Topics in randomized numerical linear algebra

This thesis studies three classes of randomized numerical linear algebra algorithms, namely: (i) randomized matrix sparsification algorithms, (ii) low-rank approximation algorithms that use randomized unitary transformations, and (iii) low-rank approximation algorithms for positive-semidefinite (PSD) matrices.

Randomized matrix sparsification algorithms set randomly chosen entries of the input matrix to zero. When the approximant is substituted for the original matrix in computations, its sparsity allows one to employ faster sparsity-exploiting algorithms. This thesis contributes bounds on the approximation error of nonuniform randomized sparsification schemes, measured in the spectral norm and two NP-hard norms that are of interest in computational graph theory and subset selection applications.

Low-rank approximations based on randomized unitary transformations have several desirable properties: they have low communication costs, are amenable to parallel implementation, and exploit the existence of fast transform algorithms. This thesis investigates the tradeoff between the accuracy and cost of generating such approximations. State-of-the-art spectral and Frobenius-norm error bounds are provided.

The last class of algorithms considered are SPSD "sketching" algorithms. Such sketches can be computed faster than approximations based on projecting onto mixtures of the columns of the matrix. The performance of several such sketching schemes is empirically evaluated using a suite of canonical matrices drawn from machine learning and data analysis applications, and a framework is developed for establishing theoretical error bounds.

In addition to studying these algorithms, this thesis extends the Matrix Laplace Transform framework to derive Chernoff and Bernstein inequalities that apply to all the eigenvalues of certain classes of random matrices. These inequalities are used to investigate the behavior of the singular values of a matrix under random sampling, and to derive convergence rates for each individual eigenvalue of a sample covariance matrix.

Generic differentiability of convex functions and monotone operators

The aim of this paper is to investigate to what extent the known theory of subdifferentiability and generic differentiability of convex functions defined on open sets can be carried out in the context of convex functions defined on not necessarily open sets. Among the main results obtained I would like to mention a Kenderov type theorem (the subdifferential at a generic point is contained in a sphere), a generic Gâteaux differentiability result in Banach spaces of class S and a generic Fréchet differentiability result in Asplund spaces. At least two methods can be used to prove these results: first, a direct one, and second, a more general one, based on the theory of monotone operators. Since this last theory was previously developed essentially for monotone operators defined on open sets, it was necessary to extend it to the context of monotone operators defined on a larger class of sets, our "quasi open" sets. This is done in Chapter III. As a matter of fact, most of these results have an even more general nature and have roots in the theory of minimal usco maps, as shown in Chapter II.

1994: Optimismus nur für den Nordseesprottenbestand

The new German R.V. "Walther Herwig III" joined the International North Sea Bottom Trawl Survey (IBTS) coordinated in the frame or ICES from February 18 to March 22 1994. A total of 354 valid half hour tows were made by seven research vessels from the different ICES members in order to determine the strength of incoming year classes of commercially most important fish species. Preliminary index figures for the stocks under observation of all participating vessels show that only the North Sea sprat stock developed a substantial year class in 1993. In addition, hydrographic data were collected during the survey. These data show the temperatures and salinities of the northern and central North Sea in most parts beneath the mean values during the period under investigation.

A unified strategy to ent-kauranoid natural products : total syntheses of (–)-maoecrystal Z, (–)-trichorabdal A, and (–)-longikaurin E

The diterpenoid constituents of the Isodon plants have attracted reasearchers interested in both their chemical structures and biological properties for more than a half-century. In recent years, the isolations of new members displaying previously unprecedented ring systems and highly selective biological properties have piqued interest from the synthetic community in this class of natural products.

Reported herein is the first total synthesis of such a recently isolated diterpenoid, (–)-maoecrystal Z. The principal transformations implemented in this synthesis include two highly diastereoselective radical cyclization reactions: a Sm^(II)-mediated reductive cascade cyclization, which forms two rings and establishes four new stereocenters in a single step, and a Ti^(III)-mediated reductive epoxide-acrylate coupling that yields a functionalized spirolactone product, which forms a core bicycle of maoecrystal Z.

The preparation of two additional ent-kauranoid natural products, (–)-trichorabdal A and (–)-longikaurin E, is also described from a derivative of this key spirolactone. These syntheses are additionally enabled by the palladium-mediated oxidative cyclization reaction of a silyl ketene acetal precursor that is used to install the bridgehead all-carbon quaternary stereocenter and bicyclo[3.2.1]octane present in each natural product. These studies have established a synthetic relationship among three architecturally distinct ent-kaurane diterpenoids and have forged a path for the preparation of interesting unnatural ent-kauranoid structural analogs for more thorough biological study.

