On the settling speed of dilute arrays of spheres

A method is developed to calculate the settling speed of dilute arrays of spheres for the three cases of: I, a random array of freely moving particles; II, a random array of rigidly held particles; and III, a cubic array of particles. The basic idea of the technique is to give a formal representation for the solution and then manipulate this representation in a straightforward manner to obtain the result. For infinite arrays of spheres, our results agree with the results previously found by other authors, and the analysis here appears to be simpler. This method is able to obtain more terms in the answer than was possible by Saffman's unified treatment for point particles. Some results for arbitrary two sphere distributions are presented, and an analysis of the wall effect for particles settling in a tube is given. It is expected that the method presented here can be generalized to solve other types of problems.

Determinantal hypersurface from a geometric perspective

In this paper, we give a geometric interpretation of determinantal forms, both in the case of general matrices and symmetric matrices. We will prove irreducibility of the determinantal singular loci and state its dimension. We also provide detailed description of the singular locus for small dimensions.

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.

Sunyaev-Zel'dovich observations using large-format millimeter arrays

Galaxy clusters are the largest gravitationally bound objects in the observable universe, and they are formed from the largest perturbations of the primordial matter power spectrum. During initial cluster collapse, matter is accelerated to supersonic velocities, and the baryonic component is heated as it passes through accretion shocks. This process stabilizes when the pressure of the bound matter prevents further gravitational collapse. Galaxy clusters are useful cosmological probes, because their formation progressively freezes out at the epoch when dark energy begins to dominate the expansion and energy density of the universe. A diverse set of observables, from radio through X-ray wavelengths, are sourced from galaxy clusters, and this is useful for self-calibration. The distributions of these observables trace a cluster's dark matter halo, which represents more than 80% of the cluster's gravitational potential. One such observable is the Sunyaev-Zel'dovich effect (SZE), which results when the ionized intercluster medium blueshifts the cosmic microwave background via Compton scattering. Great technical advances in the last several decades have made regular observation of the SZE possible. Resolved SZE science, such as is explored in this analysis, has benefitted from the construction of large-format camera arrays consisting of highly sensitive millimeter-wave detectors, such as Bolocam. Bolocam is a submillimeter camera, sensitive to 140 GHz and 268 GHz radiation, located at one of the best observing sites in the world: the Caltech Submillimeter Observatory on Mauna Kea in Hawaii. Bolocam fielded 144 of the original spider web NTD bolometers used in an entire generation of ground-based, balloon-borne, and satellite-borne millimeter wave instrumention. Over approximately six years, our group at Caltech has developed a mature galaxy cluster observational program with Bolocam. This thesis describes the construction of the instrument's full cluster catalog: BOXSZ. Using this catalog, I have scaled the Bolocam SZE measurements with X-ray mass approximations in an effort to characterize the SZE signal as a viable mass probe for cosmology. This work has confirmed the SZE to be a low-scatter tracer of cluster mass. The analysis has also revealed how sensitive the SZE-mass scaling is to small biases in the adopted mass approximation. Future Bolocam analysis efforts are set on resolving these discrepancies by approximating cluster mass jointly with different observational probes.

Geometric quantization and foliation reduction

A standard question in the study of geometric quantization is whether symplectic reduction interacts nicely with the quantized theory, and in particular whether “quantization commutes with reduction.” Guillemin and Sternberg first proposed this question, and answered it in the affirmative for the case of a free action of a compact Lie group on a compact Kähler manifold. Subsequent work has focused mainly on extending their proof to non-free actions and non-Kähler manifolds. For realistic physical examples, however, it is desirable to have a proof which also applies to non-compact symplectic manifolds.

In this thesis we give a proof of the quantization-reduction problem for general symplectic manifolds. This is accomplished by working in a particular wavefunction representation, associated with a polarization that is in some sense compatible with reduction. While the polarized sections described by Guillemin and Sternberg are nonzero on a dense subset of the Kähler manifold, the ones considered here are distributional, having support only on regions of the phase space associated with certain quantized, or “admissible”, values of momentum.

