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Heading Estimation via Sun Sensing for Autonomous Navigation
Get PDFIn preparation for the mission to Mars in 2020, NASA JPL and Caltech have been exploring the potential of sending a scout robot to accompany the new rover. One of the leading candidates for this scout robot is a lightweight helicopter that can fly every day for ~1 to 3 minutes. Its findings would be critical in the path planning for the rover because of its ability to see over and round local terrain elements. The inconsistent Mars magnetic field and GPS-denied environment would require the navigation system of such a vehicle to be completely overhauled. In this thesis, we present a novel technique for heading estimation for autonomous vehicles using sun sensing via fisheye camera. The approach results in accurate heading estimates within 2.4° when relying on the camera alone. If the information from the camera is fused with our sensors, the heading estimates are even more accurate. While this does not yet meet the desired error bound, it is a start with the critical flaws in the algorithm already identified in order to improve performance significantly. This lightweight solution however shows promise and does meet the weight constraints for the 1 kg Mars 2020 Helicopter Scout.</p
Knots, Trees, and Fields: Common Ground Between Physics and Mathematics
Get PDFOne main theme of this thesis is a connection between mathematical physics (in particular, the three-dimensional topological quantum field theory known as Chern-Simons theory) and three-dimensional topology. This connection arises because the partition function of Chern-Simons theory provides an invariant of three-manifolds, and the Wilson-loop observables in the theory define invariants of knots. In the first chapter, we review this connection, as well as more recent work that studies the classical limit of quantum Chern-Simons theory, leading to relations to another knot invariant known as the A-polynomial. (Roughly speaking, this invariant can be thought of as the moduli space of flat SL(2,C) connections on the knot complement.) In fact, the connection can be deepened: through an embedding into string theory, categorifications of polynomial knot invariants can be understood as spaces of BPS states.
We go on to study these homological knot invariants, and interpret spectral sequences that relate them to one another in terms of perturbations of supersymmetric theories. Our point is more general than the application to knots; in general, when one perturbs any modulus of a supersymmetric theory and breaks a symmetry, one should expect a spectral sequence to relate the BPS states of the unperturbed and perturbed theories. We consider several diverse instances of this general lesson. In another chapter, we consider connections between supersymmetric quantum mechanics and the de Rham version of homotopy theory developed by Sullivan; this leads to a new interpretation of Sullivan's minimal models, and of Massey products as vacuum states which are entangled between different degrees of freedom in these models.
We then turn to consider a discrete model of holography: a Gaussian lattice model defined on an infinite tree of uniform valence. Despite being discrete, the matching of bulk isometries and boundary conformal symmetries takes place as usual; the relevant group is PGL(2,Qp), and all of the formulas developed for holography in the context of scalar fields on fixed backgrounds have natural analogues in this setting. The key observation underlying this generalization is that the geometry underlying AdS3/CFT2 can be understood algebraically, and the base field can therefore be changed while maintaining much of the structure. Finally, we give some analysis of A-polynomials under change of base (to finite fields), bringing things full circle.</p
Organic Structure Directing Agent Free Synthesis of Small Pore Zeolite Catalysts for the Methanol-to-Olefins Reaction
Get PDFLight olefins, ethylene and propylene, are two of the highest produced petrochemicals globally. The methanol-to-olefins (MTO) reaction is a promising route for making these chemicals from non-petroleum feedstocks such as natural gas and coal that has been successfully commercialized. The catalysts employed for this reaction are typically microporous molecular sieves with Brønsted acidity, e.g., zeolites and silicoaluminophosphates, that generally require costly organic structure directing agents (OSDAs) to synthesize.
