I. The effect of N,N,N',N' -tetramethylethylenediamine on the Schlenk Equilibrium. II. The nature of the Di-Grignard reagent

I. The influence of N,N,N’,N’-tetramethylethylenediamine on the Schlenk equilibrium

The equilibrium between ethylmagnesium bromide, diethylmagnesium, and magnesium bromide has been studied by nuclear magnetic resonance spectroscopy. The interconversion of the species is very fast on the nmr time scale, and only an averaged spectrum is observed for the ethyl species. When N,N,N’,N’-tetramethylethylenediamine is added to solutions of these reagents in tetrahydrofuran, the rate of interconversion is reduced. At temperatures near -50°, two ethylmagnesium species have been observed. These are attributed to the different ethyl groups in ethylmagnesium bromide and diethylmagnesium, two of the species involved in the Schlenk equilibrium of Grignard reagents.

II. The nature of di-Grignard reagents

Di-Grignard reagents have been examined by nuclear magnetic resonance spectroscopy in an attempt to prove that dialkylmagnesium reagents are in equilibrium with alkylmagnesium halides. The di-Grignard reagents of compounds such as 1,4-dibromobutane have been investigated. The dialkylmagnesium form of this di-Grignard reagent can exist as an intramolecular cyclic species, tetramethylene-magnesium. This cyclic form would give an nmr spectrum different from that of the classical alkylmagnesium halide di-Grignard reagent. In dimethyl ether-tetrahydrofuran solutions of di-Grignard reagents containing N N,N,N’,N’-Tetramethylethylenediamine, evidence has been found for the existence of an intramolecular dialkylmagnesium species. This species is rapidly equilibrating with other forms, but at low temperatures, the rates of interconversion are reduced. Two species can be seen in the nmr spectrum at -50°. One is the cyclic species; the other is an open form.

Inversion of the carbon at the carbon-magnesium bond in di-Grignard reagents has also been studied. This process is much faster than in corresponding monofunctional Grignard reagents.

Invariants, Lie-Bäcklund operators and Bäcklund transformations

This thesis is mainly concerned with the application of groups of transformations to differential equations and in particular with the connection between the group structure of a given equation and the existence of exact solutions and conservation laws. In this respect the Lie-Bäcklund groups of tangent transformations, particular cases of which are the Lie tangent and the Lie point groups, are extensively used.

In Chapter I we first review the classical results of Lie, Bäcklund and Bianchi as well as the more recent ones due mainly to Ovsjannikov. We then concentrate on the Lie-Bäcklund groups (or more precisely on the corresponding Lie-Bäcklund operators), as introduced by Ibragimov and Anderson, and prove some lemmas about them which are useful for the following chapters. Finally we introduce the concept of a conditionally admissible operator (as opposed to an admissible one) and show how this can be used to generate exact solutions.

In Chapter II we establish the group nature of all separable solutions and conserved quantities in classical mechanics by analyzing the group structure of the Hamilton-Jacobi equation. It is shown that consideration of only Lie point groups is insufficient. For this purpose a special type of Lie-Bäcklund groups, those equivalent to Lie tangent groups, is used. It is also shown how these generalized groups induce Lie point groups on Hamilton's equations. The generalization of the above results to any first order equation, where the dependent variable does not appear explicitly, is obvious. In the second part of this chapter we investigate admissible operators (or equivalently constants of motion) of the Hamilton-Jacobi equation with polynornial dependence on the momenta. The form of the most general constant of motion linear, quadratic and cubic in the momenta is explicitly found. Emphasis is given to the quadratic case, where the particular case of a fixed (say zero) energy state is also considered; it is shown that in the latter case additional symmetries may appear. Finally, some potentials of physical interest admitting higher symmetries are considered. These include potentials due to two centers and limiting cases thereof. The most general two-center potential admitting a quadratic constant of motion is obtained, as well as the corresponding invariant. Also some new cubic invariants are found.

In Chapter III we first establish the group nature of all separable solutions of any linear, homogeneous equation. We then concentrate on the Schrodinger equation and look for an algorithm which generates a quantum invariant from a classical one. The problem of an isomorphism between functions in classical observables and quantum observables is studied concretely and constructively. For functions at most quadratic in the momenta an isomorphism is possible which agrees with Weyl' s transform and which takes invariants into invariants. It is not possible to extend the isomorphism indefinitely. The requirement that an invariant goes into an invariant may necessitate variants of Weyl' s transform. This is illustrated for the case of cubic invariants. Finally, the case of a specific value of energy is considered; in this case Weyl's transform does not yield an isomorphism even for the quadratic case. However, for this case a correspondence mapping a classical invariant to a quantum orie is explicitly found.

Chapters IV and V are concerned with the general group structure of evolution equations. In Chapter IV we establish a one to one correspondence between admissible Lie-Bäcklund operators of evolution equations (derivable from a variational principle) and conservation laws of these equations. This correspondence takes the form of a simple algorithm.