We first propose a reduction procedure for the prequantum geometric structures that “covers” symplectic reduction, and demonstrate how both symplectic and prequantum reduction can be viewed as examples of foliation reduction. Consistency of prequantum reduction imposes the above-mentioned admissibility conditions on the quantized momenta, which can be seen as analogues of the Bohr-Wilson-Sommerfeld conditions for completely integrable systems.

We then describe our reduction-compatible polarization, and demonstrate a one-to-one correspondence between polarized sections on the unreduced and reduced spaces.

Finally, we describe a factorization of the reduced prequantum bundle, suggested by the structure of the underlying reduced symplectic manifold. This in turn induces a factorization of the space of polarized sections that agrees with its usual decomposition by irreducible representations, and so proves that quantization and reduction do indeed commute in this context.

A significant omission from the proof is the construction of an inner product on the space of polarized sections, and a discussion of its behavior under reduction. In the concluding chapter of the thesis, we suggest some ideas for future work in this direction.

Seismic reflection experiments imaging the physical nature of crustal structures in southern California

Seismic reflection methods have been extensively used to probe the Earth's crust and suggest the nature of its formative processes. The analysis of multi-offset seismic reflection data extends the technique from a reconnaissance method to a powerful scientific tool that can be applied to test specific hypotheses. The treatment of reflections at multiple offsets becomes tractable if the assumptions of high-frequency rays are valid for the problem being considered. Their validity can be tested by applying the methods of analysis to full wave synthetics.

Three studies illustrate the application of these principles to investigations of the nature of the crust in southern California. A survey shot by the COCORP consortium in 1977 across the San Andreas fault near Parkfield revealed events in the record sections whose arrival time decreased with offset. The reflectors generating these events are imaged using a multi-offset three-dimensional Kirchhoff migration. Migrations of full wave acoustic synthetics having the same limitations in geometric coverage as the field survey demonstrate the utility of this back projection process for imaging. The migrated depth sections show the locations of the major physical boundaries of the San Andreas fault zone. The zone is bounded on the southwest by a near-vertical fault juxtaposing a Tertiary sedimentary section against uplifted crystalline rocks of the fault zone block. On the northeast, the fault zone is bounded by a fault dipping into the San Andreas, which includes slices of serpentinized ultramafics, intersecting it at 3 km depth. These interpretations can be made despite complications introduced by lateral heterogeneities.

In 1985 the Calcrust consortium designed a survey in the eastern Mojave desert to image structures in both the shallow and the deep crust. Preliminary field experiments showed that the major geophysical acquisition problem to be solved was the poor penetration of seismic energy through a low-velocity surface layer. Its effects could be mitigated through special acquisition and processing techniques. Data obtained from industry showed that quality data could be obtained from areas having a deeper, older sedimentary cover, causing a re-definition of the geologic objectives. Long offset stationary arrays were designed to provide reversed, wider angle coverage of the deep crust over parts of the survey. The preliminary field tests and constant monitoring of data quality and parameter adjustment allowed 108 km of excellent crustal data to be obtained.

This dataset, along with two others from the central and western Mojave, was used to constrain rock properties and the physical condition of the crust. The multi-offset analysis proceeded in two steps. First, an increase in reflection peak frequency with offset is indicative of a thinly layered reflector. The thickness and velocity contrast of the layering can be calculated from the spectral dispersion, to discriminate between structures resulting from broad scale or local effects. Second, the amplitude effects at different offsets of P-P scattering from weak elastic heterogeneities indicate whether the signs of the changes in density, rigidity, and Lame's parameter at the reflector agree or are opposed. The effects of reflection generation and propagation in a heterogeneous, anisotropic crust were contained by the design of the experiment and the simplicity of the observed amplitude and frequency trends. Multi-offset spectra and amplitude trend stacks of the three Mojave Desert datasets suggest that the most reflective structures in the middle crust are strong Poisson's ratio (σ) contrasts. Porous zones or the juxtaposition of units of mutually distant origin are indicated. Heterogeneities in σ increase towards the top of a basal crustal zone at ~22 km depth. The transition to the basal zone and to the mantle include increases in σ. The Moho itself includes ~400 m layering having a velocity higher than that of the uppermost mantle. The Moho maintains the same configuration across the Mojave despite 5 km of crustal thinning near the Colorado River. This indicates that Miocene extension there either thinned just the basal zone, or that the basal zone developed regionally after the extensional event.

