Recently, there has been a growing interest in the use of nonlocal partial differential equation (PDE) to model biological and physical phenomena. In this dissertation, we study the behavior of solutions to several nonlocal PDEs, which have both an aggregation term and a degenerate diffusion term.

Chapter 1 and Chapter 2 of this dissertation are devoted to the study of the Patlak-Keller-Segel (PKS) equation and its variations. The PKS equation is a degenerate diffusion equation with a nonlocal aggregation term, which models the collective motion of cells attracted by a self-emitted chemical substance. While the global well-posedness and finite-time blow-up criteria are well known, the asymptotic behaviors of solutions are not completely clear.

In Chapter 1, we investigate qualitative and asymptotic behavior of solutions for the PKS equation when the solution exists globally in time. The challenge in the analysis consists of the nonlocal aggregation term as well as the degeneracy of the diffusion term which generates compactly supported solutions. Using maximum-principle type arguments as well as energy argument, we prove the finite propagation property of general solutions, and several results regarding asymptotic behaviors of solutions.

In Chapter 2, we consider the PKS equation with general power-law interaction kernel, and focus on the cases where the solution blows up in finite time. We study radially symmetric finite time blow-up dynamics from both the numerical and asymptotic aspect, and show that the solution exhibits three kinds of blow-up behavior: self-similar with no mass concentrated at the core, imploding shock solution and near-self-similar blow-up with a fixed amount of mass concentrated at the core. Computation are performed for a variety of parameters using an arbitrary Lagrangian Eulerian method with adaptive mesh refinement.

Chapter 3 discusses the study on an aggregation-diffusion equation with smooth interaction kernel in the periodic domain. This equation represents the generalization to m>1 of the McKean-Vlasov equation where here the "diffusive" portion of the dynamics are governed by Porous medium self-interactions. We focus primarily on m in (1,2] with particular emphasis on m=2. In general, we establish regularity properties and, for small interaction, exponential decay to the uniform stationary solution. For m=2, we obtain essentially sharp results on the rate of decay for the entire regime up to the (sharp) transitional value of the interaction parameter.

The vertebrate heart is comprised of two types of chambers, ventricles and atria, which exhibit unique structural and contractile properties. Effective cardiac function depends upon the distinct morphological, physiological, and molecular characteristics of ventricular and atrial cardiomyocytes. Intriguingly, despite the early specification of ventricular and atrial lineages, chamber-specific features need to be actively reinforced even after myocardial differentiation is underway. By combining genetic and embryological approaches, work in zebrafish has revealed multiple players with potent influences on chamber fate specification and commitment. Notably, our prior studies in zebrafish have shown that sustained FGF signaling acts upstream of nkx2.5 to maintain ventricular chamber identity: upon inhibition of FGF signaling or loss of nkx2.5 function, ventricular cardiomyocytes extinguish expression of the ventricular gene vmhc and initiate ectopic expression of the atrial gene amhc. However, the rest of the pathway acting downstream of FGF signaling in this context remains largely unknown. Here, we show that MEK1/2-ERK1/2 signaling plays an important part in promoting ventricular identity maintenance. Pharmacological inhibition of MEK signaling results in ectopic amhc expression as well as reduced vmhc expression in ventricular cardiomyocytes. Epistasis analysis suggests that MEK signaling influences ventricular identity maintenance downstream of FGF signaling and upstream of nkx2.5 function. We find that FGF signaling and MEK signaling both influence ventricular identity maintenance over a similar timeframe, and we can detect phosphorylated ERK (pERK) in the myocardium during these stages. Intriguingly, the role of FGF-MEK activity seems to be context-dependent: some regions of the ventricle are more sensitive than others to inhibition of the FGF-MEK pathway. On the other hand, in the atrium, although endogenous levels of MEK-ERK signaling activity are not sufficient to induce ventricular traits in atrial cardiomyocytes, heightened MEK1 activity can provoke ectopic expression of vmhc. Together, our data indicate distinct functions of MEK-ERK signaling in the maintenance of ventricular and atrial identities.

A search for a low mass dark photon below 1 MeV which is radiated from a muon using proton-proton collision data at a center-of-mass energy $\sqrt{s}$ = 13 TeV is presented. The data was collected with the CERN CMS detector in 2018 corresponding to an integrated luminosity of 59.8 fb$^{-1}$. Such a low mass dark photon has no available decay channel to standard model particles, and therefore is stable. The dark photon is assumed to directly interact with detector materials through pair-production, and its small kinetic mixing leads to it depositing energy outside the CMS Electromagnetic Calorimeter. The search is performed by looking for events whose final state is composed of two muons and a photon-like shower in the CMS Hadronic Calorimeter, with the 3-body mass consistent with the Z boson. The accessible range of kinetic mixing in this search is between 0.032 to 0.32. No significant deviation from estimated backgrounds is observed with the 2018 data. The dark photon with kinetic mixing between 0.073 to 0.32 and mass lower than 1 MeV is excluded at 95\% confidence level.

This dissertation investigates the effects of phonological neighborhoods on pronunciation variation in conversational speech. Phonological neighbors are defined as words that are different in one and only one phoneme by addition, deletion and substitution. Phonological neighborhood density refers to the number of neighbors a certain word has.

Previous research has shown that phonological neighbors impede auditory perception, but facilitate lexical production. As a result, words from dense neighborhoods are harder to perceive, but easier to produce, than words from sparse neighborhoods. Following these effects, two opposite hypotheses can be formed regarding the effects of neighborhood density on pronunciation variation. The listener-oriented hypothesis predicts that high-density words will be hyperarticulated, in order to compensate for their perceptual difficulty. But the speaker-oriented hypothesis predicts that high-density words are more likely to undergo speech reduction, as other words that are easy to produce (such as high-frequency words).

To test these hypotheses, two statistical models are constructed to investigate neighborhood effects on pronunciation variation in CVC monomorphemic content words. All speech data come from the Buckeye Corpus of Conversational Speech (Pitt, Dilley, Johnson, Kiesling, Raymond, Hume, and Fosler-Lussier 2007), which contains audio recordings of 40 speaker's free-form interviews. The dataset for the current research consists of more than 500 target words, represented by over 13,000 tokens. The outcome variables in the two models are word duration and degree of vowel dispersion, respectively. Neighborhood density and average neighbor frequency are the critical predictors in both models. Other factors that might affect pronunciation variation in conversational speech, such as word frequency and predictability, are entered into the models as control factors.

The results of the model analyses show that everything else being equal, high-density words are realized with shorter durations and more centralized vowels than low-density words. These findings provide strong evidence for the speaker-oriented hypothesis. A peripheral finding in the current research is a tendency for words with high-frequency neighbors to have more dispersed vowels. However, this effect is only significant when neighbor frequency from the Hoosier mental lexicon (Nusbaum, Pisoni, and Davis 1984) is used, but not when neighbor frequency computed from the CELEX database (Baayen, Piepenbrock, and Rijn 1993) is used.

Overall, major findings of the current research support the speaker-oriented hypothesis over the listener-oriented hypothesis, which suggests that word-level pronunciation variation is more heavily influenced by features of the speaker's own production system than by the consideration of listeners' needs.