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Spectral Theory for Generalized Bounded Variation Perturbations of Orthogonal Polynomials and Schrödinger Operators
Get PDFThe purpose of this text is to present some new results in the spectral theory of orthogonal polynomials and Schrodinger operators.
These results concern perturbations of the free Schrodinger operator and of the free case for orthogonal polynomials on the unit circle (which corresponds to Verblunsky coefficients equal to 0) and the real line (which corresponds to off-diagonal Jacobi coefficients equal to 1 and diagonal Jacobi coefficients equal to 0).
The condition central to our results is that of generalized bounded variation. This class consists of finite linear combinations of sequences of rotated bounded variation with an L¹ perturbation.
This generalizes both usual bounded variation and expressions of the form λ(x) cos(φx + α) with λ(x) of bounded variation (and, in particular, with λ(x) = xγ, Wigner-von Neumann potentials) as well as their finite linear combinations.
Assuming generalized bounded variation and an Lp condition (with any finite p) on the perturbation, our results show preservation of absolutely continuous spectrum, absence of singular continuous spectrum, and that embedded pure points in the continuous spectrum can only occur in an explicit finite set.</p
Simulation Capabilities for Challenging Medical Imaging and Treatment Planning Problems
Get PDFAdvanced numerical solvers and associated simulation tools, such as, for example, numerical algorithms based on novel spectral methods, efficient time-stepping and domain meshing techniques for solution of Partial Differential Equations (PDEs) (enabling, in particular, effective resolution of extremely steep boundary layers in short computing times), can have a significant impact in the design of medical procedures. In this thesis we present three recently introduced numerical algorithms for medical problems whose performance improves significantly over those of earlier counterparts, and which can thereby provide solutions to a range of challenging computational problems for planning and design of medical treatments
Synthesis of Interlocked Molecules by Olefin Metathesis
Get PDFA large body of work in the Grubbs group has focused on the development of functional-group tolerant ruthenium alkylidene catalysts that perform a number of olefin metathesis reactions. These catalysts have seen application in a wide range of fields, including classic total synthesis as well as polymer and materials chemistry. One particular family of compounds, interlocked molecules, has benefitted greatly from these advances in catalyst stability and activity. This thesis describes several elusive and challenging interlocked architectures whose syntheses have been realized through the utilization of different types of ruthenium-catalyzed olefin metathesis reactions.
Ring-closing olefin metathesis has enabled the synthesis of a [c2]daisy-chain dimer with the ammonium binding site near the cap of the dimer. A deprotonated DCD possessing such a structural attribute will more forcefully seek to restore coordinating interactions upon reprotonation, enhancing its utility as a synthetic molecular actuator. Dimer functionalization facilitated incorporation into linear polymers, with a 48% size increase of an unbound, extended analogue of the polymer demonstrating slippage of the dimer units. Ongoing work is directed at further materials studies, in particular, exploring the synthesis of macroscopic networks containing the DCD units and analyzing the correlation between molecular-scale extension-contraction manipulations and resulting macro-scale changes.
A "clipping" approach to a polycatenated cyclic polymer, a structure that resembles a molecular "charm bracelet", has been described. The use of ring-opening metathesis polymerization of a carbamate monomer in the presence of a chain transfer agent allowed for the synthesis of a linear polymer that was subsequently functionalized and cyclized to the corresponding cyclic analogue. This cyclic polymer was characterized through a variety of techniques, and subjected to further functionalization reactions, affording a cyclic polyammonium scaffold. Diolefin polyether fragments were coordinated and "clipped" around the ammonium sites within the polymer backbone using ring-closing olefin metathesis, giving the molecular "charm bracelet". Confirmation of the interlocked nature of the product was achieved via 1H NMR spectroscopy and two-dimensional diffusion ordered NMR spectroscopy.