A fundamental approach to phase noise reduction in hybrid Si/III-V lasers

Spontaneous emission into the lasing mode fundamentally limits laser linewidths. Reducing cavity losses provides two benefits to linewidth: (1) fewer excited carriers are needed to reach threshold, resulting in less phase-corrupting spontaneous emission into the laser mode, and (2) more photons are stored in the laser cavity, such that each individual spontaneous emission event disturbs the phase of the field less. Strong optical absorption in III-V materials causes high losses, preventing currently-available semiconductor lasers from achieving ultra-narrow linewidths. This absorption is a natural consequence of the compromise between efficient electrical and efficient optical performance in a semiconductor laser. Some of the III-V layers must be heavily doped in order to funnel excited carriers into the active region, which has the side effect of making the material strongly absorbing.

This thesis presents a new technique, called modal engineering, to remove modal energy from the lossy region and store it in an adjacent low-loss material, thereby reducing overall optical absorption. A quantum mechanical analysis of modal engineering shows that modal gain and spontaneous emission rate into the laser mode are both proportional to the normalized intensity of that mode at the active region. If optical absorption near the active region dominates the total losses of the laser cavity, shifting modal energy from the lossy region to the low-loss region will reduce modal gain, total loss, and the spontaneous emission rate into the mode by the same factor, so that linewidth decreases while the threshold inversion remains constant. The total spontaneous emission rate into all other modes is unchanged.

Modal engineering is demonstrated using the Si/III-V platform, in which light is generated in the III-V material and stored in the low-loss silicon material. The silicon is patterned as a high-Q resonator to minimize all sources of loss. Fabricated lasers employing modal engineering to concentrate light in silicon demonstrate linewidths at least 5 times smaller than lasers without modal engineering at the same pump level above threshold, while maintaining the same thresholds.

Expressão do colágeno I, III e metaloproteinase nos diferentes graus de Gleason e estadio patológico do câncer prostático

O objetivo deste trabalho foi avaliar se a expressão do colágeno tipo I, III e metaloproteinase podem estar relacionadas com o grau de Gleason, estadio patológico e PSA pré-operatório, e se isto poderia servir como prognóstico de doença. O grupo de estudo incluiu espécimes de prostatectomia radical de 33 pacientes com adenocarcinoma submetidos à cirurgia no período de 2001 a 2009. Os pacientes foram divididos em 3 grupos: grau de Gleason = 6 (13 pacientes), escore de Gleason = 7 (10 pacientes), escore de Gleason ≥ 8 (10 pacientes). O tecido prostático benigno adjacente à area de câncer nos graus de Gleason foi utilizado como grupo controle. As áreas de adenocarcinoma e de tecido benigno foram selecionados sob análise microscópica e processados para colágeno I e III sob análise do gene por PCR em Tempo Real. Dez seções desparafinadas de cada grupo foram utilizados para avaliar o colágeno I, III e a imunoexpressão de metaloproteinase. Os resultados foram relacionados com o grau de Gleason, PSA pré-operatório e estadio patológico. Apesar da diferença significativa na expressão gênica de ambos colágeno I e III entre as áreas de tecido prostático benigno e tumor nas amostras de próstata Gleason = 6 (colágeno I = 0,4 0,2 vs 5 2,4, p<0,05; colágeno III = 0,2 0,06 vs 0,7 0,1, p<0,05) e grau de Gleason ≥ 8 (I = 8 3,4 vs 1,4 0,8, p<0,05; colágeno III = 1,8 0,5 vs 0,6 0,1, p<0,05), não houve correlação com grau de Gleason, PSA pré-operatório ou estadio patológico. Houve uma correlação positiva entre a expressão de metaloproteinases e grau de Gleason (r2 = 0,47). Concluindo, tem-se que a correlação positiva entre a expressão de metaloproteinases e o grau de Gleason sugere que a metaloproteinase pode ser um fator promissor para melhorar o grau de Gleason. Sua expressão e regulação não parecem estar relacionados com a degradação do colágeno. Não há correlação entre expressão de colágeno e grau de Gleason, nem a nível gênico nem protéico.

On the local Tamagawa number conjecture for Tate motives

There is a wonderful conjecture of Bloch and Kato that generalizes both the analytic Class Number Formula and the Birch and Swinnerton-Dyer conjecture. The conjecture itself was generalized by Fukaya and Kato to an equivariant formulation. In this thesis, I provide a new proof for the equivariant local Tamagawa number conjecture in the case of Tate motives for unramified fields, using Iwasawa theory and (φ,Γ)-modules, and provide some work towards extending the proof to tamely ramified fields.

