Terahertz time domain spectroscopy of amino acids and sugars

A time-domain spectrometer for use in the terahertz (THz) spectral range was designed and constructed. Due to there being few existing methods of generating and detecting THz radiation, the spectrometer is expected to have vast applications to solid, liquid, and gas phase samples. In particular, knowledge of complex organic chemistry and chemical abundances in the interstellar medium (ISM) can be obtained when compared to astronomical data. The THz spectral region is of particular interest due to reduced line density when compared to the millimeter wave spectrum, the existence of high resolution observatories, and potentially strong transitions resulting from the lowest-lying vibrational modes of large molecules.

The heart of the THz time-domain spectrometer (THz-TDS) is the ultrafast laser. Due to the femtosecond duration of ultrafast laser pulses and an energy-time uncertainty relationship, the pulses typically have a several-THz bandwidth. By various means of optical rectification, the optical pulse carrier envelope shape, i.e. intensity-time profile, can be transferred to the phase of the resulting THz pulse. As a consequence, optical pump-THz probe spectroscopy is readily achieved, as was demonstrated in studies of dye-sensitized TiO₂, as discussed in chapter 4. Detection of the terahertz radiation is commonly based on electro-optic sampling and provides full phase information. This allows for accurate determination of both the real and imaginary index of refraction, the so-called optical constants, without additional analysis. A suite of amino acids and sugars, all of which have been found in meteorites, were studied in crystalline form embedded in a polyethylene matrix. As the temperature was varied between 10 and 310 K, various strong vibrational modes were found to shift in spectral intensity and frequency. Such modes can be attributed to intramolecular, intermolecular, or phonon modes, or to some combination of the three.

The electric field of a weakly electric fish

Freshwater fish of the genus Apteronotus (family Gymnotidae) generate a weak, high frequency electric field (< 100 mV/cm, 0.5-10 kHz) which permeates their local environment. These nocturnal fish are acutely sensitive to perturbations in their electric field caused by other electric fish, and nearby objects whose impedance is different from the surrounding water. This thesis presents high temporal and spatial resolution maps of the electric potential and field on and near Apteronotus. The fish's electric field is a complicated and highly stable function of space and time. Its characteristics, such as spectral composition, timing, and rate of attenuation, are examined in terms of physical constraints, and their possible functional roles in electroreception.

Temporal jitter of the periodic field is less than 1 µsec. However, electrocyte activity is not globally synchronous along the fish 's electric organ. The propagation of electrocyte activation down the fish's body produces a rotation of the electric field vector in the caudal part of the fish. This may assist the fish in identifying nonsymmetrical objects, and could also confuse electrosensory predators that try to locate Apteronotus by following its fieldlines. The propagation also results in a complex spatiotemporal pattern of the EOD potential near the fish. Visualizing the potential on the same and different fish over timescales of several months suggests that it is stable and could serve as a unique signature for individual fish.

Measurements of the electric field were used to calculate the effects of simple objects on the fish's electric field. The shape of the perturbation or "electric image" on the fish's skin is relatively independent of a simple object's size, conductivity, and rostrocaudal location, and therefore could unambiguously determine object distance. The range of electrolocation may depend on both the size of objects and their rostrocaudal location. Only objects with very large dielectric constants cause appreciable phase shifts, and these are strongly dependent on the water conductivity.

A geometric analysis of convex demixing

Demixing is the task of identifying multiple signals given only their sum and prior information about their structures. Examples of demixing problems include (i) separating a signal that is sparse with respect to one basis from a signal that is sparse with respect to a second basis; (ii) decomposing an observed matrix into low-rank and sparse components; and (iii) identifying a binary codeword with impulsive corruptions. This thesis describes and analyzes a convex optimization framework for solving an array of demixing problems.

Our framework includes a random orientation model for the constituent signals that ensures the structures are incoherent. This work introduces a summary parameter, the statistical dimension, that reflects the intrinsic complexity of a signal. The main result indicates that the difficulty of demixing under this random model depends only on the total complexity of the constituent signals involved: demixing succeeds with high probability when the sum of the complexities is less than the ambient dimension; otherwise, it fails with high probability.