A simple strategy for a one-pot, multi-component synthesis of polyrotaxanes using acyclic diene metathesis polymerization was developed. The polyrotaxanes were characterized by traditional 1H NMR spectroscopy as well as size exclusion chromatography, and the interlocked topology was confirmed using two-dimension diffusion-ordered NMR spectroscopy. The dynamic, self-correcting nature of the ADMET polymerization was also explored through the equilibration of a capped polyammonium polymer in the presence of dibenzo-24-crown-8 ether and olefin metathesis catalysts. The efficiency and ease with which these mechanically interlocked macromolecules can be assembled should facilitate rapid modulation to achieve versatile polyrotaxane architectures.
Flexible, switchable [c2]daisy-chain dimers (DCDs) were synthesized, where the macromer ammonium binding site was adjacent to the crown-type recognition structure and separated from the cap by an alkyl chain. A DCD of this topology is expected to have an extended structure in the bound conformation (when the ammonium was coordinated to the crown). Several different macromer candidates were designed to allow access to DCDs with flexible alkyl chains between the ammonium binding site and the cap, and a number of synthetic routes were explored in an effort to access these challenging materials. While the first generation DCD structure proved to be unstable due to a labile ester linkage, work is continuing toward the development of several cap structures in an effort to replace the ester linkage with an ether linkage, which, in the second generation model systems, has proven much more stable to the acidic and basic conditions necessary to induce switching of the dimeric architecture.
One of the efforts in our lab is directed at the synthesis of 18F-labeled nanoparticles to be used as tumor imaging agents in positron emission tomography. We have been working to optimize fluorine incorporation while minimizing NP crosslinking. Because of evidence of NP side-reactions with the potassium carbonate base, we have begun to use potassium benzoate solid-state beads. To analyze the fluorinated NPs, various sorbents were explored. It was found that silica sorbents rapidly reacted and bound to the NPs, while the NPs remained unreactive and mobile on alumina. Further analysis of the NPs has been accomplished using 2D-DOSY NMR spectroscopy. Future work with the NPs will involve a systematic evaluation of the role of water on the extent of fluorination, as well as functionalization of the NPs with Cy5.5 dye for use in studies on eyes to be done in collaboration with researchers at the Mayo Clinic.</p
Quantum Mechanical Simulation and X-Ray Scattering Applied to Pressure-Induced Invar Anomaly in Magnetic Iron Alloy
Get PDFThe Invar effect has remained at the forefront of materials research since Charles-Édouard Guillaume discovered the vanishing thermal expansion of Fe-Ni alloys in 1897. More recently, a pressure-induced Invar effect was discovered in Fe-Ni alloys, and the relationship between classical and pressure induced Invar phenomena has added complexity to the century-old struggle to comprehend the microscopic origins of Invar behavior.
In this thesis I present our recent discovery of pressure-induced Invar behavior in Pd₃Fe with the ordered L1₂ structure. Nuclear forward scattering measurements show that the ferromagnetic ground state in Pd₃Fe is destabilized with pressure, collapsing around 10GPa (V/V0=0.96) to a lowspin magnetic state. From high-pressure synchrotron x-ray diffraction measurements we find a large volume collapse at ambient temperature to accompany the collapse of ferromagnetism. After the volume collapse there is a significant increase in the bulk modulus. Using nuclear resonant inelastic x-ray scattering to study the 57Fe phonon partial density of states (PDOS) at high pressures, we find the pressure-induced magnetic transition to cause an anomalous relative softening of the average phonon frequency. Heating our sample to 650K in a furnace at a pressure of 7GPa, synchrotron xray diffraction measurements reveal negligible thermal expansion from 300 to 523 K, demonstrating pressure-induced Invar behavior in Pd₃Fe.