I. Foehn winds of southern California. II. Foehn wind cyclo-genesis. III. Weather conditions associated with the Akron disaster. IV. The Los Angeles storm of December 30, 1933 to January 1, 1934

I. Foehn winds of southern California.
An investigation of the hot, dry and dust laden winds occurring in the late fall and early winter in the Los Angeles Basin and attributed in the past to the influences of the desert regions to the north revealed that these currents were of a foehn nature. Their properties were found to be entirely due to dynamical heating produced in the descent from the high level areas in the interior to the lower Los Angeles Basin. Any dust associated with the phenomenon was found to be acquired from the Los Angeles area rather than transported from the desert. It was found that the frequency of occurrence of a mild type foehn of this nature during this season was sufficient to warrant its classification as a winter monsoon. This results from the topography of the Los Angeles region which allows an easy entrance to the air from the interior by virtue of the low level mountain passes north of the area. This monsoon provides the mild winter climate of southern California since temperatures associated with the foehn currents are far higher than those experienced when maritime air from the adjacent Pacific Ocean occupies the region.

II. Foehn wind cyclo-genesis.
Intense anticyclones frequently build up over the high level regions of the Great Basin and Columbia Plateau which lie between the Sierra Nevada and Cascade Mountains to the west and the Rocky Mountains to the east. The outflow from these anticyclones produce extensive foehns east of the Rockies in the comparatively low level areas of the middle west and the Canadian provinces of Alberta and Saskatchewan. Normally at this season of the year very cold polar continental air masses are present over this territory and with the occurrence of these foehns marked discontinuity surfaces arise between the warm foehn current, which is obliged to slide over a colder mass, and the Pc air to the east. Cyclones are easily produced from this phenomenon and take the form of unstable waves which propagate along the discontinuity surface between the two dissimilar masses. A continual series of such cyclones was found to occur as long as the Great Basin anticyclone is maintained with undiminished intensity.

III. Weather conditions associated with the Akron disaster.
This situation illustrates the speedy development and propagation of young disturbances in the eastern United States during the spring of the year under the influence of the conditionally unstable tropical maritime air masses which characterise the region. It also furnishes an excellent example of the superiority of air mass and frontal methods of weather prediction for aircraft operation over the older methods based upon pressure distribution.

IV. The Los Angeles storm of December 30, 1933 to January 1, 1934.
This discussion points out some of the fundamental interactions occurring between air masses of the North Pacific Ocean in connection with Pacific Coast storms and the value of topographic and aerological considerations in predicting them. Estimates of rainfall intensity and duration from analyses of this type may be made and would prove very valuable in the Los Angeles area in connection with flood control problems.

Development of Metalloenzyme Dioxygen Reduction Cathodes

The prime thrust of this dissertation is to advance the development of fuel cell dioxygen reduction cathodes that employ some variant of multicopper oxidase enzymes as the catalyst. The low earth-abundance of platinum metal and its correspondingly high market cost has prompted a general search amongst chemists and materials scientists for reasonable alternatives to this metal for facilitating catalytic dioxygen reduction chemistry. The multicopper oxidases (MCOs), which constitute a class of enzyme that naturally catalyze the reaction O₂ + 4H⁺ + 4e^- → 2H₂O, provide a promising set of biochemical contenders for fuel cell cathode catalysts. In MCOs, a substrate reduces a copper atom at the type 1 site, where charge is then transferred to a trinuclear copper cluster consisting of a mononuclear type 2 or “normal copper” site and a binuclear type 3 copper site. Following the reduction of all four copper atoms in the enzyme, dioxygen is then reduced to water in two two-electron steps, upon binding to the trinuclear copper cluster. We identified an MCO, a laccase from the hyperthermophilic bacterium Thermus thermophilus strain HB27, as a promising candidate for cathodic fuel cell catalysis. This protein demonstrates resilience at high temperatures, exhibiting no denaturing transition at temperatures high as 95°C, conditions relevant to typical polymer electrolyte fuel cell operation.

In Chapter I of this thesis, we discuss initial efforts to physically characterize the enzyme when operating as a heterogeneous cathode catalyst. Following this, in Chapter II we then outline the development of a model capable of describing the observed electrochemical behavior of this enzyme when operating on porous carbon electrodes. Developing a rigorous mathematical framework with which to describe this system had the potential to improve our understanding of MCO electrokinetics, while also providing a level of predictive power that might guide any future efforts to fabricate MCO cathodes with optimized electrochemical performance. In Chapter III we detail efforts to reduce electrode overpotentials through site-directed mutagenesis of the inner and outer-sphere ligands of the Cu sites in laccase, using electrochemical methods and electronic spectroscopy to try and understand the resultant behavior of our mutant constructs. Finally, in Chapter IV, we examine future work concerning the fabrication of enhanced MCO cathodes, exploring the possibility of new cathode materials and advanced enzyme deposition techniques.

