Existence and weak-strong uniqueness for the Navier-Stokes-Smoluchowski system over moving domains

This dissertation concerns the well-posedness of the Navier-Stokes-Smoluchowski system. The system models a mixture of fluid and particles in the so-called bubbling regime. The compressible Navier-Stokes equations governing the evolution of the fluid are coupled to the Smoluchowski equation for the particle density at a continuum level. First, working on fixed domains, the existence of weak solutions is established using a three-level approximation scheme and based largely on the Lions-Feireisl theory of compressible fluids. The system is then posed over a moving domain. By utilizing a Brinkman-type penalization as well as penalization of the viscosity, the existence of weak solutions of the Navier-Stokes-Smoluchowski system is proved over moving domains. As a corollary the convergence of the Brinkman penalization is proved. Finally, a suitable relative entropy is defined. This relative entropy is used to establish a weak-strong uniqueness result for the Navier-Stokes-Smoluchowski system over moving domains, ensuring that strong solutions are unique in the class of weak solutions.

Strong shift equivalence, algebraic K-theory, and isolating zero-dimensional dynamics on manifolds

We study the relations of shift equivalence and strong shift equivalence for matrices over a ring $\mathcal{R}$, and establish a connection between these relations and algebraic K-theory. We utilize this connection to obtain results in two areas where the shift and strong shift equivalence relations play an important role: the study of finite group extensions of shifts of finite type, and the Generalized Spectral Conjectures of Boyle and Handelman for nonnegative matrices over subrings of the real numbers. We show the refinement of the shift equivalence class of a matrix $A$ over a ring $\mathcal{R}$ by strong shift equivalence classes over the ring is classified by a quotient $NK_{1}(\mathcal{R}) / E(A,\mathcal{R})$ of the algebraic K-group $NK_{1}(\calR)$. We use the K-theory of non-commutative localizations to show that in certain cases the subgroup $E(A,\mathcal{R})$ must vanish, including the case $A$ is invertible over $\mathcal{R}$. We use the K-theory connection to clarify the structure of algebraic invariants for finite group extensions of shifts of finite type. In particular, we give a strong negative answer to a question of Parry, who asked whether the dynamical zeta function determines up to finitely many topological conjugacy classes the extensions by $G$ of a fixed mixing shift of finite type. We apply the K-theory connection to prove the equivalence of a strong and weak form of the Generalized Spectral Conjecture of Boyle and Handelman for primitive matrices over subrings of $\mathbb{R}$. We construct explicit matrices whose class in the algebraic K-group $NK_{1}(\mathcal{R})$ is non-zero for certain rings $\mathcal{R}$ motivated by applications. We study the possible dynamics of the restriction of a homeomorphism of a compact manifold to an isolated zero-dimensional set. We prove that for $n \ge 3$ every compact zero-dimensional system can arise as an isolated invariant set for a homeomorphism of a compact $n$-manifold. In dimension two, we provide obstructions and examples.

Institutional Persistence and Change in England's Common Law: 1700-1865

Institutions are widely regarded as important, even ultimate drivers of economic growth and performance. A recent mainstream of institutional economics has concentrated on the effect of persisting, often imprecisely measured institutions and on cataclysmic events as agents of noteworthy institutional change. As a consequence, institutional change without large-scale shocks has received little attention. In this dissertation I apply a complementary, quantitative-descriptive approach that relies on measures of actually enforced institutions to study institutional persistence and change over a long time period that is undisturbed by the typically studied cataclysmic events. By placing institutional change into the center of attention one can recognize different speeds of institutional innovation and the continuous coexistence of institutional persistence and change. Specifically, I combine text mining procedures, network analysis techniques and statistical approaches to study persistence and change in England’s common law over the Industrial Revolution (1700-1865). Based on the doctrine of precedent - a peculiarity of common law systems - I construct and analyze the apparently first citation network that reflects lawmaking in England. Most strikingly, I find large-scale change in the making of English common law around the turn of the 19th century - a period free from the typically studied cataclysmic events. Within a few decades a legal innovation process with low depreciation rates (1 to 2 percent) and strong past-persistence transitioned to a present-focused innovation process with significantly higher depreciation rates (4 to 6 percent) and weak past-persistence. Comparison with U.S. Supreme Court data reveals a similar U.S. transition towards the end of the 19th century. The English and U.S. transitions appear to have unfolded in a very specific manner: a new body of law arose during the transitions and developed in a self-referential manner while the existing body of law lost influence, but remained prominent. Additional findings suggest that Parliament doubled its influence on the making of case law within the first decades after the Glorious Revolution and that England’s legal rules manifested a high degree of long-term persistence. The latter allows for the possibility that the often-noted persistence of institutional outcomes derives from the actual persistence of institutions.