In Chapter V we first establish the group nature of all Bäcklund transformations (BT) by proving that any solution generated by a BT is invariant under the action of some conditionally admissible operator. We then use an algorithm based on invariance criteria to rederive many known BT and to derive some new ones. Finally, we propose a generalization of BT which, among other advantages, clarifies the connection between the wave-train solution and a BT in the sense that, a BT may be thought of as a variation of parameters of some. special case of the wave-train solution (usually the solitary wave one). Some open problems are indicated.

Most of the material of Chapters II and III is contained in [I], [II], [III] and [IV] and the first part of Chapter V in [V].

Wire array photovoltaics

Over the past five years, the cost of solar panels has dropped drastically and, in concert, the number of installed modules has risen exponentially. However, solar electricity is still more than twice as expensive as electricity from a natural gas plant. Fortunately, wire array solar cells have emerged as a promising technology for further lowering the cost of solar.

Si wire array solar cells are formed with a unique, low cost growth method and use 100 times less material than conventional Si cells. The wires can be embedded in a transparent, flexible polymer to create a free-standing array that can be rolled up for easy installation in a variety of form factors. Furthermore, by incorporating multijunctions into the wire morphology, higher efficiencies can be achieved while taking advantage of the unique defect relaxation pathways afforded by the 3D wire geometry.

The work in this thesis shepherded Si wires from undoped arrays to flexible, functional large area devices and laid the groundwork for multijunction wire array cells. Fabrication techniques were developed to turn intrinsic Si wires into full p-n junctions and the wires were passivated with a-Si:H and a-SiNx:H. Single wire devices yielded open circuit voltages of 600 mV and efficiencies of 9%. The arrays were then embedded in a polymer and contacted with a transparent, flexible, Ni nanoparticle and Ag nanowire top contact. The contact connected >99% of the wires in parallel and yielded flexible, substrate free solar cells featuring hundreds of thousands of wires.

Building on the success of the Si wire arrays, GaP was epitaxially grown on the material to create heterostructures for photoelectrochemistry. These cells were limited by low absorption in the GaP due to its indirect bandgap, and poor current collection due to a diffusion length of only 80 nm. However, GaAsP on SiGe offers a superior combination of materials, and wire architectures based on these semiconductors were investigated for multijunction arrays. These devices offer potential efficiencies of 34%, as demonstrated through an analytical model and optoelectronic simulations. SiGe and Ge wires were fabricated via chemical-vapor deposition and reactive ion etching. GaAs was then grown on these substrates at the National Renewable Energy Lab and yielded ns lifetime components, as required for achieving high efficiency devices.

Control of wettability of carbon nanotube array by reversible dry oxidation for superhydrophobic coating and supercapacitor applications

In this thesis, dry chemical modification methods involving UV/ozone, oxygen plasma, and vacuum annealing treatments are explored to precisely control the wettability of CNT arrays. By varying the exposure time of these treatments the surface concentration of oxygenated groups adsorbed on the CNT arrays can be controlled. CNT arrays with very low amount of oxygenated groups exhibit a superhydrophobic behavior. In addition to their extremely high static contact angle, they cannot be dispersed in DI water and their impedance in aqueous electrolytes is extremely high. These arrays have an extreme water repellency capability such that a water droplet will bounce off of their surface upon impact and a thin film of air is formed on their surface as they are immersed in a deep pool of water. In contrast, CNT arrays with very high surface concentration of oxygenated functional groups exhibit an extreme hydrophilic behavior. In addition to their extremely low static contact angle, they can be dispersed easily in DI water and their impedance in aqueous electrolytes is tremendously low. Since the bulk structure of the CNT arrays are preserved during the UV/ozone, oxygen plasma, and vacuum annealing treatments, all CNT arrays can be repeatedly switched between superhydrophilic and superhydrophobic, as long as their O/C ratio is kept below 18%.

The effect of oxidation using UV/ozone and oxygen plasma treatments is highly reversible as long as the O/C ratio of the CNT arrays is kept below 18%. At O/C ratios higher than 18%, the effect of oxidation is no longer reversible. This irreversible oxidation is caused by irreversible changes to the CNT atomic structure during the oxidation process. During the oxidation process, CNT arrays undergo three different processes. For CNT arrays with O/C ratios lower than 40%, the oxidation process results in the functionalization of CNT outer walls by oxygenated groups. Although this functionalization process introduces defects, vacancies and micropores opening, the graphitic structure of the CNT is still largely intact. For CNT arrays with O/C ratios between 40% and 45%, the oxidation process results in the etching of CNT outer walls. This etching process introduces large scale defects and holes that can be obviously seen under TEM at high magnification. Most of these holes are found to be several layers deep and, in some cases, a large portion of the CNT side walls are cut open. For CNT arrays with O/C ratios higher than 45%, the oxidation process results in the exfoliation of the CNT walls and amorphization of the remaining CNT structure. This amorphization process can be implied from the disappearance of C-C sp2 peak in the XPS spectra associated with the pi-bond network.

