Ordinary mod p representations of the metaplectic cover of p-adic SL_2

We classify the genuine ordinary mod p representations of the metaplectic group SL(2,F)-tilde, where F is a p-adic field, and compute its genuine mod p spherical and Iwahori Hecke algebras. The motivation is an interest in a possible correspondence between genuine mod p representations of SL(2,F)-tilde and mod p representations of the dual group PGL(2,F), so we also compare the two Hecke algebras to the mod p spherical and Iwahori Hecke algebras of PGL(2,F). We show that the genuine mod p spherical Hecke algebra of SL(2,F)-tilde is isomorphic to the mod p spherical Hecke algebra of PGL(2,F), and that one can choose an isomorphism which is compatible with a natural, though partial, correspondence of unramified ordinary representations via the Hecke action on their spherical vectors. We then show that the genuine mod p Iwahori Hecke algebra of SL(2,F)-tilde is a subquotient of the mod p Iwahori Hecke algebra of PGL(2,F), but that the two algebras are not isomorphic. This is in contrast to the situation in characteristic 0, where by work of Savin one can recover the local Shimura correspondence for representations generated by their Iwahori fixed vectors from an isomorphism of Iwahori Hecke algebras.

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.

Self-assembly of brush polymers

The unique structure and properties of brush polymers have led to increased interest in them within the scientific community. This thesis describes studies on the self-assembly of these brush polymers.

Chapter 2 describes a study on the rapid self-assembly of brush block copolymers into nanostructures with photonic bandgaps spanning the entire visible spectrum, from ultraviolet to near infrared. Linear relationships are observed between the peak wavelengths of reflection and polymer molecular weights. This work enables "bottom-up" fabrication of photonic crystals with application-tailored bandgaps, through synthetic control of the polymer molecular weight and the method of self-assembly.

Chapter 3 details the analysis of the self-assembly of symmetrical brush block copolymers in bulk and thin films. Highly ordered lamellae with domain spacing ranging from 20 to 240 nm are obtained by varying molecular weight of the backbone. The relationship between degree of polymerization and the domain spacing is reported, and evidence is provided for how rapidly the brush block copolymers self-assemble and achieve thermodynamic equilibrium.

Chapter 4 describes investigations into where morphology transitions take place as the volume fraction of each block is varied in asymmetrical brush block copolymers. Imaging techniques are used to observe a transition from lamellar to a cylindrical morphology as the volume fraction of one of the blocks exceeds 70%. It is also shown that the asymmetric brush block copolymers can be kinetically trapped into undulating lamellar structures by drop casting the samples.

Chapter 5 explores the capability of macromolecules to interdigitate into densely grafted molecular brush copolymers using stereocomplex formation as a driving force. The stereocomplex formation between complementary linear polymers and brush copolymers is demonstrated, while the stereocomplex formation between complementary brush copolymers is shown to be restricted.

Asymptotic analysis of thin plates under normal load and horizontal edge thrust

We consider the radially symmetric nonlinear von Kármán plate equations for circular or annular plates in the limit of small thickness. The loads on the plate consist of a radially symmetric pressure load and a uniform edge load. The dependence of the steady states on the edge load and thickness is studied using asymptotics as well as numerical calculations. The von Kármán plate equations are a singular perturbation of the Fӧppl membrane equation in the asymptotic limit of small thickness. We study the role of compressive membrane solutions in the small thickness asymptotic behavior of the plate solutions.

We give evidence for the existence of a singular compressive solution for the circular membrane and show by a singular perturbation expansion that the nonsingular compressive solution approach this singular solution as the radial stress at the center of the plate vanishes. In this limit, an infinite number of folds occur with respect to the edge load. Similar behavior is observed for the annular membrane with zero edge load at the inner radius in the limit as the circumferential stress vanishes.

We develop multiscale expansions, which are asymptotic to members of this family for plates with edges that are elastically supported against rotation. At some thicknesses this approximation breaks down and a boundary layer appears at the center of the plate. In the limit of small normal load, the points of breakdown approach the bifurcation points corresponding to buckling of the nondeflected state. A uniform asymptotic expansion for small thickness combining the boundary layer with a multiscale approximation of the outer solution is developed for this case. These approximations complement the well known boundary layer expansions based on tensile membrane solutions in describing the bending and stretching of thin plates. The approximation becomes inconsistent as the clamped state is approached by increasing the resistance against rotation at the edge. We prove that such an expansion for the clamped circular plate cannot exist unless the pressure load is self-equilibrating.

Dynamics and scaling of self-excited passive vortex generators for underwater propulsion

A series of experiments was conducted on the use of a device to passively generate vortex rings, henceforth a passive vortex generator (PVG). The device is intended as a means of propulsion for underwater vehicles, as the use of vortex rings has been shown to decrease the fuel consumption of a vehicle by up to 40% Ruiz (2010).

