Logarithmic Potential Theory on Riemann Surfaces

We develop a logarithmic potential theory on Riemann surfaces which generalizes logarithmic potential theory on the complex plane. We show the existence of an equilibrium measure and examine its structure. This leads to a formula for the structure of the equilibrium measure which is new even in the plane. We then use our results to study quadrature domains, Laplacian growth, and Coulomb gas ensembles on Riemann surfaces. We prove that the complement of the support of the equilibrium measure satisfies a quadrature identity. Furthermore, our setup allows us to naturally realize weak solutions of Laplacian growth (for a general time-dependent source) as an evolution of the support of equilibrium measures. When applied to the Riemann sphere this approach unifies the known methods for generating interior and exterior Laplacian growth. We later narrow our focus to a special class of quadrature domains which we call Algebraic Quadrature Domains. We show that many of the properties of quadrature domains generalize to this setting. In particular, the boundary of an Algebraic Quadrature Domain is the inverse image of a planar algebraic curve under a meromorphic function. This makes the study of the topology of Algebraic Quadrature Domains an interesting problem. We briefly investigate this problem and then narrow our focus to the study of the topology of classical quadrature domains. We extend the results of Lee and Makarov and prove (for n ≥ 3) c ≤ 5n-5, where c and n denote the connectivity and degree of a (classical) quadrature domain. At the same time we obtain a new upper bound on the number of isolated points of the algebraic curve corresponding to the boundary and thus a new upper bound on the number of special points. In the final chapter we study Coulomb gas ensembles on Riemann surfaces.

Movement of a Spherical Brownian Particle Between Infinite Parallel Plates: Hindered Dispersion and Sedimentation

The dispersion of an isolated, spherical, Brownian particle immersed in a Newtonian fluid between infinite parallel plates is investigated. Expressions are developed for both a 'molecular' contribution to dispersion, which arises from random thermal fluctuations, and a 'convective' contribution, arising when a shear flow is applied between the plates. These expressions are evaluated numerically for all sizes of the particle relative to the bounding plates, and the method of matched asymptotic expansions is used to develop analytical expressions for the dispersion coefficients as a function of particle size to plate spacing ratio for small values of this parameter.

It is shown that both the molecular and convective dispersion coefficients decrease as the size of the particle relative to the bounding plates increase. When the particle is small compared to the plate spacing, the coefficients decrease roughly proportional to the particle size to plate spacing ratio. When the particle closely fills the space between the plates, the molecular dispersion coefficient approaches zero slowly as an inverse logarithmic function of the particle size to plate spacing ratio, and the convective dispersion coefficent approaches zero approximately proportional to the width of the gap between the edges of the sphere and the bounding plates.

Synthetic molecular machines for active self-assembly : prototype algorithms, designs, and experimental study

Computer science and electrical engineering have been the great success story of the twentieth century. The neat modularity and mapping of a language onto circuits has led to robots on Mars, desktop computers and smartphones. But these devices are not yet able to do some of the things that life takes for granted: repair a scratch, reproduce, regenerate, or grow exponentially fast–all while remaining functional.

This thesis explores and develops algorithms, molecular implementations, and theoretical proofs in the context of “active self-assembly” of molecular systems. The long-term vision of active self-assembly is the theoretical and physical implementation of materials that are composed of reconfigurable units with the programmability and adaptability of biology’s numerous molecular machines. En route to this goal, we must first find a way to overcome the memory limitations of molecular systems, and to discover the limits of complexity that can be achieved with individual molecules.

One of the main thrusts in molecular programming is to use computer science as a tool for figuring out what can be achieved. While molecular systems that are Turing-complete have been demonstrated [Winfree, 1996], these systems still cannot achieve some of the feats biology has achieved.

One might think that because a system is Turing-complete, capable of computing “anything,” that it can do any arbitrary task. But while it can simulate any digital computational problem, there are many behaviors that are not “computations” in a classical sense, and cannot be directly implemented. Examples include exponential growth and molecular motion relative to a surface.

Passive self-assembly systems cannot implement these behaviors because (a) molecular motion relative to a surface requires a source of fuel that is external to the system, and (b) passive systems are too slow to assemble exponentially-fast-growing structures. We call these behaviors “energetically incomplete” programmable behaviors. This class of behaviors includes any behavior where a passive physical system simply does not have enough physical energy to perform the specified tasks in the requisite amount of time.

