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.

THERMAL VACUUM TEST CORRELATION OF A ZERO PROPELLANT LOAD CASE THERMAL CAPACITANCE PROPELLANT GAUGING ANALYTICAL MODEL

This thesis describes the development and correlation of a thermal model that forms the foundation of a thermal capacitance spacecraft propellant load estimator. Specific details of creating the thermal model for the diaphragm propellant tank used on NASA’s Magnetospheric Multiscale spacecraft using ANSYS and the correlation process implemented are presented. The thermal model was correlated to within +/- 3 Celsius of the thermal vacuum test data, and was determined sufficient to make future propellant predictions on MMS. The model was also found to be relatively sensitive to uncertainties in applied heat flux and mass knowledge of the tank. More work is needed to improve temperature predictions in the upper hemisphere of the propellant tank where predictions were found to be 2-2.5 Celsius lower than the test data. A road map for applying the model to predict propellant loads on the actual MMS spacecraft in 2017-2018 is also presented.