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.

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.

Surveillance in Cyberspace: Applying Natural and Place Manager Surveillance to System Trespassing

Research on the criminological side of system trespassing (i.e. unlawfully gaining access to a computer system) is relatively rare and has yet to examine the effect of the presence of other users on the system during the trespassing event (i.e. the time of communication between a trespasser’s system and the infiltrated system). This thesis seeks to analyze this relationship drawing on principles of Situational Crime Prevention, Routine Activities Theory, and restrictive deterrence. Data were collected from a randomized control trial of target computers deployed on the Internet network of a large U.S. university. This study examined whether the number (one or multiple) and type (administrative or non-administrative) of computer users present on a system reduced the seriousness and frequency of trespassing. Results indicated that the type of user (administrative) produced a restrictive deterrent effect and significantly reduced the frequency and duration of trespassing events.

Jiao Tong: A Grounded Theory of Chinese International Students' Transition to American Tertiary Education

University students are more globally mobile than ever before, increasingly receiving education outside of their home countries. One significant student exchange pattern is between China and the United States; Chinese students are the largest population of international students in the U.S. (Institute of International Education, 2014). Differences between Chinese and American culture in turn influence higher education praxis in both countries, and students are enculturated into the expectations and practices of their home countries. This implies significant changes for students who must navigate cultural differences, academic expectations, and social norms during the process of transition to a system of higher education outside their home country. Despite the trends in students’ global mobility and implications for international students’ transitions, scholarship about international students does not examine students’ experiences with the transition process to a new country and system of higher education. Related models were developed with American organizations and individuals, making it unlikely that they would be culturally transferable to Chinese international students’ transitions. This study used qualitative methods to deepen the understanding of Chinese international students’ transition processes. Grounded theory methods were used to invite the narratives of 18 Chinese international students at a large public American university, analyze the data, and build a theory that reflects Chinese international students’ experiences transitioning to American university life. Findings of the study show that Chinese international students experience a complex process of transition to study in the United States. Students’ pre-departure experiences, including previous exposure to American culture, family expectations, and language preparation, informed their transition. Upon arrival, students navigate resource seeking to fulfill their practical, emotional, social, intellectual, and ideological needs. As students experienced various positive and discouraging events, they developed responses to the pivotal moments. These behaviors formed patterns in which students sought familiarity or challenge subsequent to certain events. The findings and resulting theory provide a framework through which to better understand the experiences of Chinese international students in the context of American higher education.