We consider strong wave propagation in the generalized compressible Euler equations. Our results include pairwise interactions of nonlinear waves, smooth wave propagation, formation of gradient blowup and several exact examples. In particular, we directly generalize P.Lax’s gradient blowup results for conservation laws with two variables to the generalized compressible Euler equations.

<p> Methods to extract vessel networks in medical images have been in high demand for its applications to health risk predictions. For example, vessel enhancement of retinal images has shown promises in diagnosing diabetes. Within the existing literature, multiscale vessel enhancement stands out as one of the best for its accuracy, speed, and simplicity. But like many vessel extraction techniques, the efficacy of the method is greatly hindered in the presence of noise, lighting variations, and decreased resolution. This deficiency is presents itself in retinal images and are particularly pronounced in digital photographs of human placenta.</p><p> Retinal images have a been popular data set of testing vessel extraction methods because of its simplicity in anatomical structure yet high hopes in diagnosing conditions such as diabetic retinopathy and glaucoma. Thus, the thesis will focus on the application of vessel extraction methods on retinal images. Specifically, we focus on the DRIVE and STARE database.</p><p> Also, recent placental pathology evidence has contributed to current understanding of causes of low birth weight and preterm birth, each has been linked to increased risk of later neurodevelopmental disorders. Among various factors that cause such disorders, the vessel network on the placenta has been hypothesized to offer the most clue in bridging that connection. Herein lies the most essential step of the blood vessel extraction, which has only been done manually through a laborious process.</p><p> Motivated by its ability to handle curvilinear structures, we propose the use of directional filter banks to further enhance the results obtained from the multiscale method. Validating experiments will be performed on a private database that is made available by the Placental Analytics, LLC.</p><p> It will be shown that for retinal images, the directional filter bank approach significantly improves the performance over the well-known multiscale vessel enhancement method. However, the directional filter bank approach are comparable to multiscale vessel enhancement on placentas.</p>

This dissertation concentrates on the existence, stability and dynamical properties of nonlinear waves in Bose-Einstein condensates (BECs) trapped in doublewell potentials (DWPs). The fundamental model of interest will be the nonlinear Schrödinger equation, the so-called Gross-Pitaevskii (GP) equation, contributed to the well-established mean-field description of BECs. In this context of the GP equation with DWP, a Galerkin-type few-mode approach provides us a powerful handle towards studying the stationary states and predicting the bifurcation diagram including the occurrence of spontaneous symmetry breaking (SSB). Such method and the corresponding phenomena are discussed based on a prototypical quasi-1D model in Chapter 2. The systematic analysis progresses by considering various modified models, starting with the ones involving different interatomic interactions, e.g., collisionally inhomogeneous interactions, long-range interactions, and competing of short- and long-range interactions. We observe how the basic SSB bifurcation structure persists or is appropriately modified in the presence of these interactions in Chapter 3. We also extend the study to multi-component systems, including nonlinearly coupled two-component settings and F = 1 spinor BECs (genuinely three-component settings) confined in DWPs, where besides the one-component stationary states, combined states involving two or three components appear as well, and novel SSB phenomena emerge within them. Finally the trapped stationary modes of a twodimensional (2D) GP equation with a symmetric four-well potential are explored, providing the picture of SSB in the fundamental 2D setting. These various systems are studied in Chapter 4 - 6. In all models, our analytical predictions based on the few-mode approximation are in excellent agreement with the numerical results of the full GP equations.