The fact that a phase transition between success and failure occurs in demixing is a consequence of a new inequality in conic integral geometry. Roughly speaking, this inequality asserts that a convex cone behaves like a subspace whose dimension is equal to the statistical dimension of the cone. When combined with a geometric optimality condition for demixing, this inequality provides precise quantitative information about the phase transition, including the location and width of the transition region.

Convex analysis for minimizing and learning submodular set functions

The connections between convexity and submodularity are explored, for purposes of minimizing and learning submodular set functions.

First, we develop a novel method for minimizing a particular class of submodular functions, which can be expressed as a sum of concave functions composed with modular functions. The basic algorithm uses an accelerated first order method applied to a smoothed version of its convex extension. The smoothing algorithm is particularly novel as it allows us to treat general concave potentials without needing to construct a piecewise linear approximation as with graph-based techniques.

Second, we derive the general conditions under which it is possible to find a minimizer of a submodular function via a convex problem. This provides a framework for developing submodular minimization algorithms. The framework is then used to develop several algorithms that can be run in a distributed fashion. This is particularly useful for applications where the submodular objective function consists of a sum of many terms, each term dependent on a small part of a large data set.

Lastly, we approach the problem of learning set functions from an unorthodox perspective---sparse reconstruction. We demonstrate an explicit connection between the problem of learning set functions from random evaluations and that of sparse signals. Based on the observation that the Fourier transform for set functions satisfies exactly the conditions needed for sparse reconstruction algorithms to work, we examine some different function classes under which uniform reconstruction is possible.

Financial equilibrium, voting procedures, and coalition structures in allocational mechanisms

This thesis is comprised of three chapters, each of which is concerned with properties of allocational mechanisms which include voting procedures as part of their operation. The theme of interaction between economic and political forces recurs in the three chapters, as described below.

Chapter One demonstrates existence of a non-controlling interest shareholders' equilibrium for a stylized one-period stock market economy with fewer securities than states of the world. The economy has two decision mechanisms: Owners vote to change firms' production plans across states, fixing shareholdings; and individuals trade shares and the current production / consumption good, fixing production plans. A shareholders' equilibrium is a production plan profile, and a shares / current good allocation stable for both mechanisms. In equilibrium, no (Kramer direction-restricted) plan revision is supported by a share-weighted majority, and there exists no Pareto superior reallocation.

Chapter Two addresses efficient management of stationary-site, fixed-budget, partisan voter registration drives. Sufficient conditions obtain for unique optimal registrar deployment within contested districts. Each census tract is assigned an expected net plurality return to registration investment index, computed from estimates of registration, partisanship, and turnout. Optimum registration intensity is a logarithmic transformation of a tract's index. These conditions are tested using a merged data set including both census variables and Los Angeles County Registrar data from several 1984 Assembly registration drives. Marginal registration spending benefits, registrar compensation, and the general campaign problem are also discussed.

The last chapter considers social decision procedures at a higher level of abstraction. Chapter Three analyzes the structure of decisive coalition families, given a quasitransitive-valued social decision procedure satisfying the universal domain and ITA axioms. By identifying those alternatives X* ⊆ X on which the Pareto principle fails, imposition in the social ranking is characterized. Every coaliton is weakly decisive for X* over X~X*, and weakly antidecisive for X~X* over X*; therefore, alternatives in X~X* are never socially ranked above X*. Repeated filtering of alternatives causing Pareto failure shows states in X^n*~X^((n+1))* are never socially ranked above X^((n+1))*. Limiting results of iterated application of the *-operator are also discussed.

Generic differentiability of convex functions and monotone operators

The aim of this paper is to investigate to what extent the known theory of subdifferentiability and generic differentiability of convex functions defined on open sets can be carried out in the context of convex functions defined on not necessarily open sets. Among the main results obtained I would like to mention a Kenderov type theorem (the subdifferential at a generic point is contained in a sphere), a generic Gâteaux differentiability result in Banach spaces of class S and a generic Fréchet differentiability result in Asplund spaces. At least two methods can be used to prove these results: first, a direct one, and second, a more general one, based on the theory of monotone operators. Since this last theory was previously developed essentially for monotone operators defined on open sets, it was necessary to extend it to the context of monotone operators defined on a larger class of sets, our "quasi open" sets. This is done in Chapter III. As a matter of fact, most of these results have an even more general nature and have roots in the theory of minimal usco maps, as shown in Chapter II.