As we will demonstrate and prove, a sufficiently expressive implementation of an “active” molecular self-assembly approach can achieve these behaviors. Using an external source of fuel solves part of the the problem, so the system is not “energetically incomplete.” But the programmable system also needs to have sufficient expressive power to achieve the specified behaviors. Perhaps surprisingly, some of these systems do not even require Turing completeness to be sufficiently expressive.

Building on a large variety of work by other scientists in the fields of DNA nanotechnology, chemistry and reconfigurable robotics, this thesis introduces several research contributions in the context of active self-assembly.

We show that simple primitives such as insertion and deletion are able to generate complex and interesting results such as the growth of a linear polymer in logarithmic time and the ability of a linear polymer to treadmill. To this end we developed a formal model for active-self assembly that is directly implementable with DNA molecules. We show that this model is computationally equivalent to a machine capable of producing strings that are stronger than regular languages and, at most, as strong as context-free grammars. This is a great advance in the theory of active self- assembly as prior models were either entirely theoretical or only implementable in the context of macro-scale robotics.

We developed a chain reaction method for the autonomous exponential growth of a linear DNA polymer. Our method is based on the insertion of molecules into the assembly, which generates two new insertion sites for every initial one employed. The building of a line in logarithmic time is a first step toward building a shape in logarithmic time. We demonstrate the first construction of a synthetic linear polymer that grows exponentially fast via insertion. We show that monomer molecules are converted into the polymer in logarithmic time via spectrofluorimetry and gel electrophoresis experiments. We also demonstrate the division of these polymers via the addition of a single DNA complex that competes with the insertion mechanism. This shows the growth of a population of polymers in logarithmic time. We characterize the DNA insertion mechanism that we utilize in Chapter 4. We experimentally demonstrate that we can control the kinetics of this re- action over at least seven orders of magnitude, by programming the sequences of DNA that initiate the reaction.

In addition, we review co-authored work on programming molecular robots using prescriptive landscapes of DNA origami; this was the first microscopic demonstration of programming a molec- ular robot to walk on a 2-dimensional surface. We developed a snapshot method for imaging these random walking molecular robots and a CAPTCHA-like analysis method for difficult-to-interpret imaging data.

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.

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.

Large-eddy simulations of fully developed turbulent channel and pipe flows with smooth and rough walls

Studies in turbulence often focus on two flow conditions, both of which occur frequently in real-world flows and are sought-after for their value in advancing turbulence theory. These are the high Reynolds number regime and the effect of wall surface roughness. In this dissertation, a Large-Eddy Simulation (LES) recreates both conditions over a wide range of Reynolds numbers Re_τ = O(10²)-O(10⁸) and accounts for roughness by locally modeling the statistical effects of near-wall anisotropic fine scales in a thin layer immediately above the rough surface. A subgrid, roughness-corrected wall model is introduced to dynamically transmit this modeled information from the wall to the outer LES, which uses a stretched-vortex subgrid-scale model operating in the bulk of the flow. Of primary interest is the Reynolds number and roughness dependence of these flows in terms of first and second order statistics. The LES is first applied to a fully turbulent uniformly-smooth/rough channel flow to capture the flow dynamics over smooth, transitionally rough and fully rough regimes. Results include a Moody-like diagram for the wall averaged friction factor, believed to be the first of its kind obtained from LES. Confirmation is found for experimentally observed logarithmic behavior in the normalized stream-wise turbulent intensities. Tight logarithmic collapse, scaled on the wall friction velocity, is found for smooth-wall flows when Re_τ ≥ O(10⁶) and in fully rough cases. Since the wall model operates locally and dynamically, the framework is used to investigate non-uniform roughness distribution cases in a channel, where the flow adjustments to sudden surface changes are investigated. Recovery of mean quantities and turbulent statistics after transitions are discussed qualitatively and quantitatively at various roughness and Reynolds number levels. The internal boundary layer, which is defined as the border between the flow affected by the new surface condition and the unaffected part, is computed, and a collapse of the profiles on a length scale containing the logarithm of friction Reynolds number is presented. Finally, we turn to the possibility of expanding the present framework to accommodate more general geometries. As a first step, the whole LES framework is modified for use in the curvilinear geometry of a fully-developed turbulent pipe flow, with implementation carried out in a spectral element solver capable of handling complex wall profiles. The friction factors have shown favorable agreement with the superpipe data, and the LES estimates of the Karman constant and additive constant of the log-law closely match values obtained from experiment.