The PVG was constructed out of a collapsible tube encased in a rigid, airtight box. By adjusting the pressure within the airtight box while fluid was flowing through the tube, it was possible to create a pulsed jet with vortex rings via self-excited oscillations of the collapsible tube.

A study of PVG integration into an existing autonomous underwater vehicle (AUV) system was conducted. A small AUV was used to retrofit a PVG with limited alterations to the original vehicle. The PVG-integrated AUV was used for self-propelled testing to measure the hydrodynamic (Froude) efficiency of the system. The results show that the PVG-integrated AUV had a 22% increase in the Froude efficiency using a pulsed jet over a steady jet. The maximum increase in the Froude efficiency was realized when the formation time of the pulsed jet, a nondimensional time to characterize vortex ring formation, was coincident with vortex ring pinch-off. This is consistent with previous studies that indicate that the maximization of efficiency for a pulsed jet vehicle is realized when the formation of vortex rings maximizes the vortex ring energy and size.

The other study was a parameter study of the physical dimensions of a PVG. This study was conducted to determine the effect of the tube diameter and length on the oscillation characteristics such as the frequency. By changing the tube diameter and length by factors of 3, the frequency of self-excited oscillations was found to scale as f~D_0^{-1/2} L_0^0, where D_0 is the tube diameter and L_0 the tube length. The mechanism of operation is suggested to rely on traveling waves between the tube throat and the end of the tube. A model based on this mechanism yields oscillation frequencies that are within the range observed by the experiment.

Beyond Watson and Crick : programming the self-assembly and reconfiguration of DNA nanostructures based on stacking interactions

Life is the result of the execution of molecular programs: like how an embryo is fated to become a human or a whale, or how a person’s appearance is inherited from their parents, many biological phenomena are governed by genetic programs written in DNA molecules. At the core of such programs is the highly reliable base pairing interaction between nucleic acids. DNA nanotechnology exploits the programming power of DNA to build artificial nanostructures, molecular computers, and nanomachines. In particular, DNA origami—which is a simple yet versatile technique that allows one to create various nanoscale shapes and patterns—is at the heart of the technology. In this thesis, I describe the development of programmable self-assembly and reconfiguration of DNA origami nanostructures based on a unique strategy: rather than relying on Watson-Crick base pairing, we developed programmable bonds via the geometric arrangement of stacking interactions, which we termed stacking bonds. We further demonstrated that such bonds can be dynamically reconfigurable.

The first part of this thesis describes the design and implementation of stacking bonds. Our work addresses the fundamental question of whether one can create diverse bond types out of a single kind of attractive interaction—a question first posed implicitly by Francis Crick while seeking a deeper understanding of the origin of life and primitive genetic code. For the creation of multiple specific bonds, we used two different approaches: binary coding and shape coding of geometric arrangement of stacking interaction units, which are called blunt ends. To construct a bond space for each approach, we performed a systematic search using a computer algorithm. We used orthogonal bonds to experimentally implement the connection of five distinct DNA origami nanostructures. We also programmed the bonds to control cis/trans configuration between asymmetric nanostructures.

The second part of this thesis describes the large-scale self-assembly of DNA origami into two-dimensional checkerboard-pattern crystals via surface diffusion. We developed a protocol where the diffusion of DNA origami occurs on a substrate and is dynamically controlled by changing the cationic condition of the system. We used stacking interactions to mediate connections between the origami, because of their potential for reconfiguring during the assembly process. Assembling DNA nanostructures directly on substrate surfaces can benefit nano/microfabrication processes by eliminating a pattern transfer step. At the same time, the use of DNA origami allows high complexity and unique addressability with six-nanometer resolution within each structural unit.

The third part of this thesis describes the use of stacking bonds as dynamically breakable bonds. To break the bonds, we used biological machinery called the ParMRC system extracted from bacteria. The system ensures that, when a cell divides, each daughter cell gets one copy of the cell’s DNA by actively pushing each copy to the opposite poles of the cell. We demonstrate dynamically expandable nanostructures, which makes stacking bonds a promising candidate for reconfigurable connectors for nanoscale machine parts.

Crystals that count! Physical principles and experimental investigations of DNA tile self-assembly

Algorithmic DNA tiles systems are fascinating. From a theoretical perspective, they can result in simple systems that assemble themselves into beautiful, complex structures through fundamental interactions and logical rules. As an experimental technique, they provide a promising method for programmably assembling complex, precise crystals that can grow to considerable size while retaining nanoscale resolution. In the journey from theoretical abstractions to experimental demonstrations, however, lie numerous challenges and complications.