Vibrational pooling and constrained equilibration on surfaces

In this thesis, we provide a statistical theory for the vibrational pooling and fluorescence time dependence observed in infrared laser excitation of CO on an NaCl surface. The pooling is seen in experiment and in computer simulations. In the theory, we assume a rapid equilibration of the quanta in the substrate and minimize the free energy subject to the constraint at any time t of a fixed number of vibrational quanta N(t). At low incident intensity, the distribution is limited to one- quantum exchanges with the solid and so the Debye frequency of the solid plays a key role in limiting the range of this one-quantum domain. The resulting inverted vibrational equilibrium population depends only on fundamental parameters of the oscillator (ω_e and ω_eχ_e) and the surface (ω_D and T). Possible applications and relation to the Treanor gas phase treatment are discussed. Unlike the solid phase system, the gas phase system has no Debye-constraining maximum. We discuss the possible distributions for arbitrary N-conserving diatom-surface pairs, and include application to H:Si(111) as an example.

Computations are presented to describe and analyze the high levels of infrared laser-induced vibrational excitation of a monolayer of absorbed ¹³CO on a NaCl(100) surface. The calculations confirm that, for situations where the Debye frequency limited n domain restriction approximately holds, the vibrational state population deviates from a Boltzmann population linearly in n. Nonetheless, the full kinetic calculation is necessary to capture the result in detail.

We discuss the one-to-one relationship between N and γ and the examine the state space of the new distribution function for varied γ. We derive the Free Energy, F = NγkT − kTln(∑P_n), and effective chemical potential, μn ≈ γkT, for the vibrational pool. We also find the anti correlation of neighbor vibrations leads to an emergent correlation that appears to extend further than nearest neighbor.

Optimal uncertainty quantification via convex optimization and relaxation

Many engineering applications face the problem of bounding the expected value of a quantity of interest (performance, risk, cost, etc.) that depends on stochastic uncertainties whose probability distribution is not known exactly. Optimal uncertainty quantification (OUQ) is a framework that aims at obtaining the best bound in these situations by explicitly incorporating available information about the distribution. Unfortunately, this often leads to non-convex optimization problems that are numerically expensive to solve.

This thesis emphasizes on efficient numerical algorithms for OUQ problems. It begins by investigating several classes of OUQ problems that can be reformulated as convex optimization problems. Conditions on the objective function and information constraints under which a convex formulation exists are presented. Since the size of the optimization problem can become quite large, solutions for scaling up are also discussed. Finally, the capability of analyzing a practical system through such convex formulations is demonstrated by a numerical example of energy storage placement in power grids.

When an equivalent convex formulation is unavailable, it is possible to find a convex problem that provides a meaningful bound for the original problem, also known as a convex relaxation. As an example, the thesis investigates the setting used in Hoeffding's inequality. The naive formulation requires solving a collection of non-convex polynomial optimization problems whose number grows doubly exponentially. After structures such as symmetry are exploited, it is shown that both the number and the size of the polynomial optimization problems can be reduced significantly. Each polynomial optimization problem is then bounded by its convex relaxation using sums-of-squares. These bounds are found to be tight in all the numerical examples tested in the thesis and are significantly better than Hoeffding's bounds.

Molecular mechanism of sulfated carbohydrate recognition: structural and biochemical studies of the cysteine-rich domain of mannose receptor

Mannose receptor (MR) is widely expressed on macrophages, immature dendritic cells, and a variety of epithelial and endothelial cells. It is a 180 kD type I transmembrane receptor whose extracellular region consists of three parts: the amino-terminal cysteine-rich domain (Cys-MR); a fibronectin type II-like domain; and a series of eight tandem C-type lectin carbohydrate recognition domains (CRDs). Two portions of MR have distinct carbohydrate recognition properties: Cys-MR recognizes sulfated carbohydrates and the tandem CRD region binds terminal mannose, fucose, and N-acetyl-glucosamine (GlcNAc). The dual carbohydrate binding specificity allows MR to interact with sulfated and nonsulfated polysaccharide chains, and thereby facilitating the involvement of MR in immunological and physiological processes. The immunological functions of MR include antigen capturing (through binding non-sulfated carbohydrates) and antigen targeting (through binding sulfated carbohydrates), and the physiological roles include rapid clearance of circulatory luteinizing hormone (LH), which bears polysaccharide chains terminating with sulfated and non-sulfated carbohydrates.