Multipole moments in general relativity and dynamical perturbations of black-hole magnetospheres

This thesis consists of two parts. In Part I, we develop a multipole moment formalism in general relativity and use it to analyze the motion and precession of compact bodies. More specifically, the generic, vacuum, dynamical gravitational field of the exterior universe in the vicinity of a freely moving body is expanded in positive powers of the distance r away from the body's spatial origin (i.e., in the distance r from its timelike-geodesic world line). The expansion coefficients, called "external multipole moments,'' are defined covariantly in terms of the Riemann curvature tensor and its spatial derivatives evaluated on the body's central world line. In a carefully chosen class of de Donder coordinates, the expansion of the external field involves only integral powers of r ; no logarithmic terms occur. The expansion is used to derive higher-order corrections to previously known laws of motion and precession for black holes and other bodies. The resulting laws of motion and precession are expressed in terms of couplings of the time derivatives of the body's quadrupole and octopole moments to the external moments, i.e., to the external curvature and its gradient.

In part II, we study the interaction of magnetohydrodynamic (MHD) waves in a black-hole magnetosphere with the "dragging of inertial frames" effect of the hole's rotation - i.e., with the hole's "gravitomagnetic field." More specifically: we first rewrite the laws of perfect general relativistic magnetohydrodynamics (GRMHD) in 3+1 language in a general spacetime, in terms of quantities (magnetic field, flow velocity, ...) that would be measured by the ''fiducial observers” whose world lines are orthogonal to (arbitrarily chosen) hypersurfaces of constant time. We then specialize to a stationary spacetime and MHD flow with one arbitrary spatial symmetry (e.g., the stationary magnetosphere of a Kerr black hole); and for this spacetime we reduce the GRMHD equations to a set of algebraic equations. The general features of the resulting stationary, symmetric GRMHD magnetospheric solutions are discussed, including the Blandford-Znajek effect in which the gravitomagnetic field interacts with the magnetosphere to produce an outflowing jet. Then in a specific model spacetime with two spatial symmetries, which captures the key features of the Kerr geometry, we derive the GRMHD equations which govern weak, linealized perturbations of a stationary magnetosphere with outflowing jet. These perturbation equations are then Fourier analyzed in time t and in the symmetry coordinate x, and subsequently solved numerically. The numerical solutions describe the interaction of MHD waves with the gravitomagnetic field. It is found that, among other features, when an oscillatory external force is applied to the region of the magnetosphere where plasma (e⁺e^-) is being created, the magnetosphere responds especially strongly at a particular, resonant, driving frequency. The resonant frequency is that for which the perturbations appear to be stationary (time independent) in the common rest frame of the freshly created plasma and the rotating magnetic field lines. The magnetosphere of a rotating black hole, when buffeted by nonaxisymmetric magnetic fields anchored in a surrounding accretion disk, might exhibit an analogous resonance. If so then the hole's outflowing jet might be modulated at resonant frequencies ω=(m/2) Ω_H where m is an integer and Ω_H is the hole's angular velocity.

Mean-speed measurements in two-dimensional, incompressible, fully-developed turbulent channel flow

Mean velocity profiles were measured in the 5” x 60” wind channel of the turbulence laboratory at the GALCIT, by the use of a hot-wire anemometer. The repeatability of results was established, and the accuracy of the instrumentation estimated. Scatter of experimental results is a little, if any, beyond this limit, although some effects might be expected to arise from variations in atmospheric humidity, no account of this factor having been taken in the present work. Also, slight unsteadiness in flow conditions will be responsible for some scatter.

Irregularities of a hot-wire in close proximity to a solid boundary at low speeds were observed, as have already been found by others.

That Kármán’s logarithmic law holds reasonably well over the main part of a fully developed turbulent flow was checked, the equation u/u_t = 6.0 + 6.25 log₁₀ y^ut/v being obtained, and, as has been previously the case, the experimental points do not quite form one straight line in the region where viscosity effects are small. The values of the constants for this law for the best over-all agreement were determined and compared with those obtained by others.

The range of Reynolds numbers used (based on half-width of channel) was from 20,000 to 60,000.

Observation and interpretation of tectonic strain release mechanisms

In four chapters various aspects of earthquake source are studied.

