Spectral density of first order Piecewise linear systems excited by white noise

The Fokker-Planck (FP) equation is used to develop a general method for finding the spectral density for a class of randomly excited first order systems. This class consists of systems satisfying stochastic differential equations of form ẋ + f(x) = m/Ʃ/j = 1 h_j(x)n_j(t) where f and the h_j are piecewise linear functions (not necessarily continuous), and the n_j are stationary Gaussian white noise. For such systems, it is shown how the Laplace-transformed FP equation can be solved for the transformed transition probability density. By manipulation of the FP equation and its adjoint, a formula is derived for the transformed autocorrelation function in terms of the transformed transition density. From this, the spectral density is readily obtained. The method generalizes that of Caughey and Dienes, J. Appl. Phys., 32.11.

This method is applied to 4 subclasses: (1) m = 1, h₁ = const. (forcing function excitation); (2) m = 1, h₁ = f (parametric excitation); (3) m = 2, h₁ = const., h₂ = f, n₁ and n₂ correlated; (4) the same, uncorrelated. Many special cases, especially in subclass (1), are worked through to obtain explicit formulas for the spectral density, most of which have not been obtained before. Some results are graphed.

Dealing with parametrically excited first order systems leads to two complications. There is some controversy concerning the form of the FP equation involved (see Gray and Caughey, J. Math. Phys., 44.3); and the conditions which apply at irregular points, where the second order coefficient of the FP equation vanishes, are not obvious but require use of the mathematical theory of diffusion processes developed by Feller and others. These points are discussed in the first chapter, relevant results from various sources being summarized and applied. Also discussed is the steady-state density (the limit of the transition density as t → ∞).

Existence, uniqueness, and stability of solutions of a class of nonlinear partial differential equations

In this study we investigate the existence, uniqueness and asymptotic stability of solutions of a class of nonlinear integral equations which are representations for some time dependent non- linear partial differential equations. Sufficient conditions are established which allow one to infer the stability of the nonlinear equations from the stability of the linearized equations. Improved estimates of the domain of stability are obtained using a Liapunov Functional approach. These results are applied to some nonlinear partial differential equations governing the behavior of nonlinear continuous dynamical systems.

Nonlinear optics in planar silica-on-silicon disk resonators

Optical frequency combs (OFCs) provide direct phase-coherent link between optical and RF frequencies, and enable precision measurement of optical frequencies. In recent years, a new class of frequency combs (microcombs) have emerged based on parametric frequency conversions in dielectric microresonators. Micocombs have large line spacing from 10's to 100's GHz, allowing easy access to individual comb lines for arbitrary waveform synthesis. They also provide broadband parametric gain bandwidth, not limited by specific atomic or molecular transitions in conventional OFCs. The emerging applications of microcombs include low noise microwave generation, astronomical spectrograph calibration, direct comb spectroscopy, and high capacity telecommunications.

In this thesis, research is presented starting with the introduction of a new type of chemically etched, planar silica-on-silicon disk resonator. A record Q factor of 875 million is achieved for on-chip devices. A simple and accurate approach to characterize the FSR and dispersion of microcavities is demonstrated. Microresonator-based frequency combs (microcombs) are demonstrated with microwave repetition rate less than 80 GHz on a chip for the first time. Overall low threshold power (as low as 1 mW) of microcombs across a wide range of resonator FSRs from 2.6 to 220 GHz in surface-loss-limited disk resonators is demonstrated. The rich and complex dynamics of microcomb RF noise are studied. High-coherence, RF phase-locking of microcombs is demonstrated where injection locking of the subcomb offset frequencies are observed by pump-detuning-alignment. Moreover, temporal mode locking, featuring subpicosecond pulses from a parametric 22 GHz microcomb, is observed. We further demonstrated a shot-noise-limited white phase noise of microcomb for the first time. Finally, stabilization of the microcomb repetition rate is realized by phase lock loop control.

For another major nonlinear optical application of disk resonators, highly coherent, simulated Brillouin lasers (SBL) on silicon are also demonstrated, with record low Schawlow-Townes noise less than 0.1 Hz^2/Hz for any chip-based lasers and low technical noise comparable to commercial narrow-linewidth fiber lasers. The SBL devices are efficient, featuring more than 90% quantum efficiency and threshold as low as 60 microwatts. Moreover, novel properties of the SBL are studied, including cascaded operation, threshold tuning, and mode-pulling phenomena. Furthermore, high performance microwave generation using on-chip cascaded Brillouin oscillation is demonstrated. It is also robust enough to enable incorporation as the optical voltage-controlled-oscillator in the first demonstration of a photonic-based, microwave frequency synthesizer. Finally, applications of microresonators as frequency reference cavities and low-phase-noise optomechanical oscillators are presented.

Finding needles in the haystack : a search for AM CVn systems using the Palomar Transient Factory

The AM CVn systems are a rare class of ultra-compact astrophysical binaries. With orbital periods of under an hour and as short as five minutes, they are among the closest known binary star systems and their evolution has direct relevance to the type Ia supernova rate and the white dwarf binary population. However, their faint and rare nature has made population studies of these systems difficult and several studies have found conflicting results.