Essays on Information Frictions, Prices and Unemployment

I investigate the effects of information frictions in price setting decisions. I show that firms' output prices and wages are less sensitive to aggregate economic conditions when firms and workers cannot perfectly understand (or know) the aggregate state of the economy. Prices and wages respond with a lag to aggregate innovations because agents learn slowly about those changes, and this delayed adjustment in prices makes output and unemployment more sensitive to aggregate shocks. In the first chapter of this dissertation, I show that workers' noisy information about the state of the economy help us to explain why real wages are sluggish. In the context of a search and matching model, wages do not immediately respond to a positive aggregate shock because workers do not (yet) have enough information to demand higher wages. This increases firms' incentives to post more vacancies, and it makes unemployment volatile and sensitive to aggregate shocks. This mechanism is robust to two major criticisms of existing theories of sluggish wages and volatile unemployment: the flexibility of wages for new hires and the cyclicality of the opportunity cost of employment. Calibrated to U.S. data, the model explains 60% of the overall unemployment volatility. Consistent with empirical evidence, the response of unemployment to TFP shocks predicted by my model is large, hump-shaped, and peaks one year after the TFP shock, while the response of the aggregate wage is weak and delayed, peaking after two years. In the second chapter of this dissertation, I study the role of information frictions and inventories in firms' price setting decisions in the context of a monetary model. In this model, intermediate goods firms accumulate output inventories, observe aggregate variables with one period lag, and observe their nominal input prices and demand at all times. Firms face idiosyncratic shocks and cannot perfectly infer the state of nature. After a contractionary nominal shock, nominal input prices go down, and firms accumulate inventories because they perceive some positive probability that the nominal price decline is due to a good productivity shock. This prevents firms' prices from decreasing and makes current profits, households' income, and aggregate demand go down. According to my model simulations, a 1% decrease in the money growth rate causes output to decline 0.17% in the first quarter and 0.38% in the second followed by a slow recovery to the steady state. Contractionary nominal shocks also have significant effects on total investment, which remains 1% below the steady state for the first 6 quarters.

CONTROL OF SIZE AND MAGNETIC PROPERTIES OF CARBOXYL-FUNCTIONALIZED MAGNETITE PARTICLES: SYNTHESIS PROCEDURES, CHARACTERIZATIONS, AND APPLICATIONS

Magnetic nanoparticles (MNPs) are known for the unique properties conferred by their small size and have found wide application in food safety analyses. However, their high surface energy and strong magnetization often lead to aggregation, compromising their functions. In this study, iron oxide magnetic particles (MPs) over the range of nano to micro size were synthesized, from which particles with less aggregation and excellent magnetic properties were obtained. MPs were synthesized via three different hydrothermal procedures, using poly (acrylic acid) (PAA) of different molecular weight (Mw) as the stabilizer. The particle size, morphology, and magnetic properties of the MPs from these synthesis procedures were characterized and compared. Among the three syntheses, one-step hydrothermal synthesis demonstrated the highest yield and most efficient magnetic collection of the resulting PAA-coated magnetic microparticles (PAA-MMPs, >100 nm). Iron oxide content of these PAA-MMPs was around 90%, and the saturation magnetization ranged from 70.3 emu/g to 57.0 emu/g, depending on the Mw of PAA used. In this approach, the particles prepared using PAA with Mw of 100K g/mol exhibited super-paramagnetic behavior with ~65% lower coercivity and remanence compared to others. They were therefore less susceptible to aggregation and remained remarkably water-dispersible even after one-month storage. Three applications involving PAA-MMPs from one-step hydrothermal synthesis were explored: food proteins and enzymes immobilization, antibody conjugation for pathogen capture, and magnetic hydrogel film fabrication. These studies demonstrated their versatile functions as well as their potential applications in the food science area.