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.

ENGINEERING DIGITAL SHARING PLATFORM TO CREATE SOCIAL CONTAGION: EVIDENCE FROM LARGE SCALE RANDOMIZED FIELD EXPERIMENTS

Peer-to-peer information sharing has fundamentally changed customer decision-making process. Recent developments in information technologies have enabled digital sharing platforms to influence various granular aspects of the information sharing process. Despite the growing importance of digital information sharing, little research has examined the optimal design choices for a platform seeking to maximize returns from information sharing. My dissertation seeks to fill this gap. Specifically, I study novel interventions that can be implemented by the platform at different stages of the information sharing. In collaboration with a leading for-profit platform and a non-profit platform, I conduct three large-scale field experiments to causally identify the impact of these interventions on customers’ sharing behaviors as well as the sharing outcomes. The first essay examines whether and how a firm can enhance social contagion by simply varying the message shared by customers with their friends. Using a large randomized field experiment, I find that i) adding only information about the sender’s purchase status increases the likelihood of recipients’ purchase; ii) adding only information about referral reward increases recipients’ follow-up referrals; and iii) adding information about both the sender’s purchase as well as the referral rewards increases neither the likelihood of purchase nor follow-up referrals. I then discuss the underlying mechanisms. The second essay studies whether and how a firm can design unconditional incentive to engage customers who already reveal willingness to share. I conduct a field experiment to examine the impact of incentive design on sender’s purchase as well as further referral behavior. I find evidence that incentive structure has a significant, but interestingly opposing, impact on both outcomes. The results also provide insights about senders’ motives in sharing. The third essay examines whether and how a non-profit platform can use mobile messaging to leverage recipients’ social ties to encourage blood donation. I design a large field experiment to causally identify the impact of different types of information and incentives on donor’s self-donation and group donation behavior. My results show that non-profits can stimulate group effect and increase blood donation, but only with group reward. Such group reward works by motivating a different donor population. In summary, the findings from the three studies will offer valuable insights for platforms and social enterprises on how to engineer digital platforms to create social contagion. The rich data from randomized experiments and complementary sources (archive and survey) also allows me to test the underlying mechanism at work. In this way, my dissertation provides both managerial implication and theoretical contribution to the phenomenon of peer-to-peer information sharing.

An Investigation of Child and Adolescent Dental Sealant Predictors, NHANES 2011-2012

Objective: To examine sociodemographic and dental factors for associations with dental sealant placement in children and adolescents aged 6-18 years old. Methods: Secondary data analysis of 2011-2012 NHANES data was conducted. Multiple logistic regression models were used to assess relationships between predictor variables and sealant presence. Results: More than a third (37.1%) of children and adolescents have at least one sealant present; 67.9% of children compared with 40.4% of adolescents. Racial/ethnic differences exist, with Non-Hispanic black youth having the lowest odds of having sealants. Sealant placement odds vary by presence of dental home; the magnitude of the odds varies by age group. Those with untreated decay have lower odds of having sealants than those who do not have untreated decay (child OR: 2.6, 95% CI: 1.83-3.72; adolescent OR: 3.9, 95% CI: 2.59-6.07). Conclusion: Disparities exist in odds of sealant prevalence across racial/ethnic groups, income levels, and dental disease and visit characteristics. Further research is necessary to understand the reasons for these differences and to inform future interventions.

Observed Social Problem Solving and Friendship Quality in Children with a Traumatic Brain Injury

Children who have experienced a traumatic brain injury (TBI) are at risk for a variety of maladaptive cognitive, behavioral and social outcomes (Yeates et al., 2007). Research involving the social problem solving (SPS) abilities of children with TBI indicates a preference for lower level strategies when compared to children who have experienced an orthopedic injury (OI; Hanten et al., 2008, 2011). Research on SPS in non-injured populations has highlighted the significance of the identity of the social partner (Rubin et al., 2006). Within the pediatric TBI literature few studies have utilized friends as the social partner in SPS contexts, and fewer have used in-vivo SPS assessments. The current study aimed to build on existing research of SPS in children with TBI by utilizing an observational coding scheme to capture in-vivo problem solving behaviors between children with TBI and a best friend. The current study included children with TBI (n = 41), children with OI (n = 43), and a non-injured typically developing group (n = 41). All participants were observed completing a task with a friend and completed a measure of friendship quality. SPS was assessed using an observational coding scheme that captured SPS goals, strategies, and outcomes. It was expected children with TBI would produce fewer successes, fewer direct strategies, and more avoidant strategies. ANOVAs tested for group differences in SPS successes, direct strategies and avoidant strategies. Analyses were run to see if positive or negative friendship quality moderated the relation between group type and SPS behaviors. Group differences were found between the TBI and non-injured group in the SPS direct strategy of commands. No group differences were found for other SPS outcome variables of interest. Moderation analyses partially supported study hypotheses regarding the effect of friendship quality as a moderator variable. Additional analyses examined SPS goal-strategy sequencing and grouped SPS goals into high cost and low cost categories. Results showed a trend supporting the hypothesis that children with TBI had fewer SPS successes, especially with high cost goals, compared to the other two groups. Findings were discussed highlighting the moderation results involving children with severe TBI.