The impact behavior of water droplet impinging on superhydrophobic CNT arrays in a low viscosity regime is investigated for the first time. Here, the experimental data are presented in the form of several important impact behavior characteristics including critical Weber number, volume ratio, restitution coefficient, and maximum spreading diameter. As observed experimentally, three different impact regimes are identified while another impact regime is proposed. These regimes are partitioned by three critical Weber numbers, two of which are experimentally observed. The volume ratio between the primary and the secondary droplets is found to decrease with the increase of Weber number in all impact regimes other than the first one. In the first impact regime, this is found to be independent of Weber number since the droplet remains intact during and subsequent to the impingement. Experimental data show that the coefficient of restitution decreases with the increase of Weber number in all impact regimes. The rate of decrease of the coefficient of restitution in the high Weber number regime is found to be higher than that in the low and moderate Weber number. Experimental data also show that the maximum spreading factor increases with the increase of Weber number in all impact regimes. The rate of increase of the maximum spreading factor in the high Weber number regime is found to be higher than that in the low and moderate Weber number. Phenomenological approximations and interpretations of the experimental data, as well as brief comparisons to the previously proposed scaling laws, are shown here.

Dry oxidation methods are used for the first time to characterize the influence of oxidation on the capacitive behavior of CNT array EDLCs. The capacitive behavior of CNT array EDLCs can be tailored by varying their oxygen content, represented by their O/C ratio. The specific capacitance of these CNT arrays increases with the increase of their oxygen content in both KOH and Et4NBF4/PC electrolytes. As a result, their gravimetric energy density increases with the increase of their oxygen content. However, their gravimetric power density decreases with the increase of their oxygen content. The optimally oxidized CNT arrays are able to withstand more than 35,000 charge/discharge cycles in Et4NBF4/PC at a current density of 5 A/g while only losing 10% of their original capacitance.

A measurement of the proton elastic form factors for 1 ≤ Q^2 ≤ 3 (GeV/c)^2

We report measurements of the proton form factors, G^p_E and G^p_M, extracted from elastic electron scattering in the range 1 ≤ Q^2 ≤ 3 (GeV/c)^2 with uncertainties of <15% in G^p_E and <3% in G^p_M. The results for G^p_E are somewhat larger than indicated by most theoretical parameterizations. The ratio of Pauli and Dirac form factors, Q^2(F^p_2/F^p_1), is lower in value and demonstrates less Q^2 dependence than these parameterizations have indicated. Comparisons are made to theoretical models, including those based on perturbative QCD, vector-meson dominance, QCD sum rules, and diquark constituents to the proton. A global extraction of the form factors, including previous elastic scattering measurements, is also presented.

Designing conformational control of human tissue transglutaminase

Understanding the mechanisms of enzymes is crucial for our understanding of their role in biology and for designing methods to perturb or harness their activities for medical treatments, industrial processes, or biological engineering. One aspect of enzymes that makes them difficult to fully understand is that they are in constant motion, and these motions and the conformations adopted throughout these transitions often play a role in their function.

Traditionally, it has been difficult to isolate a protein in a particular conformation to determine what role each form plays in the reaction or biology of that enzyme. A new technology, computational protein design, makes the isolation of various conformations possible, and therefore is an extremely powerful tool in enabling a fuller understanding of the role a protein conformation plays in various biological processes.

One such protein that undergoes large structural shifts during different activities is human type II transglutaminase (TG2). TG2 is an enzyme that exists in two dramatically different conformational states: (1) an open, extended form, which is adopted upon the binding of calcium, and (2) a closed, compact form, which is adopted upon the binding of GTP or GDP. TG2 possess two separate active sites, each with a radically different activity. This open, calcium-bound form of TG2 is believed to act as a transglutaminse, where it catalyzes the formation of an isopeptide bond between the sidechain of a peptide-bound glutamine and a primary amine. The closed, GTP-bound conformation is believed to act as a GTPase. TG2 is also implicated in a variety of biological and pathological processes.

To better understand the effects of TG2’s conformations on its activities and pathological processes, we set out to design variants of TG2 isolated in either the closed or open conformations. We were able to design open-locked and closed-biased TG2 variants, and use these designs to unseat the current understanding of the activities and their concurrent conformations of TG2 and explore each conformation’s role in celiac disease models. This work also enabled us to help explain older confusing results in regards to this enzyme and its activities. The new model for TG2 activity has immense implications for our understanding of its functional capabilities in various environments, and for our ability to understand which conformations need to be inhibited in the design of new drugs for diseases in which TG2’s activities are believed to elicit pathological effects.

First principles based multiparadigm modeling of electronic structures and dynamics

Electronic structures and dynamics are the key to linking the material composition and structure to functionality and performance.