This thesis explores an alternative, low cost method for synthesizing small pore zeolite catalysts for the MTO reaction without the use of OSDAs. Small pore zeolites are first synthesized in the absence of OSDAs. The resulting high-aluminum materials are then converted into useful catalysts via high temperature (500-800°C) steam treatments that extract a portion of the framework aluminum, thereby modifying the acid site concentration, pore structure and catalytic behavior of the materials. This synthesis method is demonstrated on three small pore zeolite structures that are prepared without using OSDAs: CHA, RHO and KFI. In the as-synthesized forms, these materials deactivate rapidly when evaluated as catalysts for the MTO reaction due to their high aluminum contents. Upon steam treatment, however, improved catalyst lifetimes and olefin selectivities are observed that are attributed to a decrease in the total Brønsted acid site concentration and the creation of mesopores that facilitate transport of reactants and products.
Improvements in the activity were observed for all three of the zeolites chosen for investigation with CHA-type zeolites performing best, though differences in olefin selectivities were observed. Poisoning of acid sites located in the mesopores and on external surface of the steamed zeolites did not change the observed product distribution, suggesting that these differences do not arise from secondary reactions of olefins and instead may be attributed to differences in the pore structures.
Overall, the successful demonstration of this catalyst preparation method on three different zeolite structures suggests that it may be a useful route for converting any small pore zeolite that can be synthesized without using OSDAs into catalysts that may be useful for reactions like MTO.</p
Circuit Cavity Electromechanics in the Quantum Regime
Get PDFGenerating nonclassical states of a macroscopic mechanical object has been a subject of considerable interest. It offers a route toward fundamental test of quantum mechanics in an unexplored regime. However, a macroscopic quantum state is very susceptible to decoherence due to the environment. One way to generate robust quantum states is quantum reservoir engineering. In this work, we utilize the reservoir engineering scheme to generate a steady quantum squeezed state of a micron-scale mechanical oscillator in an electromechanical system. Together with the backaction evading measurement technique, we demonstrate a quantum nondemolition measurement of the mechanical quadratures to characterize the quantum squeezed state. By measuring the quadrature variances of the mechanical motion, more than 3dB squeezing below the zero-point level has been achieved
Automated Macro-scale Causal Hypothesis Formation Based on Micro-scale Observation
Get PDFThis book introduces new concepts at the intersection of machine learning, causal inference and philosophy of science: the macrovariable cause and effect. Methods for learning such from microvariable data are introduced. The learning process proposes a minimal number of guided experiments that recover the macrovariable cause from observational data.
Mathematical definitions of a micro- and macro- scale manipulation, an observational and causal partition, and a subsidiary variable are given. These concepts provide a link to previous work in causal inference and machine learning.
The main theoretical result is the Causal Coarsening Theorem, a new insight into the measure-theoretic structure of probability spaces and structural equation models. The theorem provides grounds for automatic causal hypothesis formation from data. Other results concern the minimality and sufficiency of representations created in accordance with the theorem.
Finally, this book proposes the first algorithms for supervised and unsupervised causal macrovariable discovery. These algorithms bridge large-scale, multidimensional machine learning and causal inference. In an application to climate science, the algorithms re-discover a known causal mechanism as a viable causal hypothesis. In a psychophysical experiment, the algorithms learn to minimally change visual stimuli to achieve a desired effect on human perception.</p
An Optofluidic Ring Resonator Platform for Rapid and Robust Sensing
Get PDFRing resonators show great potential as sensitive optical detectors for studies of biomolecular interactions, drug screening, and point-of-care diagnostics. They are sensitive to minute changes in the refractive index of the surrounding medium, which enables them to detect and quantify sub-femtomolar concentrations of target molecules. This thesis investigates the advantages of an optofluidic ring resonator platform that employs a differential measurement scheme for reducing environmental noise due to temperature and pressure fluctuations. Through simulations and experiments, I determine the sensitivity of the platform to changes in the target analyte concentration and to environmental noise, and demonstrate the benefits of employing a second, reference, ring resonator.