In this thesis, to examine these challenges, we consider the physical principles behind DNA tile self-assembly. We survey recent progress in experimental algorithmic self-assembly, and explain the simple physical models behind this progress. Using direct observation of individual tile attachments and detachments with an atomic force microscope, we test some of the fundamental assumptions of the widely-used kinetic Tile Assembly Model, obtaining results that fit the model to within error. We then depart from the simplest form of that model, examining the effects of DNA sticky end sequence energetics on tile system behavior. We develop theoretical models, sequence assignment algorithms, and a software package, StickyDesign, for sticky end sequence design.

As a demonstration of a specific tile system, we design a binary counting ribbon that can accurately count from a programmable starting value and stop growing after overflowing, resulting in a single system that can construct ribbons of precise and programmable length. In the process of designing the system, we explain numerous considerations that provide insight into more general tile system design, particularly with regards to tile concentrations, facet nucleation, the construction of finite assemblies, and design beyond the abstract Tile Assembly Model.

Finally, we present our crystals that count: experimental results with our binary counting system that represent a significant improvement in the accuracy of experimental algorithmic self-assembly, including crystals that count perfectly with 5 bits from 0 to 31. We show some preliminary experimental results on the construction of our capping system to stop growth after counters overflow, and offer some speculation on potential future directions of the field.

Self-healing Techniques for RF and mm-Wave Transmitters and Receivers

With continuing advances in CMOS technology, feature sizes of modern Silicon chip-sets have gone down drastically over the past decade. In addition to desktops and laptop processors, a vast majority of these chips are also being deployed in mobile communication devices like smart-phones and tablets, where multiple radio-frequency integrated circuits (RFICs) must be integrated into one device to cater to a wide variety of applications such as Wi-Fi, Bluetooth, NFC, wireless charging, etc. While a small feature size enables higher integration levels leading to billions of transistors co-existing on a single chip, it also makes these Silicon ICs more susceptible to variations. A part of these variations can be attributed to the manufacturing process itself, particularly due to the stringent dimensional tolerances associated with the lithographic steps in modern processes. Additionally, RF or millimeter-wave communication chip-sets are subject to another type of variation caused by dynamic changes in the operating environment. Another bottleneck in the development of high performance RF/mm-wave Silicon ICs is the lack of accurate analog/high-frequency models in nanometer CMOS processes. This can be primarily attributed to the fact that most cutting edge processes are geared towards digital system implementation and as such there is little model-to-hardware correlation at RF frequencies.

All these issues have significantly degraded yield of high performance mm-wave and RF CMOS systems which often require multiple trial-and-error based Silicon validations, thereby incurring additional production costs. This dissertation proposes a low overhead technique which attempts to counter the detrimental effects of these variations, thereby improving both performance and yield of chips post fabrication in a systematic way. The key idea behind this approach is to dynamically sense the performance of the system, identify when a problem has occurred, and then actuate it back to its desired performance level through an intelligent on-chip optimization algorithm. We term this technique as self-healing drawing inspiration from nature's own way of healing the body against adverse environmental effects. To effectively demonstrate the efficacy of self-healing in CMOS systems, several representative examples are designed, fabricated, and measured against a variety of operating conditions.

We demonstrate a high-power mm-wave segmented power mixer array based transmitter architecture that is capable of generating high-speed and non-constant envelope modulations at higher efficiencies compared to existing conventional designs. We then incorporate several sensors and actuators into the design and demonstrate closed-loop healing against a wide variety of non-ideal operating conditions. We also demonstrate fully-integrated self-healing in the context of another mm-wave power amplifier, where measurements were performed across several chips, showing significant improvements in performance as well as reduced variability in the presence of process variations and load impedance mismatch, as well as catastrophic transistor failure. Finally, on the receiver side, a closed-loop self-healing phase synthesis scheme is demonstrated in conjunction with a wide-band voltage controlled oscillator to generate phase shifter local oscillator (LO) signals for a phased array receiver. The system is shown to heal against non-idealities in the LO signal generation and distribution, significantly reducing phase errors across a wide range of frequencies.

Effects of self energy of the ions on the double layer structure and properties at the dielectric interface

Although numerous theoretical efforts have been put forth, a systematic, unified and predictive theoretical framework that is able to capture all the essential physics of the interfacial behaviors of ions, such as the Hofmeister series effect, Jones-Ray effect and the salt effect on the bubble coalescence remain an outstanding challenge. The most common approach to treating electrostatic interactions in the presence of salt ions is the Poisson-Boltzmann (PB) theory. However, there are many systems for which the PB theory fails to offer even a qualitative explanation of the behavior, especially for ions distributed in the vicinity of an interface with dielectric contrast between the two media (like the water-vapor/oil interface). A key factor missing in the PB theory is the self energy of the ion.