An essential issue in developing semiconductor devices for photovoltaics is to design materials with optimal band gaps and relative positioning of band levels. Approximate DFT methods have been justified to predict band gaps from KS/GKS eigenvalues, but the accuracy is decisively dependent on the choice of XC functionals. We show here for CuInSe₂ and CuGaSe₂, the parent compounds of the promising CIGS solar cells, conventional LDA and GGA obtain gaps of 0.0-0.01 and 0.02-0.24 eV (versus experimental values of 1.04 and 1.67 eV), while the historically first global hybrid functional, B3PW91, is surprisingly the best, with band gaps of 1.07 and 1.58 eV. Furthermore, we show that for 27 related binary and ternary semiconductors, B3PW91 predicts gaps with a MAD of only 0.09 eV, which is substantially better than all modern hybrid functionals, including B3LYP (MAD of 0.19 eV) and screened hybrid functional HSE06 (MAD of 0.18 eV).

The laboratory performance of CIGS solar cells (> 20% efficiency) makes them promising candidate photovoltaic devices. However, there remains little understanding of how defects at the CIGS/CdS interface affect the band offsets and interfacial energies, and hence the performance of manufactured devices. To determine these relationships, we use the B3PW91 hybrid functional of DFT with the AEP method that we validate to provide very accurate descriptions of both band gaps and band offsets. This confirms the weak dependence of band offsets on surface orientation observed experimentally. We predict that the CBO of perfect CuInSe₂/CdS interface is large, 0.79 eV, which would dramatically degrade performance. Moreover we show that band gap widening induced by Ga adjusts only the VBO, and we find that Cd impurities do not significantly affect the CBO. Thus we show that Cu vacancies at the interface play the key role in enabling the tunability of CBO. We predict that Na further improves the CBO through electrostatically elevating the valence levels to decrease the CBO, explaining the observed essential role of Na for high performance. Moreover we find that K leads to a dramatic decrease in the CBO to 0.05 eV, much better than Na. We suggest that the efficiency of CIGS devices might be improved substantially by tuning the ratio of Na to K, with the improved phase stability of Na balancing phase instability from K. All these defects reduce interfacial stability slightly, but not significantly.

A number of exotic structures have been formed through high pressure chemistry, but applications have been hindered by difficulties in recovering the high pressure phase to ambient conditions (i.e., one atmosphere and room temperature). Here we use dispersion-corrected DFT (PBE-ulg flavor) to predict that above 60 GPa the most stable form of N₂O (the laughing gas in its molecular form) is a 1D polymer with an all-nitrogen backbone analogous to cis-polyacetylene in which alternate N are bonded (ionic covalent) to O. The analogous trans-polymer is only 0.03-0.10 eV/molecular unit less stable. Upon relaxation to ambient conditions both polymers relax below 14 GPa to the same stable non-planar trans-polymer, accompanied by possible electronic structure transitions. The predicted phonon spectrum and dissociation kinetics validate the stability of this trans-poly-NNO at ambient conditions, which has potential applications as a new type of conducting polymer with all-nitrogen chains and as a high-energy oxidizer for rocket propulsion. This work illustrates in silico materials discovery particularly in the realm of extreme conditions.

Modeling non-adiabatic electron dynamics has been a long-standing challenge for computational chemistry and materials science, and the eFF method presents a cost-efficient alternative. However, due to the deficiency of FSG representation, eFF is limited to low-Z elements with electrons of predominant s-character. To overcome this, we introduce a formal set of ECP extensions that enable accurate description of p-block elements. The extensions consist of a model representing the core electrons with the nucleus as a single pseudo particle represented by FSG, interacting with valence electrons through ECPs. We demonstrate and validate the ECP extensions for complex bonding structures, geometries, and energetics of systems with p-block character (C, O, Al, Si) and apply them to study materials under extreme mechanical loading conditions.

Despite its success, the eFF framework has some limitations, originated from both the design of Pauli potentials and the FSG representation. To overcome these, we develop a new framework of two-level hierarchy that is a more rigorous and accurate successor to the eFF method. The fundamental level, GHA-QM, is based on a new set of Pauli potentials that renders exact QM level of accuracy for any FSG represented electron systems. To achieve this, we start with using exactly derived energy expressions for the same spin electron pair, and fitting a simple functional form, inspired by DFT, against open singlet electron pair curves (H₂ systems). Symmetric and asymmetric scaling factors are then introduced at this level to recover the QM total energies of multiple electron pair systems from the sum of local interactions. To complement the imperfect FSG representation, the AMPERE extension is implemented, and aims at embedding the interactions associated with both the cusp condition and explicit nodal structures. The whole GHA-QM+AMPERE framework is tested on H element, and the preliminary results are promising.

MEMS for Diabetic Retinopathy

As the worldwide prevalence of diabetes mellitus continues to increase, diabetic retinopathy remains the leading cause of visual impairment and blindness in many developed countries. Between 32 to 40 percent of about 246 million people with diabetes develop diabetic retinopathy. Approximately 4.1 million American adults 40 years and older are affected by diabetic retinopathy. This glucose-induced microvascular disease progressively damages the tiny blood vessels that nourish the retina, the light-sensitive tissue at the back of the eye, leading to retinal ischemia (i.e., inadequate blood flow), retinal hypoxia (i.e., oxygen deprivation), and retinal nerve cell degeneration or death. It is a most serious sight-threatening complication of diabetes, resulting in significant irreversible vision loss, and even total blindness.