A crucial step in achieving versatile biosensing platforms is the integration of the high sensitivity resonators into platforms with short assay times and robust chemical functionalization. I focus on the development of robust chemistry protocols for depositing linker silane layers for biomolecular interaction studies. Moreover, since the fluid handling scheme strongly influences the response time of the platform, I design and test two fluidic platforms integrated with elastomeric valves that show excellent response times. To further increase the response time of the sensing platform, I explore the effects of a patterned channel geometry on the enhancement of mass transport to the sensor in low Reynolds number flows.</p
Recovering Structured Low-rank Operators Using Nuclear Norms
Get PDFThis work considers the problem of recovering matrices and operators from limited and/or noisy observations. Whereas matrices result from summing tensor products of vectors, operators result from summing tensor products of matrices. These constructions lead to viewing both matrices and operators as the sum of "simple" rank-1 factors.
A popular line of work in this direction is low-rank matrix recovery, i.e., using linear measurements of a matrix to reconstruct it as the sum of few rank-1 factors. Rank minimization problems are hard in general, and a popular approach to avoid them is convex relaxation. Using the trace norm as a surrogate for rank, the low-rank matrix recovery problem becomes convex.
While the trace norm has received much attention in the literature, other convexifications are possible. This thesis focuses on the class of nuclear norms—a class that includes the trace norm itself. Much as the trace norm is a convex surrogate for the matrix rank, other nuclear norms provide convex complexity measures for additional matrix structure. Namely, nuclear norms measure the structure of the factors used to construct the matrix.
Transitioning to the operator framework allows for novel uses of nuclear norms in recovering these structured matrices. In particular, this thesis shows how to lift structured matrix factorization problems to rank-1 operator recovery problems. This new viewpoint allows nuclear norms to measure richer types of structures present in matrix factorizations.
This work also includes a Python software package to model and solve structured operator recovery problems. Systematic numerical experiments in operator denoising demonstrate the effectiveness of nuclear norms in recovering structured operators. In particular, choosing a specific nuclear norm that corresponds to the underlying factor structure of the operator improves the performance of the recovery procedures when compared, for instance, to the trace norm.
Applications in hyperspectral imaging and self-calibration demonstrate the additional flexibility gained by utilizing operator (as opposed to matrix) factorization models.</p
Theory of Dynamical Processes in Semiconductor Quantum Dots
Get PDFThe focus of this dissertation is on the theory of the electronic dynamical processes in semiconductor quantum dots (QDs). The first part of the dissertation introduces the calculation method of electronic eigenstates used through the dissertation, the sp3s* tight-binding (TB) method, and the application of the symmetry-adapted linear combination (SALC) of atomic orbitals to the TB method. The combination of the SALC and TB method reduces the computational load, and generates reliable electronic eigenstates and eigenvalues of Wurtzite CdSe QDs. The second part of the dissertation uses the calculated eigenstates and eigenvalues of CdSe QDs, whose band gap states are removed by a passivation layer, to calculate various kinds of physical properties, such as the structure, the permanent dipole moment, the band gap, the molecular orbitals, the density of states (DOS), and the absorption spectrum. These calculated results are compared with the respective experimental measurements in further discussions. The last part of the dissertation focuses on the studies of the size-dependent trend of the Auger electron-hole recombination process that causes the semiconductor QDs to remain in the dark state, including the cases of a negative trion, a positive trion, and a biexciton, in semiconductor QDs. The rates of these Auger processes are expressed in the form of Fermi’s golden rule, where the Coulombic interaction between the two electrons is the operator. Although the calculated results shows larger size dependence than that of the experimental findings, the literature of recent experiments and theories points out potential remedies to the discrepancy by modifying the current computational setting and theory in the dissertation
Applied Computational Topology for Point Clouds and Sparse Timeseries Data
Get PDFThe proliferation of sensors and advancement of technology has led to the production and collection of unprecedented amounts of data in recent years. The data are often noisy, non-linear, and high-dimensional, and the effectiveness of traditional tools may be limited. Thus, the technological advances that enable the ubiquitous collection of data from the cosmological scale to the subatomic scale also necessitate the development of complementary tools that address the new nature of the data.
Recently, there has been much interest in and success with developing topologically-motivated techniques for data analysis. These approaches are especially useful when a topological method is sensitive to large- and small-scale features that might not be detected by methods that require a level of geometric detail that is not provided by the data or by methods that may obscure geometric features, such as principal component analysis (PCA), multi–dimensional scaling (MDS), and cluster analysis.