Density functional theory calculations identify a ferromagnetic ground state in Pd₃Fe with large moments at the Fe sites. These calculations show that the application of pressure counteracts the band-filling effect of Pd. By tuning the position of the top of the 3d band with respect to the Fermi level, pressure-induced Invar behavior resembles classical Invar behavior that is controlled by chemical composition. This insight marks the first step towards a unification of our understanding vii of classical and pressure-induced Invar behavior. Pressure drives the majority-spin t2g antibonding electronic states closer to the Fermi level. The transition to the low-spin state occurs as these t2g states move across the Fermi level, transferring charge to the minority-spin eg nonbonding electronic states. This charge transfer reduces the internal electronic pressure in the material, giving a volume reduction in the low-spin state. The movement of the t2g states with increasing pressure results in a greater number of states at the Fermi level, increasing screening efficiency and softening the first nearest-neighbor Fe-Pd longitudinal force constants in the low-spin state. The measured and calculated magnetic transition pressures differ significantly, despite sharing similar elastic properties in both the ferromagnetic and low-spin states. The magnitude of the disagreement between theoretical and experimental magnetic transition pressures suggests a spin-disordered state exists at high pressures in Pd₃Fe. A shape discrepancy between the calculated and measured high-pressure Fe PDOS suggests significant short-range spin correlations exist in this spin-disordered state.</p
Total Synthesis of Cyanthiwigin Natural Products via Double Asymmetric Catalytic Alkylation and Investigations into the Nature of Double Asymmetric Processes
Get PDFSince the initial isolation of the cyathane molecules in 1970, considerable synthetic interest has been invested into the preparation of these diterpenoid natural products. Owing to the biological activity and intriguing molecular architecture of these compounds, the members of the cyathane family of natural products have emerged as appealing targets for total synthesis. After a brief summary of the isolation and bioactivity properties of these diterpene compounds, previous synthetic efforts toward these molecules are reviewed.
A concise and versatile approach toward the preparation of the cyanthiwigin family of cyathane natural products is described. By leveraging a unique double asymmetric catalytic alkylation procedure it is possible to quickly establish two of the most critical stereocenters of the cyanthiwigin framework with high levels of selectivity and expediency. The synthesis additionally employs a tandem ring-opening and cross-metathesis reaction, and an aldehyde-olefin radical cyclization process, to rapidly arrive at the tricyclic cyathane core of the cyanthiwigin molecules. From this unifying intermediate, the preparation of cyanthiwigins B, F, and G are attained swiftly and without the need for protecting groups.
The nature of double asymmetric transformations is investigated from a historical, mathematical, and experimental perspective. The initial findings of Langenbeck and Horeau concerning the enantioenriching effects of scalemic duplication are described, with a specific focus on the impact of this phenomenon on total synthesis. A thorough mathematical examination, based on the work of Kagan, is then presented for situations involving double asymmetric transformations of prochiral starting materials. Expressions relating the final quantities of the stereoisomeric products to the intermediary selectivity of each stereoselective process are presented based on these formulae.
Finally, experiments designed to probe the selectivity of each stage of stereoselective bond construction in a double asymmetric process are presented. The compiled results are scrutinized in keeping with the previously derived equations, and these findings are analyzed to understand the nature of the double asymmetric processes in question.</p
Interpretation and Scaling of Positional Information During Development
Get PDFCells in a developing animal require information about their relative position in order to function and differentiate appropriately. In the classical view, cellular positional information is interpreted from the concentration of chemical signals known as morphogens. However, recent studies have questioned the ability of morphogens to establish gene expression patterns in a concentration-dependent manner. Here we combine theoretical tools and experimental work in Drosophila melanogaster to investigate the mechanisms by which positional information is interpreted from a morphogen gradient and the ability of patterns to scale with respect to the size of the system.
First, we study how a concentration gradient of the signaling molecule Hedgehog establishes multiple patterns of gene expression along the anterior-posterior axis of the Drosophila wing disc. Using mathematical modeling as a hypotheses-generating tool, we predicted that positional information cannot be explained by different concentration thresholds from a static Hedgehog gradient. Instead, we propose that cells take into account their history of Hedgehog signaling exposure to determine patterns of gene expression. We provide experimental evidence that supports our model and conclude that gradient dynamics, resulting from the gene network architecture of the Hedgehog signaling pathway, determine pattern formation in the wing disc.