Unfortunately, although current treatments of diabetic retinopathy (i.e., laser therapy, vitrectomy surgery and anti-VEGF therapy) can reduce vision loss, they only slow down but cannot stop the degradation of the retina. Patients require repeated treatment to protect their sight. The current treatments also have significant drawbacks. Laser therapy is focused on preserving the macula, the area of the retina that is responsible for sharp, clear, central vision, by sacrificing the peripheral retina since there is only limited oxygen supply. Therefore, laser therapy results in a constricted peripheral visual field, reduced color vision, delayed dark adaptation, and weakened night vision. Vitrectomy surgery increases the risk of neovascular glaucoma, another devastating ocular disease, characterized by the proliferation of fibrovascular tissue in the anterior chamber angle. Anti-VEGF agents have potential adverse effects, and currently there is insufficient evidence to recommend their routine use.

In this work, for the first time, a paradigm shift in the treatment of diabetic retinopathy is proposed: providing localized, supplemental oxygen to the ischemic tissue via an implantable MEMS device. The retinal architecture (e.g., thickness, cell densities, layered structure, etc.) of the rabbit eye exposed to ischemic hypoxic injuries was well preserved after targeted oxygen delivery to the hypoxic tissue, showing that the use of an external source of oxygen could improve the retinal oxygenation and prevent the progression of the ischemic cascade.

The proposed MEMS device transports oxygen from an oxygen-rich space to the oxygen-deficient vitreous, the gel-like fluid that fills the inside of the eye, and then to the ischemic retina. This oxygen transport process is purely passive and completely driven by the gradient of oxygen partial pressure (pO₂). Two types of devices were designed. For the first type, the oxygen-rich space is underneath the conjunctiva, a membrane covering the sclera (white part of the eye), beneath the eyelids and highly permeable to oxygen in the atmosphere when the eye is open. Therefore, sub-conjunctival pO₂ is very high during the daytime. For the second type, the oxygen-rich space is inside the device since pure oxygen is needle-injected into the device on a regular basis.

To prevent too fast or too slow permeation of oxygen through the device that is made of parylene and silicone (two widely used biocompatible polymers in medical devices), the material properties of the hybrid parylene/silicone were investigated, including mechanical behaviors, permeation rates, and adhesive forces. Then the thicknesses of parylene and silicone became important design parameters that were fine-tuned to reach the optimal oxygen permeation rate.

The passive MEMS oxygen transporter devices were designed, built, and tested in both bench-top artificial eye models and in-vitro porcine cadaver eyes. The 3D unsteady saccade-induced laminar flow of water inside the eye model was modeled by computational fluid dynamics to study the convective transport of oxygen inside the eye induced by saccade (rapid eye movement). The saccade-enhanced transport effect was also demonstrated experimentally. Acute in-vivo animal experiments were performed in rabbits and dogs to verify the surgical procedure and the device functionality. Various hypotheses were confirmed both experimentally and computationally, suggesting that both the two types of devices are very promising to cure diabetic retinopathy. The chronic implantation of devices in ischemic dog eyes is still underway.

The proposed MEMS oxygen transporter devices can be also applied to treat other ocular and systemic diseases accompanied by retinal ischemia, such as central retinal artery occlusion, carotid artery disease, and some form of glaucoma.

A Direct Approach to Robustness Optimization

This dissertation reformulates and streamlines the core tools of robustness analysis for linear time invariant systems using now-standard methods in convex optimization. In particular, robust performance analysis can be formulated as a primal convex optimization in the form of a semidefinite program using a semidefinite representation of a set of Gramians. The same approach with semidefinite programming duality is applied to develop a linear matrix inequality test for well-connectedness analysis, and many existing results such as the Kalman-Yakubovich--Popov lemma and various scaled small gain tests are derived in an elegant fashion. More importantly, unlike the classical approach, a decision variable in this novel optimization framework contains all inner products of signals in a system, and an algorithm for constructing an input and state pair of a system corresponding to the optimal solution of robustness optimization is presented based on this information. This insight may open up new research directions, and as one such example, this dissertation proposes a semidefinite programming relaxation of a cardinality constrained variant of the H ∞ norm, which we term sparse H ∞ analysis, where an adversarial disturbance can use only a limited number of channels. Finally, sparse H ∞ analysis is applied to the linearized swing dynamics in order to detect potential vulnerable spots in power networks.

Effects of density differences on lateral mixing in open-channel flows

This study investigates lateral mixing of tracer fluids in turbulent open-channel flows when the tracer and ambient fluids have different densities. Longitudinal dispersion in flows with longitudinal density gradients is investigated also.

