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.

The Real-Quaternionic Indicator of Irreducible Self-Conjugate Representations of Real Reductive Algebraic Groups and A Comment on the Local Langlands Correspondence of GL(2, F )

The real-quaternionic indicator, also called the $\delta$ indicator, indicates if a self-conjugate representation is of real or quaternionic type. It is closely related to the Frobenius-Schur indicator, which we call the $\varepsilon$ indicator. The Frobenius-Schur indicator $\varepsilon(\pi)$ is known to be given by a particular value of the central character. We would like a similar result for the $\delta$ indicator. When $G$ is compact, $\delta(\pi)$ and $\varepsilon(\pi)$ coincide. In general, they are not necessarily the same. In this thesis, we will give a relation between the two indicators when $G$ is a real reductive algebraic group. This relation also leads to a formula for $\delta(\pi)$ in terms of the central character. For the second part, we consider the construction of the local Langlands correspondence of $GL(2,F)$ when $F$ is a non-Archimedean local field with odd residual characteristics. By re-examining the construction, we provide new proofs to some important properties of the correspondence. Namely, the construction is independent of the choice of additive character in the theta correspondence.

Relations among Topic Knowledge, Individual Interest, and Relational Reasoning, and Critical Thinking in Maternity Nursing

Critical thinking in learners is a goal of educators and professional organizations in nursing as well as other professions. However, few studies in nursing have examined the role of the important individual difference factors topic knowledge, individual interest, and general relational reasoning strategies in predicting critical thinking. In addition, most previous studies have used domain-general, standardized measures, with inconsistent results. Moreover, few studies have investigated critical thinking across multiple levels of experience. The major purpose of this study was to examine the degree to which topic knowledge, individual interest, and relational reasoning predict critical thinking in maternity nurses. For this study, 182 maternity nurses were recruited from national nursing listservs explicitly chosen to capture multiple levels of experience from prelicensure to very experienced nurses. The three independent measures included a domain-specific Topic Knowledge Assessment (TKA), consisting of 24 short-answer questions, a Professed and Engaged Interest Measure (PEIM), with 20 questions indicating level of interest and engagement in maternity nursing topics and activities, and the Test of Relational Reasoning (TORR), a graphical selected response measure with 32 items organized in scales corresponding to four forms of relational reasoning: analogy, anomaly, antithesis, and antinomy. The dependent measure was the Critical Thinking Task in Maternity Nursing (CT2MN), composed of a clinical case study providing cues with follow-up questions relating to nursing care. These questions align with the cognitive processes identified in a commonly-used definition of critical thinking in nursing. Reliable coding schemes for the measures were developed for this study. Key findings included a significant correlation between topic knowledge and individual interest. Further, the three individual difference factors explained a significant proportion of the variance in critical thinking with a large effect size. While topic knowledge was the strongest predictor of critical thinking performance, individual interest had a moderate significant effect, and relational reasoning had a small but significant effect. The findings suggest that these individual difference factors should be included in future studies of critical thinking in nursing. Implications for nursing education, research, and practice are discussed.