Second, we introduce a theoretical formalism to study the role of morphogen gradient dynamics in developmental patterning. Given a mathematical model of pattern formation, we define and compute parameter perturbations that leave invariant the steady-state distribution of the relevant morphogen. We propose that this approach can be used as a tool to design genetic experiments that assay the function of morphogen dynamics.
Lastly, we use dorsal-ventral patterning of the early Drosophila embryo as a model to study scaling of gene expression patterns with respect to natural variations in axis length, that is, the ability to establish positional information relative to the size of the system. We provide evidence that gene expression patterns that depend on the maternal factor Dorsal, scale along the dorsal-ventral axis. Our data suggest that scaling in this system is a gene-dependent rather than a position-dependent property. We propose that the mechanisms for scaling depend on feedback interactions downstream of Dorsal.</p
Transcriptional Regulation by the Numbers
Get PDFRecent decades have seen dramatic advances in our ability to make quantitative measurements of the level of gene expression in organisms of all types. The data resulting from these experiments has raised the need for quantitative models that go beyond the verbal and cartoon-level descriptions that have been so useful in developing a qualitative picture of the nature of gene expression. The improvement in our quantitative description of regulatory networks and our corresponding ability to rewire these networks at will has led many to argue for an analogy between biological regulatory networks and their electronic counterparts. In the electronic setting, we can predict the output current given knowledge of the input voltage and the parameters characterizing the circuit. However, this has so far been nothing more than a hopeful analogy since the input-output functions of most quantitative models of transcriptional regulation are based on phenomenological fits with little-to-no connection to the microscopic parameters of the system. This thesis sharpens this analogy by presenting an integrated approach to understanding transcriptional regulation in bacteria in terms of the microscopic parameters involved in the decision-making processes. This is achieved by a three-pronged approach consisting of theoretical models, in vivo measurements and single-molecule experiments in vitro.
The theoretical analysis is based upon two different families of models aimed at describing the output of several regulatory architectures as a function of their input parameters. Thermodynamic models of transcriptional regulation are used to predict the mean level of gene expression of several bacterial promoter architectures as a function of the concentration of the intervening regulatory proteins and their binding energies to DNA and to the associated transcriptional machinery. In recent years, however, an increasing body of work has been performed where levels of gene expression are quantified in single cells and sometimes even at the single molecule level. These measurements have revealed that "noise" in gene expression can play a significant role in decision-making processes in systems ranging from bacteria to mammalian cells. Stochastic models of transcriptional regulation predict this variability in gene expression as a function of the microscopic parameter of the system. Unlike thermodynamic models, however, the predictions from stochastic models are dependent on the rate constants describing the regulatory circuit of interest. A complete set of models that predict input-output functions of regulatory systems in bacteria as a function of not only equilibrium parameters, but also probabilities of transition between different regulatory states is presented.
The second half of the thesis complements the theoretical analyses by presenting several experiments aimed at testing the various predictions generated by these models. One of the experiments is carried out in vivo and aims to test the theoretical predictions for the input-output function of simple repression in terms of its microscopic parameters such as the concentration of repressor inside the cell and its binding energy to DNA. By quantifying the output level of gene expression as a function of the intracellular absolute concentration of repressor it is shown that our models can account for the level of gene expression as a function of the input parameters over several orders of magnitude. The simple repression motif is also explored experimentally using a second method based upon evaluating fluctuations in the partitioning of regulatory proteins during the cell division process. A third set of experiments performed at the single-molecule level in vitro show how a particular repressor protein binds to DNA at two different sites and loops the intervening DNA.</p
MEMS Electrolytic Inchworms for Movable Neural Probe Applications
Get PDFOver decades of cortical neural prosthesis, it was found that "movable" neural probes are important to track neurons for long-term, reliable prostheses. This is challenging because the ideal movable probes require low voltage, small power, bidirectional/latchable movement, and large total traveling distance. The device should also be small enough to entirely fit under the skull after implantation. Many different devices have been demonstrated to move neural probes, but none of them satisfies all the actuation and size requirements.