Lateral mixing was studied in a laboratory flume by introducing fluid tracers at the ambient flow velocity continuously and uniformly across a fraction of the flume width and over the entire depth of the ambient flow. Fluid samples were taken to obtain concentration distributions in cross-sections at various distances, x, downstream from the tracer source. The data were used to calculate variances of the lateral distributions of the depth-averaged concentration. When there was a difference in density between the tracer and the ambient fluids, lateral mixing close to the source was enhanced by density-induced secondary flows; however, far downstream where the density gradients were small, lateral mixing rates were independent of the initial density difference. A dimensional analysis of the problem and the data show that the normalized variance is a function of only three dimensionless numbers, which represent: (1) the x-coordinate, (2) the source width, and (3) the buoyancy flux from the source.

A simplified set of equations of motion for a fluid with a horizontal density gradient was integrated to give an expression for the density-induced velocity distribution. The dispersion coefficient due to this velocity distribution was also obtained. Using this dispersion coefficient in an analysis for predicting lateral mixing rates in the experiments of this investigation gave only qualitative agreement with the data. However, predicted longitudinal salinity distributions in an idealized laboratory estuary agree well with published data.

Deep near-infrared spectroscopy of high-redshift galaxies: the physical growth of passive systems

The assembly history of massive galaxies is one of the most important aspects of galaxy formation and evolution. Although we have a broad idea of what physical processes govern the early phases of galaxy evolution, there are still many open questions. In this thesis I demonstrate the crucial role that spectroscopy can play in a physical understanding of galaxy evolution. I present deep near-infrared spectroscopy for a sample of high-redshift galaxies, from which I derive important physical properties and their evolution with cosmic time. I take advantage of the recent arrival of efficient near-infrared detectors to target the rest-frame optical spectra of z > 1 galaxies, from which many physical quantities can be derived. After illustrating the applications of near-infrared deep spectroscopy with a study of star-forming galaxies, I focus on the evolution of massive quiescent systems.

Most of this thesis is based on two samples collected at the W. M. Keck Observatory that represent a significant step forward in the spectroscopic study of z > 1 quiescent galaxies. All previous spectroscopic samples at this redshift were either limited to a few objects, or much shallower in terms of depth. Our first sample is composed of 56 quiescent galaxies at 1 < z < 1.6 collected using the upgraded red arm of the Low Resolution Imaging Spectrometer (LRIS). The second consists of 24 deep spectra of 1.5 < z < 2.5 quiescent objects observed with the Multi-Object Spectrometer For Infra-Red Exploration (MOSFIRE). Together, these spectra span the critical epoch 1 < z < 2.5, where most of the red sequence is formed, and where the sizes of quiescent systems are observed to increase significantly.

We measure stellar velocity dispersions and dynamical masses for the largest number of z > 1 quiescent galaxies to date. By assuming that the velocity dispersion of a massive galaxy does not change throughout its lifetime, as suggested by theoretical studies, we match galaxies in the local universe with their high-redshift progenitors. This allows us to derive the physical growth in mass and size experienced by individual systems, which represents a substantial advance over photometric inferences based on the overall galaxy population. We find a significant physical growth among quiescent galaxies over 0 < z < 2.5 and, by comparing the slope of growth in the mass-size plane dlogR_e/dlogM_∗ with the results of numerical simulations, we can constrain the physical process responsible for the evolution. Our results show that the slope of growth becomes steeper at higher redshifts, yet is broadly consistent with minor mergers being the main process by which individual objects evolve in mass and size.

By fitting stellar population models to the observed spectroscopy and photometry we derive reliable ages and other stellar population properties. We show that the addition of the spectroscopic data helps break the degeneracy between age and dust extinction, and yields significantly more robust results compared to fitting models to the photometry alone. We detect a clear relation between size and age, where larger galaxies are younger. Therefore, over time the average size of the quiescent population will increase because of the contribution of large galaxies recently arrived to the red sequence. This effect, called progenitor bias, is different from the physical size growth discussed above, but represents another contribution to the observed difference between the typical sizes of low- and high-redshift quiescent galaxies. By reconstructing the evolution of the red sequence starting at z ∼ 1.25 and using our stellar population histories to infer the past behavior to z ∼ 2, we demonstrate that progenitor bias accounts for only half of the observed growth of the population. The remaining size evolution must be due to physical growth of individual systems, in agreement with our dynamical study.

Finally, we use the stellar population properties to explore the earliest periods which led to the formation of massive quiescent galaxies. We find tentative evidence for two channels of star formation quenching, which suggests the existence of two independent physical mechanisms. We also detect a mass downsizing, where more massive galaxies form at higher redshift, and then evolve passively. By analyzing in depth the star formation history of the brightest object at z > 2 in our sample, we are able to put constraints on the quenching timescale and on the properties of its progenitor.

A consistent picture emerges from our analyses: massive galaxies form at very early epochs, are quenched on short timescales, and then evolve passively. The evolution is passive in the sense that no new stars are formed, but significant mass and size growth is achieved by accreting smaller, gas-poor systems. At the same time the population of quiescent galaxies grows in number due to the quenching of larger star-forming galaxies. This picture is in agreement with other observational studies, such as measurements of the merger rate and analyses of galaxy evolution at fixed number density.

