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This thesis introduces a variational formulation for a family of kinetic reaction-diffusion and their connection to Lagrangian dynamical systems. Such a formulation uses a new class of transportation costs between positive measures, and it generalizes the notion of gradient flows. We use this class to build solutions to reaction-diffusion equations with drift subject to general Dirichlet boundary condition via an extension of De Giorgi's interpolation method for the entropy functional. In 2010, Alessio Figalli and Nicola Gigli introduced a transportation cost that can be used to obtain parabolic equations with drift subject to Dirichlet boundary condition. However, the drift and the boundary condition are coupled in their work. The costs we introduce allow the drift and the boundary condition to be decoupled. Additionally, we use this variational formulation to obtain well-posedness, stability, and convergence to equilibrium for the homogeneous Vicsek model and to show the emergence of phase concentration for the Kuramoto Sakaguchi equation subject to a strong coupling force. Provided this coupling force is sufficiently large, we show that there exists a time-dependent interval such that the **oscillator's** probability density converges to zero uniformly in its complement. The length of this interval is quantified as a function of the coupling force and the diameter of the support of the natural **frequency** distribution. By doing this, we show that the diameter of the interval can be made arbitrarily small by choosing the force sufficiently large.

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How the scattering dynamics of a quantum system is affected by an application of a time-periodic driving field, such as coherent electromagnetic radiation, has been a subject of increasing importance in the past three decades. Time-periodic fields can profoundly alter the dynamics of matter in ways that are relevant to the design of semiconductor structures, such as quantum dots and superlattices, that have possible applications to quantum computation and quantum information processing. This thesis is a theoretical treatment of a localized one-dimensional quantum system that is subject to an external driving field, such as an **oscillating** electric field of a laser or of some microwave radiation, that **oscillates** periodically in time. When not subject to the driving field, the system is characterized by a potential energy function that is a finite one-dimensional square well (i.e. a 1D quantum potential well with finite depth and a flat bottom. Also, the zero level of potential energy is set such that the potential in the well’s exterior is zero). To this well, we specifically introduce a driving field that causes the well’s bottom to bend from a flat shape to a V-shape, such that the potential energy as a function of position is held fixed at the endpoints of the well’s region of space and forms the spike of the letter “V” at the midpoint of the well. Our chosen external field makes the V-shaped bottom perpetually bend up and down, causing the potential energy in the region of the well to vary periodically in time. In this study, we analyze how a plane wave that propagates toward the region of our well and is with a fixed incident energy not only scatters into outgoing plane waves with the same energy — propagating in both directions of one-dimensional space — but is also induced by the driving field to have nonzero probabilities of transition to infinitely many other states with different energies. Due to the time-periodicity of our potential, an incoming wave-particle (i.e. a matter wave) of a given energy can only access energies that differ by integer multiples of ħω from its original energy, where ω is the angular **frequency** of the **oscillation** of the potential. In the case that the time-periodic driving is due to an electromagnetic field **oscillating** at an angular **frequency** ω, we can explain the transitions in which a particle can only gain or lose an amount of energy that is an integer multiple of ħω as follows: The electromagnetic field induces the particle to absorb or emit photons, each carrying a quantum of energy equal to an integer multiple of ħω. The analysis to be revealed comprises three main achievements, all of which would help one to accomplish suitable computational precision, accuracy, and efficiency when making a prediction about a scattering phenomenon in our chosen system. The first achievement is that my research supervisor and I managed to solve the Schrӧdinger equation for this system analytically (i.e. exactly), so for a choice of four intervals that together constitute one-dimensional space, we were able to find the actual space of (complex-valued) explicit solutions to the equation in every interval separately. Within that space of solutions, if we only consider functions that are continuously differentiable, square-integrable, and somewhere nonzero (that is, nonzero solutions for which their partial derivatives with respect to spatial variables exist and are continuous everywhere and the squares of their absolute values are integrable over all space), then we obtain the set of all possible states that a particle can access in our quantum system. By accurately finding the space of solutions to the Schrӧdinger equation and by imposing the requirement that a wavefunction must be continuously differentiable, we were able to deduce into what superposition of states a given incoming plane wave must scatter into in order to form a state that is possible to include in a physically allowed superposition of states. Accounting for the fact that a physically acceptable solution must be at least continuously differentiable is directly related to the second success of the study, which is to derive a formula enabling me to find the scattering matrix (or S-matrix), a mathematical construct that relates the incoming plane waves to the states of definite energy outside the well into which they scatter. Despite the complexity of the solutions, I managed to exploit the reflection symmetry of the system about the center of the well and other simplifying properties to come up with expressions for four matrix blocks that constitute a matrix that contains the S-matrix, such that all four expressions involve the same eight matrices.This in turn led to the third achievement, which was that I set up an efficient method for using computational software (in my case, I used Wolfram Mathematica 11.0) to find the elements of the S-matrix. Such a method was suitably fast at generating high-quality graphs of moduli squared of some matrix elements as functions of incident “energy” (actually, quasienergy, as we shall see later). Those graphs revealed the probability for an incoming wave (with a fixed energy and a unit probability current) to both transmit through the region of the well and to transition to a state, such as an outgoing wave or a negative-energy state, of some fixed energy. The energy of the new state did not necessarily have to be equal to the incident energy. Given these computational freedoms, I created a demo of my method by constructing these transmission graphs for a specific set of parameters expressed in Hartree atomic units: well width of 2, particle mass µ = 1, amplitude U0 = 0.5 and angular **frequency** ω = 4 of the **oscillation** of the V-shaped bottom, and unperturbed well depth V0 = 10. When I compared the different graphs of some of the combined transmission and transition probabilities provided by the elements of the S-matrix, I noticed two incident energies for which transmission resonances occur. Next, I exploited those resonances to determine some energies of quasibound states, which are states in which a particle’s probability is localized inside the region of the well for a finite amount of time. (In contrast, bound states have their probability localized for an infinite amount of time). For the version of our system without the external field (i.e. with U0 = 0), the bound state energies for the first and second excited states are -6.7791 and -3.0542 Hartrees, respectively, while for the driven system, I found that the corresponding energies of quasibound states were -6.7788 and -3.0534 Hartrees. The fact that these energies of quasibound states for the perturbed system are quite close to those of the unperturbed system is an indication that the chosen **oscillation** strength U0 = 0.5 Hartrees is weak enough to preserve many of the general properties of the unperturbed system.