This thesis presents our work on actuators for movable neural probes that combine MEMS technology with an electrolytic actuation mechanism. Each inchworm is based on two electrolytic balloon actuators. The actuators rely on gas generation by electrolysis inside a sealed balloon, which causes its expansion. When electrolysis is stopped, gas recombination and permeation across the balloon membrane cause the balloon to relax. Electrolytic actuation, although slow, has several advantages: low power, low voltage, and ability to provide large force and displacement. The balloons have been characterized and their behavior mathematically modeled. Innovative salt-shell-based and hydrogel-based processes have been developed to fabricate the balloons and to allow their replenishment by osmosis.
Two balloons are combined into a bidirectional inchworm mechanism. Large traveling distance can be obtained in multiple cycles, the only constraint being the probe length. Displacement of a silicon probe and of a commercial metal probe have been demonstrated in both directions, with a displacement per cycle between 0.5 um and 75 um. The voltage required to drive electrolysis is typically around 3.5 V, with peak power per balloon around 100 uW. The devices were tested in air, water, and saline.
Closed-loop control of the inchworm may be needed for accurate positioning of the probe, and monitoring of the pressure inside the balloons represents a possible source of feedback from the inchworm. Parylene-membrane pressure sensors that are suitable for integration inside balloon actuators have been demonstrated.</p
Towards Open Ended Learning: Budgets, Model Selection, and Representation
Get PDFBiological organisms learn to recognize visual categories continuously over the course of their lifetimes. This impressive capability allows them to adapt to new circumstances as they arise, and to flexibly incorporate new object categories as they are discovered. Inspired by this capability, we seek to create artificial recognition systems that can learn in a similar fashion.
We identify a number of characteristics that define this Open Ended learning capability. Open Ended learning is unsupervised: object instances need not be explicitly labeled with a category indicator during training. Learning occurs incrementally as experience ensues; there is no training period that is distinct from operation and the categorization system must operate and update itself in a timely fashion with limited computational resources. Open Ended learning systems must flexibly adapt the number of categories as new evidence is uncovered.
Having identified these requirements, we develop Open Ended categorization systems based on probabilistic graphical models and study their properties. From the perspective of building practical systems, the most challenging requirement of Open Ended learning is that it must be carried out in an unsupervised fashion. We then study the question of how best to represent data items and categories in unsupervised learning algorithms in order to extend their domain of application.
Finally, we conclude that continuously learning categorization systems are likely to require human intervention and supervision for some time to come, which suggests research in how best to structure machine-human interactions. We end this thesis by studying a system that reverses the typical role of human and machine in most learning systems. In Crowd Clustering, humans perform the fundamental image categorization tasks, and the machine learning system evaluates and aggregates the results of human workers.</p
The Photoelectrochemistry of Solution Grown Zinc Oxide Nanowire Arrays
Get PDFZnO nanowire arrays were synthesized by a simple wet chemical method, and the effects of substrate, solution composition, and time on the attributes of the wire arrays were explored. Optimized wire arrays were used as photoanodes for water oxidation and in dye-sensitized solar cells with fast redox couples. The results for the wire arrays as photoanodes for water oxidation were compared to single crystals. Both ZnO electrodes exhibited poor cathodic kinetics with the aqueous solution, resulting in a non-ideal behavior of the semiconductor-liquid junction and substantial losses in the fill factor. Surprisingly, the wire arrays approached the efficiency of the single crystal, 0.18% vs 0.22% respectively. In the dye senitized solar cell, the ZnO nanowires developed a Schottky junction and allowed the use of fast redox couples. Unfortunately, the efficiencies measured were low, but results suggest the potential for substantial gains in the efficiency and versatility of the dye-sensitized solar cell