Our work explores topological data analysis through two frameworks.
In the first part, we provide a tool for detecting material coherence from a set of spatially sparse particle trajectories via the study of a map induced on homology by the braid corresponding to the motion of particles. While the theory of coherent structures has received a great deal of attention and benefited from many advances in recent years, many of these techniques are limited when the data are sparse. We demonstrate through various examples that our work provides a practical and scalable tool for identifying coherent sets from a sparse set of particle trajectories using eigenanalysis.
In the second part, we formalize the local-to-global structure captured by topology in the setting of point clouds. We extend existing tools in topological data analysis and provide a theoretical framework for studying topological features of a point cloud over a range of resolutions, enabling the analysis of topological features using statistical methods. We apply our tools to the analysis of high-dimensional geospatial sensor data and provide a statistic for quantifying climate anomalies.</p
Fundamentals of Thermocapillary Sculpting of Liquid Nanofilms and Applications to Thin Film Micro-Optics
Get PDFThis doctoral thesis describes experimental work conducted as part of ongoing efforts to identify and understand the source of linear instability in ultrathin liquid films subject to large variations in surface temperature along the air/liquid interface. Previous theoretical efforts by various groups have identified three possible physical mechanisms for instability, including an induced surface charge model, an acoustic phonon model, and a thermocapillary model. The observed instability manifests as the spontaneous formation of arrays of nano/microscale liquid protrusions arising from an initially flat nanofilm, whose organization is characterized by a distinct in-plane wavelength and associated out-of-plane growth rate. Although long range order is somewhat difficult to achieve due to thin film defects incurred during preparation, the instability tends toward hexagonal symmetry within periodic domains achieved for a geometry in which the nanofilm is held in close proximity to a cooled, proximate, parallel, and featureless substrate.
In this work, data obtained from a previous experimental setup is analyzed and it is shown how key improvements in image processing and analysis, coupled with more accurate finite element simulations of thermal profiles, lead to more accurate identification of the fastest growing unstable mode at early times. This fastest growing mode is governed by linear instability and exponential growth. This work was followed by re-examination of real time interference fringes using differential colorimetry to quantify the actual rate of growth of the fastest growing peaks within the protrusion arrays. These initial studies and lingering questions led to the introduction of a new and improved experimental setup, which was redesigned to yield larger and more reproducible data sets. Corresponding improvements to the image analysis process allowed for the measurement of both the wavelength and growth rate of the fastest growing mode simultaneously. These combined efforts establish that the dominant source of instability is attributable to large thermocapillary stresses. For the geometry in which the nanofilm surface is held in close proximity to a cooled and parallel substrate, the instability leads to a runaway process, characterized by exponential growth, in which the film is attracted to the cooled target until contact is achieved.
The second part of this thesis describes fabrication and characterization of microlens arrays and linear waveguide structures using a similar experimental setup. However, instead of relying on the native instability observed, formation and growth of liquid shapes and protrusions is triggered by pre-patterning the cooled substrate with a desired mask for replication. These preformed cooled patterns, held in close proximity to an initially flat liquid nanofilm, induce a strong non-linear response via consequent patterned thermocapillary stresses imposed along the air/liquid interface. Once the desired film shapes are achieved, the transverse thermal gradient is removed and the micro-optical components are affixed in place naturally by the resultant rapid solidification. The use of polymer nanofilms with low glass transition temperatures, such as polystyrene, facilitated rapid solidification, while providing good optical response. Surface characterization of the resulting micro-optical components was accomplished by scanning white light interferometry, which evidences formation of ultrasmooth surfaces ideal for optical applications. Finally, linear waveguides were created by this thermocapillary sculpting technique and their optical performance characterized. In conclusion, these measurements highlight the true source of instability in this geometry, and the fabrication demonstrations pave the way for harnessing this knowledge for the design and creation of novel micro-optical devices.</p