 and 9. Lysine residues 18 and 23 in H3 are acetylated in the direction from lysine 23 to lysine 18 based on the observation that if lysine 18 is acetylated, lysine 23 also acetylated. Lysine 14 was determined to be partially acetylated (two thirds), partially methylated (one third) by high accuracy mass measurement. Lysine 4 of H3 is un- and mono-methylated. Lysine residues 9, 27 and 36 in H3 are un-, mono-, di- and trimethylated while lysine 79, a newly found methylation site for H3, is un, mono- and dimethylated. H2B is partially methylated (about 50%) at lysine 30 with a rough distribution of 1 : 0.4 : 0.3 : 0.2 for un-, mono-, di- and tri-methylated forms. (Abstract shortened by UMI.)