Chapter I

Surface displacements that followed the Parkfield, 1966, earthquakes were measured for two years with six small-scale geodetic networks straddling the fault trace. The logarithmic rate and the periodic nature of the creep displacement recorded on a strain meter made it possible to predict creep episodes on the San Andreas fault. Some individual earthquakes were related directly to surface displacement, while in general, slow creep and aftershock activity were found to occur independently. The Parkfield earthquake is interpreted as a buried dislocation.

Chapter II

The source parameters of earthquakes between magnitude 1 and 6 were studied using field observations, fault plane solutions, and surface wave and S-wave spectral analysis. The seismic moment, M_O, was found to be related to local magnitude, M_L, by log M_O = 1.7 M_L + 15.1. The source length vs magnitude relation for the San Andreas system found to be: M_L = 1.9 log L - 6.7. The surface wave envelope parameter AR gives the moment according to log M_O = log AR₃₀₀ + 30.1, and the stress drop, τ, was found to be related to the magnitude by τ = 0.54 M - 2.58. The relation between surface wave magnitude M_S and M_L is proposed to be M_S = 1.7 M_L - 4.1. It is proposed to estimate the relative stress level (and possibly the strength) of a source-region by the amplitude ratio of high-frequency to low-frequency waves. An apparent stress map for Southern California is presented.

Chapter III

Seismic triggering and seismic shaking are proposed as two closely related mechanisms of strain release which explain observations of the character of the P wave generated by the Alaskan earthquake of 1964, and distant fault slippage observed after the Borrego Mountain, California earthquake of 1968. The Alaska, 1964, earthquake is shown to be adequately described as a series of individual rupture events. The first of these events had a body wave magnitude of 6.6 and is considered to have initiated or triggered the whole sequence. The propagation velocity of the disturbance is estimated to be 3.5 km/sec. On the basis of circumstantial evidence it is proposed that the Borrego Mountain, 1968, earthquake caused release of tectonic strain along three active faults at distances of 45 to 75 km from the epicenter. It is suggested that this mechanism of strain release is best described as "seismic shaking."

Chapter IV

The changes of apparent stress with depth are studied in the South American deep seismic zone. For shallow earthquakes the apparent stress is 20 bars on the average, the same as for earthquakes in the Aleutians and on Oceanic Ridges. At depths between 50 and 150 km the apparent stresses are relatively high, approximately 380 bars, and around 600 km depth they are again near 20 bars. The seismic efficiency is estimated to be 0.1. This suggests that the true stress is obtained by multiplying the apparent stress by ten. The variation of apparent stress with depth is explained in terms of the hypothesis of ocean floor consumption.

Part I. Approximate Hartree-Fock wavefunctions one-electron properties and electronic structure of the water molecule. Part II. Perturbation-variational calculation of the nuclear spin-spin isotope coupling constant in HD

Part I

Several approximate Hartree-Fock SCF wavefunctions for the ground electronic state of the water molecule have been obtained using an increasing number of multicenter s, p, and d Slater-type atomic orbitals as basis sets. The predicted charge distribution has been extensively tested at each stage by calculating the electric dipole moment, molecular quadrupole moment, diamagnetic shielding, Hellmann-Feynman forces, and electric field gradients at both the hydrogen and the oxygen nuclei. It was found that a carefully optimized minimal basis set suffices to describe the electronic charge distribution adequately except in the vicinity of the oxygen nucleus. Our calculations indicate, for example, that the correct prediction of the field gradient at this nucleus requires a more flexible linear combination of p-orbitals centered on this nucleus than that in the minimal basis set. Theoretical values for the molecular octopole moment components are also reported.

Part II

The perturbation-variational theory of R. M. Pitzer for nuclear spin-spin coupling constants is applied to the HD molecule. The zero-order molecular orbital is described in terms of a single 1s Slater-type basis function centered on each nucleus. The first-order molecular orbital is expressed in terms of these two functions plus one singular basis function each of the types e^-r/r and e^-r ln r centered on one of the nuclei. The new kinds of molecular integrals were evaluated to high accuracy using numerical and analytical means. The value of the HD spin-spin coupling constant calculated with this near-minimal set of basis functions is J_HD = +96.6 cps. This represents an improvement over the previous calculated value of +120 cps obtained without using the logarithmic basis function but is still considerably off in magnitude compared with the experimental measurement of J_HD = +43 0 ± 0.5 cps.