A generalized Hausdorff dimension for functions and sets

Let E be a compact subset of the n-dimensional unit cube, 1_n, and let C be a collection of convex bodies, all of positive n-dimensional Lebesgue measure, such that C contains bodies with arbitrarily small measure. The dimension of E with respect to the covering class C is defined to be the number

d_C(E) = sup(β:H_{β, C}(E) > 0),

where H_{β, C} is the outer measure

inf(Ʃm(C_i)^β:UC_i Ↄ E, C_i ϵ C) .

Only the one and two-dimensional cases are studied. Moreover, the covering classes considered are those consisting of intervals and rectangles, parallel to the coordinate axes, and those closed under translations. A covering class is identified with a set of points in the left-open portion, 1’_n, of 1_n, whose closure intersects 1_n - 1’_n. For n = 2, the outer measure H_{β, C} is adopted in place of the usual:

Inf(Ʃ(diam. (C_i))^β: UC_i Ↄ E, C_i ϵ C),

for the purpose of studying the influence of the shape of the covering sets on the dimension d_C(E).

If E is a closed set in 1₁, let M(E) be the class of all non-decreasing functions μ(x), supported on E with μ(x) = 0, x ≤ 0 and μ(x) = 1, x ≥ 1. Define for each μ ϵ M(E),

d_C(μ) = lim/c → inf/0 log ∆μ(c)/log c , (c ϵ C)

where ∆μ(c) = v/x (μ(x+c) – μ(x)). It is shown that

d_C(E) = sup (d_C(μ):μ ϵ M(E)).

This notion of dimension is extended to a certain class Ӻ of sub-additive functions, and the problem of studying the behavior of d_C(E) as a function of the covering class C is reduced to the study of d_C(f) where f ϵ Ӻ. Specifically, the set of points in 1₁,

(*) {d_B(F), d_C(f)): f ϵ Ӻ}

is characterized by a comparison of the relative positions of the points of B and C. A region of the form (*) is always closed and doubly-starred with respect to the points (0, 0) and (1, 1). Conversely, given any closed region in 1₂, doubly-starred with respect to (0, 0) and (1, 1), there are covering classes B and C such that (*) is exactly that region. All of the results are shown to apply to the dimension of closed sets E. Similar results can be obtained when a finite number of covering classes are considered.

In two dimensions, the notion of dimension is extended to the class M, of functions f(x, y), non-decreasing in x and y, supported on 1₂ with f(x, y) = 0 for x · y = 0 and f(1, 1) = 1, by the formula

d_C(f) = lim/s · t → inf/0 log ∆f(s, t)/log s · t , (s, t) ϵ C

where

∆f(s, t) = V/x, y (f(x+s, y+t) – f(x+s, y) – f(x, y+t) + f(x, t)).

A characterization of the equivalence d_C1(f) = d_C2(f) for all f ϵ M, is given by comparison of the gaps in the sets of products s · t and quotients s/t, (s, t) ϵ C_i (I = 1, 2).

Development of roll waves in open channels

This study is concerned with some of the properties of roll waves that develop naturally from a turbulent uniform flow in a wide rectangular channel on a constant steep slope . The wave properties considered were depth at the wave crest, depth at the wave trough, wave period, and wave velocity . The primary focus was on the mean values and standard deviations of the crest depths and wave periods at a given station and how these quantities varied with distance along the channel.

The wave properties were measured in a laboratory channel in which roll waves developed naturally from a uniform flow . The Froude number F (F = u_n/√gh_n, u_n = normal velocity , h_n = normal depth, g =acceleration of gravity) ranged from 3. 4 to 6. 0 for channel slopes S_o of . 05 and . 12 respectively . In the initial phase of their development the roll waves appeared as small amplitude waves with a continuous water surface profile . These small amplitude waves subsequently developed into large amplitude shock waves. Shock waves were found to overtake and combine with other shock waves with the result that the crest depth of the combined wave was larger than the crest depths before the overtake. Once roll waves began to develop, the mean value of the crest depths h_nmax increased with distance . Once the shock waves began to overtake, the mean wave period T_av increased approximately linearly with distance.

For a given Froude number and channel slope the observed quantities h^-_max/h_n , T' (T' = S_o T_av √g/h_n), and the standard deviations of h^-_max/h_n and T', could be expressed as unique functions of l/h_n (l = distance from beginning of channel) for the two-fold change in h_n occurring in the observed flows . A given value of h^-_max/h_n occurred at smaller values of l/h_n as the Froude number was increased. For a given value of h /hh^-_max/h_n the growth rate of δh^-_max/h^-_maxδl of the shock waves increased as the Froude number was increased.