In this thesis, we develop a self-consistent theory that treats the electrostatic self energy (including both the short-range Born solvation energy and the long-range image charge interactions), the nonelectrostatic contribution of the self energy, the ion-ion correlation and the screening effect systematically in a single framework. By assuming a finite charge spread of the ion instead of using the point-charge model, the self energy obtained by our theory is free of the divergence problems and gives a continuous self energy across the interface. This continuous feature allows ions on the water side and the vapor/oil side of the interface to be treated in a unified framework. The theory involves a minimum set of parameters of the ion, such as the valency, radius, polarizability of the ions, and the dielectric constants of the medium, that are both intrinsic and readily available. The general theory is first applied to study the thermodynamic property of the bulk electrolyte solution, which shows good agreement with the experiment result for predicting the activity coefficient and osmotic coefficient.

Next, we address the effect of local Born solvation energy on the bulk thermodynamics and interfacial properties of electrolyte solution mixtures. We show that difference in the solvation energy between the cations and anions naturally gives rise to local charge separation near the interface, and a finite Galvani potential between two coexisting solutions. The miscibility of the mixture can either increases or decreases depending on the competition between the solvation energy and translation entropy of the ions. The interfacial tension shows a non-monotonic dependence on the salt concentration: it increases linearly with the salt concentration at higher concentrations, and decreases approximately as the square root of the salt concentration for dilute solutions, which is in agreement with the Jones-Ray effect observed in experiment.

Next, we investigate the image effects on the double layer structure and interfacial properties near a single charged plate. We show that the image charge repulsion creates a depletion boundary layer that cannot be captured by a regular perturbation approach. The correct weak-coupling theory must include the self-energy of the ion due to the image charge interaction. The image force qualitatively alters the double layer structure and properties, and gives rise to many non-PB effects, such as nonmonotonic dependence of the surface energy on concentration and charge inversion. The image charge effect is then studied for electrolyte solutions between two plates. For two neutral plates, we show that depletion of the salt ions by the image charge repulsion results in short-range attractive and long-range repulsive forces. If cations and anions are of different valency, the asymmetric depletion leads to the formation of an induced electrical double layer. For two charged plates, the competition between the surface charge and the image charge effect can give rise to like- charge attraction.

Then, we study the inhomogeneous screening effect near the dielectric interface due to the anisotropic and nonuniform ion distribution. We show that the double layer structure and interfacial properties is drastically affected by the inhomogeneous screening if the bulk Debye screening length is comparable or smaller than the Bjerrum length. The width of the depletion layer is characterized by the Bjerrum length, independent of the salt concentration. We predict that the negative adsorption of ions at the interface increases linearly with the salt concentration, which cannot be captured by either the bulk screening approximation or the WKB approximation. For asymmetric salt, the inhomogeneous screening enhances the charge separation in the induced double layer and significantly increases the value of the surface potential.

Finally, to account for the ion specificity, we study the self energy of a single ion across the dielectric interface. The ion is considered to be polarizable: its charge distribution can be self-adjusted to the local dielectric environment to minimize the self energy. Using intrinsic parameters of the ions, such as the valency, radius, and polarizability, we predict the specific ion effect on the interfacial affinity of halogen anions at the water/air interface, and the strong adsorption of hydrophobic ions at the water/oil interface, in agreement with experiments and atomistic simulations.

The theory developed in this work represents the most systematic theoretical technique for weak-coupling electrolytes. We expect the theory to be more useful for studying a wide range of structural and dynamic properties in physicochemical, colloidal, soft-matter and biophysical systems.

I. Non-linear effects in a self-consistent calculation of SU_3 symmetry breaking in strong interaction coupling constants. II. Direct-channel reggeization of strong interaction scattering amplitudes

The thesis is divided into two parts. Part I generalizes a self-consistent calculation of residue shifts from SU₃ symmetry, originally performed by Dashen, Dothan, Frautschi, and Sharp, to include the effects of non-linear terms. Residue factorizability is used to transform an overdetermined set of equations into a variational problem, which is designed to take advantage of the redundancy of the mathematical system. The solution of this problem automatically satisfies the requirement of factorizability and comes close to satisfying all the original equations.

Part II investigates some consequences of direct channel Regge poles and treats the problem of relating Reggeized partial wave expansions made in different reaction channels. An analytic method is introduced which can be used to determine the crossed-channel discontinuity for a large class of direct-channel Regge representations, and this method is applied to some specific representations.

It is demonstrated that the multi-sheeted analytic structure of the Regge trajectory function can be used to resolve apparent difficulties arising from infinitely rising Regge trajectories. Also discussed are the implications of large collections of "daughter trajectories."

Two things are of particular interest: first, the threshold behavior in direct and crossed channels; second, the potentialities of Reggeized representations for us in self-consistent calculations. A new representation is introduced which surpasses previous formulations in these two areas, automatically satisfying direct-channel threshold constraints while being capable of reproducing a reasonable crossed channel discontinuity. A scalar model is investigated for low energies, and a relation is obtained between the mass of the lowest bound state and the slope of the Regge trajectory.