We have crystallized and determined the X-ray structures of unliganded Cys-MR (2.0 Å) and Cys-MR complexed with different ligands, including Hepes (1.7 Å), 4SO_4-N-Acetylgalactosamine (4SO_4-GalNAc; 2.2 Å), 3SO_4-Lewis^x (2.2 Å), 3S04-Lewis^a (1.9 Å), and 6SO_4-GalNAc (2.5 Å). The overall structure of Cys-MR consists of 12 anti-parallel β-strands arranged in three lobes with approximate three fold internal symmetry. The structure contains three disulfide bonds, formed by the six cysteines in the Cys-MR sequence. The ligand-binding site is located in a neutral pocket within the third lobe, in which the sulfate group of ligand is buried. Our results show that optimal binding is achieved by a carbohydrate ligand with a sulfate group that anchors the ligand by forming numerous hydrogen bonds and a sugar ring that makes ring-stacking interactions with Trpll7 of CysMR. Using a fluorescence-based assay, we characterized the binding affinities between CysMR and its ligands, and rationalized the derived affinities based upon the crystal structures. These studies reveal the mechanism of sulfated carbohydrate recognition by Cys-MR and facilitate our understanding of the role of Cys-MR in MR recognition of its ligands.

Engineered discoidin domain from factor VIII binds αvβ3 integrin with antibody-like affinity

Alternative scaffolds are non-antibody proteins that can be engineered to bind new targets. They have found useful niches in the therapeutic space due to their smaller size and the ease with which they can be engineered to be bispecific. We sought a new scaffold that could be used for therapeutic ends and chose the C2 discoidin domain of factor VIII, which is well studied and of human origin. Using yeast surface display, we engineered the C2 domain to bind to αvβ3 integrin with a 16 nM affinity while retaining its thermal stability and monomeric nature. We obtained a crystal structure of the engineered domain at 2.1 Å resolution. We have christened this discoidin domain alternative scaffold the “discobody.”

Efficient methods for stochastic optimal control

The Hamilton Jacobi Bellman (HJB) equation is central to stochastic optimal control (SOC) theory, yielding the optimal solution to general problems specified by known dynamics and a specified cost functional. Given the assumption of quadratic cost on the control input, it is well known that the HJB reduces to a particular partial differential equation (PDE). While powerful, this reduction is not commonly used as the PDE is of second order, is nonlinear, and examples exist where the problem may not have a solution in a classical sense. Furthermore, each state of the system appears as another dimension of the PDE, giving rise to the curse of dimensionality. Since the number of degrees of freedom required to solve the optimal control problem grows exponentially with dimension, the problem becomes intractable for systems with all but modest dimension.

In the last decade researchers have found that under certain, fairly non-restrictive structural assumptions, the HJB may be transformed into a linear PDE, with an interesting analogue in the discretized domain of Markov Decision Processes (MDP). The work presented in this thesis uses the linearity of this particular form of the HJB PDE to push the computational boundaries of stochastic optimal control.

This is done by crafting together previously disjoint lines of research in computation. The first of these is the use of Sum of Squares (SOS) techniques for synthesis of control policies. A candidate polynomial with variable coefficients is proposed as the solution to the stochastic optimal control problem. An SOS relaxation is then taken to the partial differential constraints, leading to a hierarchy of semidefinite relaxations with improving sub-optimality gap. The resulting approximate solutions are shown to be guaranteed over- and under-approximations for the optimal value function. It is shown that these results extend to arbitrary parabolic and elliptic PDEs, yielding a novel method for Uncertainty Quantification (UQ) of systems governed by partial differential constraints. Domain decomposition techniques are also made available, allowing for such problems to be solved via parallelization and low-order polynomials.