A laboratory channel was also used to measure the wave properties of periodic permanent roll waves. For a given Froude number and channel slope the h^-_max/h_n vs. T' relation did not agree with a theory in which the weight of the shock front was neglected. After the theory was modified to include this weight, the observed values of h^-_max/h_n were within an average of 6.5 percent of the predicted values, and the maximum discrepancy was 13.5 percent.

For h^-_max/h_n sufficiently large (h^-_max/h_n > approximately 1.5) it was found that the h^-_max/h_n vs. T' relation for natural roll waves was practically identical to the h^-_max/h_n vs. T' relation for periodic permanent roll waves at the same Froude number and slope. As a result of this correspondence between periodic and natural roll waves, the growth rate δh^-_max/h^-_maxδl of shock waves was predicted to depend on the channel slope, and this slope dependence was observed in the experiments.

Optomechanical Inertial Sensors and Feedback Cooling

The optomechanical interaction is an extremely powerful tool with which to measure mechanical motion. The displacement resolution of chip-scale optomechanical systems has been measured on the order of 1⁄10th of a proton radius. So strong is this optomechanical interaction that it has recently been used to remove almost all thermal noise from a mechanical resonator and observe its quantum ground-state of motion starting from cryogenic temperatures.

In this work, chapter 1 describes the basic physics of the canonical optomechanical system, optical measurement techniques, and how the optomechanical interaction affects the coupled mechanical resonator. In chapter 2, we describe our techniques for realizing this canonical optomechanical system in a chip-scale form factor.

In chapter 3, we describe an experiment where we used radiation pressure feedback to cool a mesoscopic mechanical resonator near its quantum ground-state from room-temperature. We cooled the resonator from a room temperature phonon occupation of <n> = 6.5 million to an occupation of <n> = 66, which means the resonator is in its ground state approximately 2% of the time, while being coupled to a room-temperature thermal environment. At the time of this work, this is the closest a mesoscopic mechanical resonator has been to its ground-state of motion at room temperature, and this work begins to open the door to room-temperature quantum control of mechanical objects.

Chapter 4 begins with the realization that the displacement resolutions achieved by optomechanical systems can surpass those of conventional MEMS sensors by an order of magnitude or more. This provides the motivation to develop and calibrate an optomechanical accelerometer with a resolution of approximately 10 micro-g/rt-Hz over a bandwidth of approximately 30 kHz. In chapter 5, we improve upon the performance and practicality of this sensor by greatly increasing the test mass size, investigating and reducing low-frequency noise, and incorporating more robust optical coupling techniques and capacitive wavelength tuning. Finally, in chapter 6 we present our progress towards developing another optomechanical inertial sensor - a gyroscope.

Smooth Banach spaces and approximations

If E and F are real Banach spaces let C^p,q(E, F) O ≤ q ≤ p ≤ ∞, denote those maps from E to F which have p continuous Frechet derivatives of which the first q derivatives are bounded. A Banach space E is defined to be C^p,q smooth if C^p,q(E,R) contains a nonzero function with bounded support. This generalizes the standard C^p smoothness classification.

If an L^p space, p ≥ 1, is C^q smooth then it is also C^q,q smooth so that in particular L^p for p an even integer is C^∞,∞ smooth and L^p for p an odd integer is C^p-1,p-1 smooth. In general, however, a C^p smooth B-space need not be C^p,p smooth. C_o is shown to be a non-C^2,2 smooth B-space although it is known to be C^∞ smooth. It is proved that if E is C^p,1 smooth then C_o(E) is C^p,1 smooth and if E has an equivalent C^p norm then c_o(E) has an equivalent C^p norm.

Various consequences of C^p,q smoothness are studied. If f ϵ C^p,q(E,F), if F is C^p,q smooth and if E is non-C^p,q smooth, then the image under f of the boundary of any bounded open subset U of E is dense in the image of U. If E is separable then E is C^p,q smooth if and only if E admits C^p,q partitions of unity; E is C^p,psmooth, p ˂∞, if and only if every closed subset of E is the zero set of some C^P function.

f ϵ C^q(E,F), 0 ≤ q ≤ p ≤ ∞, is said to be C_p,q approximable on a subset U of E if for any ϵ ˃ 0 there exists a g ϵ C^p(E,F) satisfying

sup/xϵU, O≤k≤q ‖ D^k f(x) - D^k g(x) ‖ ≤ ϵ.

It is shown that if E is separable and C^p,q smooth and if f ϵ C^q(E,F) is C_p,q approximable on some neighborhood of every point of E, then F is C_p,q approximable on all of E.

In general it is unknown whether an arbitrary function in C¹(l², R) is C_2,1 approximable and an example of a function in C¹(l², R) which may not be C_2,1 approximable is given. A weak form of C_∞,q, q≥1, to functions in C^q(l², R) is proved: Let {U_α} be a locally finite cover of l² and let {T_α} be a corresponding collection of Hilbert-Schmidt operators on l². Then for any f ϵ C^q(l²,F) such that for all α

sup ‖ D^k(f(x)-g(x))[T_αh]‖ ≤ 1.

xϵU_α,‖h‖≤1, 0≤k≤q