Geometric integration applied to moving mesh methods and degenerate Lagrangians

Moving mesh methods (also called r-adaptive methods) are space-adaptive strategies used for the numerical simulation of time-dependent partial differential equations. These methods keep the total number of mesh points fixed during the simulation, but redistribute them over time to follow the areas where a higher mesh point density is required. There are a very limited number of moving mesh methods designed for solving field-theoretic partial differential equations, and the numerical analysis of the resulting schemes is challenging. In this thesis we present two ways to construct r-adaptive variational and multisymplectic integrators for (1+1)-dimensional Lagrangian field theories. The first method uses a variational discretization of the physical equations and the mesh equations are then coupled in a way typical of the existing r-adaptive schemes. The second method treats the mesh points as pseudo-particles and incorporates their dynamics directly into the variational principle. A user-specified adaptation strategy is then enforced through Lagrange multipliers as a constraint on the dynamics of both the physical field and the mesh points. We discuss the advantages and limitations of our methods. The proposed methods are readily applicable to (weakly) non-degenerate field theories---numerical results for the Sine-Gordon equation are presented.

In an attempt to extend our approach to degenerate field theories, in the last part of this thesis we construct higher-order variational integrators for a class of degenerate systems described by Lagrangians that are linear in velocities. We analyze the geometry underlying such systems and develop the appropriate theory for variational integration. Our main observation is that the evolution takes place on the primary constraint and the 'Hamiltonian' equations of motion can be formulated as an index 1 differential-algebraic system. We then proceed to construct variational Runge-Kutta methods and analyze their properties. The general properties of Runge-Kutta methods depend on the 'velocity' part of the Lagrangian. If the 'velocity' part is also linear in the position coordinate, then we show that non-partitioned variational Runge-Kutta methods are equivalent to integration of the corresponding first-order Euler-Lagrange equations, which have the form of a Poisson system with a constant structure matrix, and the classical properties of the Runge-Kutta method are retained. If the 'velocity' part is nonlinear in the position coordinate, we observe a reduction of the order of convergence, which is typical of numerical integration of DAEs. We also apply our methods to several models and present the results of our numerical experiments.

Geometric elasticity for graphics, simulation, and computation

We develop new algorithms which combine the rigorous theory of mathematical elasticity with the geometric underpinnings and computational attractiveness of modern tools in geometry processing. We develop a simple elastic energy based on the Biot strain measure, which improves on state-of-the-art methods in geometry processing. We use this energy within a constrained optimization problem to, for the first time, provide surface parameterization tools which guarantee injectivity and bounded distortion, are user-directable, and which scale to large meshes. With the help of some new generalizations in the computation of matrix functions and their derivative, we extend our methods to a large class of hyperelastic stored energy functions quadratic in piecewise analytic strain measures, including the Hencky (logarithmic) strain, opening up a wide range of possibilities for robust and efficient nonlinear elastic simulation and geometry processing by elastic analogy.

Formation mechanisms of uniform arrays of periodic nanoparticles and nanoripples on 6H-SiC crystal surface induced by femtosecond laser ablation

Uniform arrays of periodic nanoparticles with 80-nm period are formed on 6H-SiC crystal irradiated by circularly polarized 400-nm femtosecond laser pulses. In order to understand the formation mechanism, the morphology evolvement as a function of laser pulse energy and number is studied. Periodic nanoripples are also formed on the sample surface irradiated by linearly polarized 400-, 510- and 800-nm femtosecond laser pulses. All these results support well the mechanism that second-harmonic generation plays an important role in the formation of periodic nanostructures.