The optimization-based SOS technique is then contrasted with the Separated Representation (SR) approach from the applied mathematics community. The technique allows for systems of equations to be solved through a low-rank decomposition that results in algorithms that scale linearly with dimensionality. Its application in stochastic optimal control allows for previously uncomputable problems to be solved quickly, scaling to such complex systems as the Quadcopter and VTOL aircraft. This technique may be combined with the SOS approach, yielding not only a numerical technique, but also an analytical one that allows for entirely new classes of systems to be studied and for stability properties to be guaranteed.

The analysis of the linear HJB is completed by the study of its implications in application. It is shown that the HJB and a popular technique in robotics, the use of navigation functions, sit on opposite ends of a spectrum of optimization problems, upon which tradeoffs may be made in problem complexity. Analytical solutions to the HJB in these settings are available in simplified domains, yielding guidance towards optimality for approximation schemes. Finally, the use of HJB equations in temporal multi-task planning problems is investigated. It is demonstrated that such problems are reducible to a sequence of SOC problems linked via boundary conditions. The linearity of the PDE allows us to pre-compute control policy primitives and then compose them, at essentially zero cost, to satisfy a complex temporal logic specification.

Convex model predictive control for vehicular systems

In this work, the author presents a method called Convex Model Predictive Control (CMPC) to control systems whose states are elements of the rotation matrices SO(n) for n = 2, 3. This is done without charts or any local linearization, and instead is performed by operating over the orbitope of rotation matrices. This results in a novel model predictive control (MPC) scheme without the drawbacks associated with conventional linearization techniques such as slow computation time and local minima. Of particular emphasis is the application to aeronautical and vehicular systems, wherein the method removes many of the trigonometric terms associated with these systems’ state space equations. Furthermore, the method is shown to be compatible with many existing variants of MPC, including obstacle avoidance via Mixed Integer Linear Programming (MILP).

Time-Domain TeraHertz Spectroscopy and Observational Probes of Prebiotic Interstellar Gas and Ice Chemistry

Understanding the origin of life on Earth has long fascinated the minds of the global community, and has been a driving factor in interdisciplinary research for centuries. Beyond the pioneering work of Darwin, perhaps the most widely known study in the last century is that of Miller and Urey, who examined the possibility of the formation of prebiotic chemical precursors on the primordial Earth [1]. More recent studies have shown that amino acids, the chemical building blocks of the biopolymers that comprise life as we know it on Earth, are present in meteoritic samples, and that the molecules extracted from the meteorites display isotopic signatures indicative of an extraterrestrial origin [2]. The most recent major discovery in this area has been the detection of glycine (NH₂CH₂COOH), the simplest amino acid, in pristine cometary samples returned by the NASA STARDUST mission [3]. Indeed, the open questions left by these discoveries, both in the public and scientific communities, hold such fascination that NASA has designated the understanding of our "Cosmic Origins" as a key mission priority.

Despite these exciting discoveries, our understanding of the chemical and physical pathways to the formation of prebiotic molecules is woefully incomplete. This is largely because we do not yet fully understand how the interplay between grain-surface and sub-surface ice reactions and the gas-phase affects astrophysical chemical evolution, and our knowledge of chemical inventories in these regions is incomplete. The research presented here aims to directly address both these issues, so that future work to understand the formation of prebiotic molecules has a solid foundation from which to work.

From an observational standpoint, a dedicated campaign to identify hydroxylamine (NH₂OH), potentially a direct precursor to glycine, in the gas-phase was undertaken. No trace of NH₂OH was found. These observations motivated a refinement of the chemical models of glycine formation, and have largely ruled out a gas-phase route to the synthesis of the simplest amino acid in the ISM. A molecular mystery in the case of the carrier of a series of transitions was resolved using observational data toward a large number of sources, confirming the identity of this important carbon-chemistry intermediate B11244 as l-C₃H⁺ and identifying it in at least two new environments. Finally, the doubly-nitrogenated molecule carbodiimide HNCNH was identified in the ISM for the first time through maser emission features in the centimeter-wavelength regime.