I. Spectroscopic observation of discrete sites in simple liquids. II. Infrared intensity perturbations of hydrogen halide fundamentals in liquid xenon. III. Rotation of HCl in solid rare gases.

Part I

The spectrum of dissolved mercury atoms in simple liquids has been shown to be capable of revealing information concerning local structures in these liquids.

Part II

Infrared intensity perturbations in simple solutions have been shown to involve more detailed interaction than just dielectric polarization. No correlation has been found between frequency shifts and intensity enhancements.

Part III

Evidence for perturbed rotation of HCl in rare gas matrices has been found. The magnitude of the barrier to rotation is concluded to be of order of 30 cm^(-1).

Nanofabrication for On-Chip Optical Levitation, Atom-Trapping, and Superconducting Quantum Circuits

Researchers have spent decades refining and improving their methods for fabricating smaller, finer-tuned, higher-quality nanoscale optical elements with the goal of making more sensitive and accurate measurements of the world around them using optics. Quantum optics has been a well-established tool of choice in making these increasingly sensitive measurements which have repeatedly pushed the limits on the accuracy of measurement set forth by quantum mechanics. A recent development in quantum optics has been a creative integration of robust, high-quality, and well-established macroscopic experimental systems with highly-engineerable on-chip nanoscale oscillators fabricated in cleanrooms. However, merging large systems with nanoscale oscillators often require them to have extremely high aspect-ratios, which make them extremely delicate and difficult to fabricate with an "experimentally reasonable" repeatability, yield and high quality. In this work we give an overview of our research, which focused on microscopic oscillators which are coupled with macroscopic optical cavities towards the goal of cooling them to their motional ground state in room temperature environments. The quality factor of a mechanical resonator is an important figure of merit for various sensing applications and observing quantum behavior. We demonstrated a technique for pushing the quality factor of a micromechanical resonator beyond conventional material and fabrication limits by using an optical field to stiffen and trap a particular motional mode of a nanoscale oscillator. Optical forces increase the oscillation frequency by storing most of the mechanical energy in a nearly loss-less optical potential, thereby strongly diluting the effects of material dissipation. By placing a 130 nm thick SiO₂ pendulum in an optical standing wave, we achieve an increase in the pendulum center-of-mass frequency from 6.2 to 145 kHz. The corresponding quality factor increases 50-fold from its intrinsic value to a final value of Q_m = 5.8(1.1) x 10⁵, representing more than an order of magnitude improvement over the conventional limits of SiO₂ for a pendulum geometry. Our technique may enable new opportunities for mechanical sensing and facilitate observations of quantum behavior in this class of mechanical systems. We then give a detailed overview of the techniques used to produce high-aspect-ratio nanostructures with applications in a wide range of quantum optics experiments. The ability to fabricate such nanodevices with high precision opens the door to a vast array of experiments which integrate macroscopic optical setups with lithographically engineered nanodevices. Coupled with atom-trapping experiments in the Kimble Lab, we use these techniques to realize a new waveguide chip designed to address ultra-cold atoms along lithographically patterned nanobeams which have large atom-photon coupling and near 4π Steradian optical access for cooling and trapping atoms. We describe a fully integrated and scalable design where cold atoms are spatially overlapped with the nanostring cavities in order to observe a resonant optical depth of d₀ ≈ 0.15. The nanodevice illuminates new possibilities for integrating atoms into photonic circuits and engineering quantum states of atoms and light on a microscopic scale. We then describe our work with superconducting microwave resonators coupled to a phononic cavity towards the goal of building an integrated device for quantum-limited microwave-to-optical wavelength conversion. We give an overview of our characterizations of several types of substrates for fabricating a low-loss high-frequency electromechanical system. We describe our electromechanical system fabricated on a Si₃N₄ membrane which consists of a 12 GHz superconducting LC resonator coupled capacitively to the high frequency localized modes of a phononic nanobeam. Using our suspended membrane geometry we isolate our system from substrates with significant loss tangents, drastically reducing the parasitic capacitance of our superconducting circuit to ≈ 2.5$ fF. This opens up a number of possibilities in making a new class of low-loss high-frequency electromechanics with relatively large electromechanical coupling. We present our substrate studies, fabrication methods, and device characterization.