Thesis (Ph.D.)--University of Central Florida, 2008. / Adviser: Zhihua Qu. Includes bibliographical references (p. 117-121).

Thesis (Ph.D..(Mathematics and Applied Mathematics)) -- University of Pretoria, 2009. / Summary and abstract in English. Includes bibliographical references.

Thesis (Ph. D.)--University of Texas at Austin, 2002. / Vita. Includes bibliographical references. Available also from UMI Company.

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 2000. / Includes bibliographical references (p. 59-61).

Theses (M.A.)--Marshall University, 2006. / Title from document title page. Includes abstract. Document formatted into pages: contains vii, 72 pages. Bibliography: p. 70-72.

Theses (M.A.)--Marshall University, 2005. / Title from document title page. Includes abstract. Document formatted into pages: contains vi, 38 pages. Bibliography: page 38.

The object of this thesis is the study of the decidability properties of linear dynamical systems, which have fundamental ties to theoretical computer science, software verification, linear hybrid systems, and control theory. In particular, we describe a method for deciding the termination of simple linear loops, partly solving a 10-year-old open problem of Tiwari (2004) and Braverman (2006). We also study the membership problem for semigroups of matrix exponentials, which we show to be undecidable in general by reduction from Hilbert's Tenth Problem, and decidable for all instances where the matrices defining the semigroup commute. In turn, this entails the undecidability of the generalised versions of the Continuous Orbit and Skolem Problems to a multi-matrix setting. We also study point-to-point controllability for linear time-invariant systems, which is a central problem in control theory. For discrete-time systems, we show that this problem is undecidable when the set of controls is non-convex, and at least as hard as the Skolem Problem even when it is a convex polytope; for continuous-time systems, we show that this problem reduces to the Continuous Orbit Problem when the set of controls is a linear subspace, which entails decidability. Finally, we show how to decide whether all solutions of a given linear ordinary differential equation starting in a given convex polytope eventually leave it; this problem, which we call the "Polytope Escape Problem'', relates to the liveness of states in linear hybrid automata. Our results rely on a number of theorems from number theory, logic, and algebra, which we introduce in a self-contained way in the preamble to this thesis, together with a few new mathematical results of independent interest.

Systems composed of dynamical networks - such as the human body with its biological networks or the global economic network consisting of regional clusters - often exhibit complicated collective dynamics. Three fundamental processes that are typically present are failure, damage spread, and recovery. Here we develop a model for such systems and find phase diagrams for single and interacting networks. By investigating networks with a small number of nodes, where finite-size effects are pronounced, we describe the spontaneous recovery phenomenon present in these systems. In the case of interacting networks the phase diagram is very rich and becomes increasingly more complex as the number of interacting networks increases. In the simplest example of two interacting networks we find two critical points, four triple points, ten allowed transitions, and two forbidden transitions, as well as complex hysteresis loops. Remarkably, we find that triple points play the dominant role in constructing the optimal repairing strategy in damaged interacting systems. To test our model, we analyze an example of real interacting financial networks and find evidence of rapid dynamical transitions between well-defined states, in agreement with the predictions of our model.

Herz–Schur multipliers of a locally compact group have a well developed theory coming from a large literature; they have proved very useful in the study of the reduced C∗-algebra of a locally compact group. There is also a rich connection to Schur multipliers,which have been studied since the early twentieth century, and have a large number of applications. We develop a theory of Herz–Schur multipliers of a C∗-dynamical system, extending the classical Herz–Schur multipliers, making Herz–Schur multiplier techniques available to study a much larger class of C∗-algebras. Furthermore, we will also introduce and study generalised Schur multipliers, and derive links between these two notions which extend the classical results describing Herz–Schur multipliers in terms of Schur multipliers. This theory will be developed in as much generality as possible, with reference to the classical motivation. After introducing all the necessary concepts we begin the investigation by defining generalised Schur multipliers. The main result is a dilation type characterisation of these multipliers; we also show how such multipliers can be represented using HilbertC∗-modules. Next we introduce and study generalised Herz–Schur multipliers, first extending a classical result involving the representation theory of SU(2), before studying how such functions are related to our generalised Schur multipliers. We give a characterisation of generalised Herz–Schur multipliers as a certain class of the generalised Schur multipliers, and obtain a description of precisely which Schur multipliers belong to this class. Finally, we consider some ways in which the generalised multipliers can arise; firstly, from the classical multipliers which provide our motivation, secondly, from the Haagerup tensor product of a C∗-algebra with itself, and finally from positivity considerations. We show that our theory behaves well with respect to positivity and give conditions under which our multipliers are automatically positive in a natural sense.

abstract: Persistence theory provides a mathematically rigorous answer to the question of population survival by establishing an initial-condition- independent positive lower bound for the long-term value of the population size. This study focuses on the persistence of discrete semiflows in infinite-dimensional state spaces that model the year-to-year dynamics of structured populations. The map which encapsulates the population development from one year to the next is approximated at the origin (the extinction state) by a linear or homogeneous map. The (cone) spectral radius of this approximating map is the threshold between extinction and persistence. General persistence results are applied to three particular models: a size-structured plant population model, a diffusion model (with both Neumann and Dirichlet boundary conditions) for a dispersing population of males and females that only mate and reproduce once during a very short season, and a rank-structured model for a population of males and females. / Dissertation/Thesis / Ph.D. Mathematics 2014