I undertook a survey for AM CVn systems using the Palomar Transient Factory (PTF) astrophysical synoptic survey by exploiting the "outbursts" these systems undergo. Such events result in an increase in luminosity by a factor of up to two-hundred and are detectable in time-domain photometric data of AM CVn systems. My search resulted in the discovery of eight new systems, over 20% of the current known population. More importantly, this search was done in a systematic fashion, which allows for a population study properly accounting for biases.

Apart from the discovery of new systems, I used the time-domain data from the PTF and other synoptic surveys to better understand the long-term behavior of these systems. This analysis of the photometric behavior of the majority of known AM CVn systems has shown changes in their behavior at longer time scales than have previously been observed. This has allowed me to find relationships between the outburst properties of an individual system and its orbital period.

Even more importantly, the systematically selected sample together with these properties have allowed me to conduct a population study of the AM CVn systems. I have shown that the latest published estimates of the AM CVn system population, a factor of fifty below theoretical estimates, are consistent with the sample of systems presented here. This is particularly noteworthy since my population study is most sensitive to a different orbital period regime than earlier surveys. This confirmation of the population density will allow the AM CVn systems population to be used in the study of other areas of astrophysics.

Computationally Guided Monomerization of Red Fluorescent Proteins of the Class Anthozoa

Red fluorescent proteins (RFPs) have attracted significant engineering focus because of the promise of near infrared fluorescent proteins, whose light penetrates biological tissue, and which would allow imaging inside of vertebrate animals. The RFP landscape, which numbers ~200 members, is mostly populated by engineered variants of four native RFPs, leaving the vast majority of native RFP biodiversity untouched. This is largely due to the fact that native RFPs are obligate tetramers, limiting their usefulness as fusion proteins. Monomerization has imposed critical costs on these evolved tetramers, however, as it has invariably led to loss of brightness, and often to many other adverse effects on the fluorescent properties of the derived monomeric variants. Here we have attempted to understand why monomerization has taken such a large toll on Anthozoa class RFPs, and to outline a clear strategy for their monomerization. We begin with a structural study of the far-red fluorescence of AQ143, one of the furthest red emitting RFPs. We then try to separate the problem of stable and bright fluorescence from the design of a soluble monomeric β-barrel surface by engineering a hybrid protein (DsRmCh) with an oligomeric parent that had been previously monomerized, DsRed, and a pre-stabilized monomeric core from mCherry. This allows us to use computational design to successfully design a stable, soluble, fluorescent monomer. Next we took HcRed, which is a previously unmonomerized RFP that has far-red fluorescence (λemission = 633 nm) and attempted to monomerize it making use of lessons learned from DsRmCh. We engineered two monomeric proteins by pre-stabilizing HcRed’s core, then monomerizing in stages, making use of computational design and directed evolution techniques such as error-prone mutagenesis and DNA shuffling. We call these proteins mGinger0.1 (λem = 637 nm / Φ = 0.02) and mGinger0.2 (λem = 631 nm Φ = 0.04). They are the furthest red first generation monomeric RFPs ever developed, are significantly thermostabilized, and add diversity to a small field of far-red monomeric FPs. We anticipate that the techniques we describe will be facilitate future RFP monomerization, and that further core optimization of the mGingers may allow significant improvements in brightness.

On the set of eigenvalues of a class of equimodular matrices

The structure of the set ϐ(A) of all eigenvalues of all complex matrices (elementwise) equimodular with a given n x n non-negative matrix A is studied. The problem was suggested by O. Taussky and some aspects have been studied by R. S. Varga and B.W. Levinger.

If every matrix equimodular with A is non-singular, then A is called regular. A new proof of the P. Camion-A.J. Hoffman characterization of regular matrices is given.

The set ϐ(A) consists of m ≤ n closed annuli centered at the origin. Each gap, ɤ, in this set can be associated with a class of regular matrices with a (unique) permutation, π(ɤ). The association depends on both the combinatorial structure of A and the size of the a_ii. Let A be associated with the set of r permutations, π₁, π₂,…, π_r, where each gap in ϐ(A) is associated with one of the π_k. Then r ≤ n, even when the complement of ϐ(A) has n+1 components. Further, if π(ɤ) is the identity, the real boundary points of ɤ are eigenvalues of real matrices equimodular with A. In particular, if A is essentially diagonally dominant, every real boundary point of ϐ(A) is an eigenvalues of a real matrix equimodular with A.

Several conjectures based on these results are made which if verified would constitute an extension of the Perron-Frobenius Theorem, and an algebraic method is introduced which unites the study of regular matrices with that of ϐ(A).

Class two p groups as fixed point free automorphism groups

Suppose that AG is a solvable group with normal subgroup G where (|A|, |G|) = 1. Assume that A is a class two odd p group all of whose irreducible representations are isomorphic to subgroups of extra special p groups. If p^c ≠ r^d + 1 for any c = 1, 2 and any prime r where r^2d+1 divides |G| and if C_G(A) = 1 then the Fitting length of G is bounded by the power of p dividing |A|.

The theorem is proved by applying a fixed point theorem to a reduction of the Fitting series of G. The fixed point theorem is proved by reducing a minimal counter example. IF R is an extra spec r subgroup of G fixed by A₁, a subgroup of A, where A₁ centralizes D(R), then all irreducible characters of A₁R which are nontrivial on Z(R) are computed. All nonlinear characters of a class two p group are computed.