In the laboratory, a TeraHertz Time-Domain Spectrometer was constructed to obtain the experimental spectra necessary to search for solid-phase species in the ISM in the THz region of the spectrum. These investigations have shown a striking dependence on large-scale, long-range (i.e. lattice) structure of the ices on the spectra they present in the THz. A database of molecular spectra has been started, and both the simplest and most abundant ice species, which have already been identified, as well as a number of more complex species, have been studied. The exquisite sensitivity of the THz spectra to both the structure and thermal history of these ices may lead to better probes of complex chemical and dynamical evolution in interstellar environments.

Convex Relaxation for Low-Dimensional Representation: Phase Transitions and Limitations

There is a growing interest in taking advantage of possible patterns and structures in data so as to extract the desired information and overcome the curse of dimensionality. In a wide range of applications, including computer vision, machine learning, medical imaging, and social networks, the signal that gives rise to the observations can be modeled to be approximately sparse and exploiting this fact can be very beneficial. This has led to an immense interest in the problem of efficiently reconstructing a sparse signal from limited linear observations. More recently, low-rank approximation techniques have become prominent tools to approach problems arising in machine learning, system identification and quantum tomography.

In sparse and low-rank estimation problems, the challenge is the inherent intractability of the objective function, and one needs efficient methods to capture the low-dimensionality of these models. Convex optimization is often a promising tool to attack such problems. An intractable problem with a combinatorial objective can often be "relaxed" to obtain a tractable but almost as powerful convex optimization problem. This dissertation studies convex optimization techniques that can take advantage of low-dimensional representations of the underlying high-dimensional data. We provide provable guarantees that ensure that the proposed algorithms will succeed under reasonable conditions, and answer questions of the following flavor:

More specifically, i) Focusing on linear inverse problems, we generalize the classical error bounds known for the least-squares technique to the lasso formulation, which incorporates the signal model. ii) We show that intuitive convex approaches do not perform as well as expected when it comes to signals that have multiple low-dimensional structures simultaneously. iii) Finally, we propose convex relaxations for the graph clustering problem and give sharp performance guarantees for a family of graphs arising from the so-called stochastic block model. We pay particular attention to the following aspects. For i) and ii), we aim to provide a general geometric framework, in which the results on sparse and low-rank estimation can be obtained as special cases. For i) and iii), we investigate the precise performance characterization, which yields the right constants in our bounds and the true dependence between the problem parameters.

General-domain compressible Navier-Stokes solvers exhibiting quasi-unconditional stability and high-order accuracy in space and time

This thesis presents a new class of solvers for the subsonic compressible Navier-Stokes equations in general two- and three-dimensional spatial domains. The proposed methodology incorporates: 1) A novel linear-cost implicit solver based on use of higher-order backward differentiation formulae (BDF) and the alternating direction implicit approach (ADI); 2) A fast explicit solver; 3) Dispersionless spectral spatial discretizations; and 4) A domain decomposition strategy that negotiates the interactions between the implicit and explicit domains. In particular, the implicit methodology is quasi-unconditionally stable (it does not suffer from CFL constraints for adequately resolved flows), and it can deliver orders of time accuracy between two and six in the presence of general boundary conditions. In fact this thesis presents, for the first time in the literature, high-order time-convergence curves for Navier-Stokes solvers based on the ADI strategy---previous ADI solvers for the Navier-Stokes equations have not demonstrated orders of temporal accuracy higher than one. An extended discussion is presented in this thesis which places on a solid theoretical basis the observed quasi-unconditional stability of the methods of orders two through six. The performance of the proposed solvers is favorable. For example, a two-dimensional rough-surface configuration including boundary layer effects at Reynolds number equal to one million and Mach number 0.85 (with a well-resolved boundary layer, run up to a sufficiently long time that single vortices travel the entire spatial extent of the domain, and with spatial mesh sizes near the wall of the order of one hundred-thousandth the length of the domain) was successfully tackled in a relatively short (approximately thirty-hour) single-core run; for such discretizations an explicit solver would require truly prohibitive computing times. As demonstrated via a variety of numerical experiments in two- and three-dimensions, further, the proposed multi-domain parallel implicit-explicit implementations exhibit high-order convergence in space and time, useful stability properties, limited dispersion, and high parallel efficiency.