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El Niño-Southern **Oscillation**

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White dwarf stars are the remnant products of the vast majority of Galactic stellar evolution. They are compact objects that serve as remote laboratories for studying high energy/density physics. The outer regions of hydrogen-atmosphere (DA) white dwarfs become convective and able to drive global, nonradial, gravity-mode pulsations below roughly 12,500 K. The pulsations propagate through and are affected by the interior structures of these stars. The **oscillations** cause a pulsating star to exhibit brightness variations at its characteristic **frequencies** as a physical system. These **frequencies** can be measured through Fourier analysis of time series photometric observations. I have focused my studies on new pulsational phenomena near the cool and low-mass edges of the DA white dwarf instability strip, using extensive space-based data from the Kepler spacecraft and the K2 mission, as well as high-speed ground-based photometry from the 2.1-meter Otto Struve Telescope at McDonald Observatory (where I have personally observed 225 nights). The extensive short-cadence (1-min exposures) light curve of the first DAV (DA variable) identified within the original Kepler field of view provided one of the most complete and sensitive records of white dwarf pulsations ever. The light curve also revealed a new, completely unexpected outburst-like phenomenon. I detected 178 instances of significant brightness enhancement in 20 months of observations of the cool DAV KIC 4552982. Recurring with a quasi-period of 2.7 days, the outbursts last 4–25 hours and increase the stellar flux by up to 17%. I estimate the energy of each outburst to be of-order 10³³ ergs. After the Kepler spacecraft suffered the loss of a second reaction wheel in May 2013, it began the K2 mission, visiting new fields along the ecliptic roughly every 80 days. This allowed us to increase the number of DAVs with extensive space-based photometry, and we quickly discovered a second, more dramatic example of this new outburst behavior in PG 1149+057 (Hermes et al. 2015b). I have led the efforts to characterize the outbursts in DAVs ever since and have detected these events in eight DAVs through K2 Campaign 10. Notably, spectroscopic effective temperature constraints place all known members of this new outbursting class of DAV near the cool (red) edge of the instability strip. With a growing outbursting class of DAV, we begin to study their ensemble outburst properties to inform a theory of their physical mechanism. Much of my work from McDonald Observatory has continued in the recent tradition of discovering and characterizing new pulsating extremely low-mass (ELM) white dwarfs. After identifying candidate ELM variables (ELMVs) from the ELM Survey catalog and parameters from model fits to the Sloan Digital Sky Survey spectroscopic data, I obtained time series photometric observations on the 2.1-meter Otto Struve telescope. I published SDSS J1618+3854 as the sixth member of this new class of variable star. However, most of the variability that I measured for this project was inconsistent with expectations for cooling track ELM white dwarfs. This includes long pulsation periods, high pulsation amplitudes, long eclipse timescales, and an overabundance of photometric variables that are not in confirmed short-period binaries from time series radial velocity measurements. Either the surface gravities of another class of star are being systematically overestimated from model fits to hydrogen line profiles in stellar spectra, or these observations are revealing an unexpectedly large population of recently formed pre-ELM white dwarfs. In total, I have discovered and characterized the variability of nine new pulsating stars in the spectroscopic parameter space of ELM white dwarfs, and I also developed an improved framework for interpreting measurements of tidally induced ellipsoidal variations in photometric binaries. Beyond these main results of my thesis on extreme pulsating white dwarfs, I have also explored the limits of the detectability of stellar pulsations in extreme photometric data sets. I analyze long-cadence (30-minute) K2 observations of two fairly typical DAVs in one such study, where the pulsations are severely undersampled. While accurate **frequency** determinations are nontrivial in such cases, I am able to recover the super-Nyquist **frequencies** of some pulsation modes with full K2 precision with the help of a few hours of ground-based observations. The space-based data, in turn, enables me to select the intrinsic **frequency** from the complex alias structure of multi-night ground-based data, providing a practical demonstration of the importance of carefully considering the spectral window. I apply what I have learned about undersampled data to anticipate upcoming pulsating star science in the next generation of synoptic time domain photometric surveys such as the Zwicky Transient Facility and the Large Synoptic Survey Telescope.

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