The fundamental goal, in preparing this thesis, is two-fold. First, the author shows the systematic development of the basic theory of Dynamic Programming. Secondly, the author demonstrates the applicability of this theory in the actual problem solving of Dynamic Programming. A very careful balance has been maintained in the setting of the theory and in its applicability of solving problems. Basic theory has been established and expanded systematically and explained vividly. However, it is short and precise. Nevertheless, it presents the tools to handle highly complex problems in Dynamic Programming. Also, the theory is used to prove a general case of an N-stage optimization which demonstrates the sufficiency and capability of the theory.

Indiana University-Purdue University Indianapolis (IUPUI) / Much is known about periodic orbits in dynamical systems of continuous interval maps. Of note is the theorem of Sharkovsky. In 1964 he proved that, for a continuous map $f$ on $\mathbb{R}$, the existence of periodic orbits of certain periods force the existence of periodic orbits of certain other periods. Unfortunately there is currently no analogue of this theorem for maps of $\mathbb{R}$ which are not continuous. Here we consider discontinuous interval maps of a particular variety, namely piecewise monotone interval maps. We observe how the presence of a given periodic orbit forces other periodic orbits, as well as the direct analogue of Sharkovsky's theorem in special families of piecewise monotone maps. We conclude by investigating the entropy of piecewise linear maps. Among particular one parameter families of piecewise linear maps, entropy remains constant even as the parameter varies. We provide a simple geometric explanation of this phenomenon known as entropy locking.

Symbolic dynamics, and in particular β-expansions, are a ubiquitous tool in studying more complicated dynamical systems. Applications include number theory, fractals, information theory, and data storage. In this thesis we will explore the basics of dynamical systems with a special focus on topological dynamics. We then examine symbolic dynamics and β-transformations through the lens of sequence spaces. We discuss observations from recent literature about how matching (the property that the itinerary of 0 and 1 coincide after some number of iterations) is linked to when Tβ,⍺ generates a subshift of finite type. We prove the set of ⍺ in the parameter space for which Tβ,⍺ exhibits matching is symmetric and analyze some examples where the symmetry is both apparent and useful in finding a dense set of ⍺ for which Tβ,⍺ generates a subshift of finite type.

In this thesis we investigate cosmological models more general than the isotropic and homogeneous Friedmann-Lemaître models. We focus on cosmologies with one spatial degree of freedom, whose matter content consists of a perfect fluid and the cosmological constant. We formulate the Einstein field equations as a system of quasilinear first order partial differential equations, using scale-invariant variables. The primary goal is to study the dynamics in the two asymptotic regimes, i. e. near the initial singularity and at late times. We highlight the role of spatially homogeneous dynamics as the background dynamics, and analyze the inhomogeneous aspect of the dynamics. We perform a variety of numerical simulations to support our analysis and to explore new phenomena.

This thesis consists of two parts. In the first part, which is expository, abstract theory of one-parameter operator is studied semi-groups. We develop in detail the necessary Banach space and Banach algebra theories of integration, differentiation, and series, and then give a careful rigorous proof of the exponential function characterization of continuous one-parameter operator semigroups. In the second part, which is applied and has new result, we discuss some related topics in dynamical systems. In general the linearizations give a reliable description of the non-linear orbits near the equilibrium points (the Hartman-Grobman theorem), thus illustrating the importance of linear semigroups. The aim of qualitative analysis of differential equations (DE) is to understand the qualitative behaviour (such as, for example, the long-term behaviour as $t\rightarrow \infty$) of typical solutions of the DE. The flow in the direction of increasing time defines a semigroup. As an application we study Einstein-Aether Cosmological models using dynamical systems theory.

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The field of mathematical neuroscience is concerned with the modeling and interpretation of neuronal dynamics and associated phenomena. Neurons can be modeled individually, in small groups, or collectively as a large network. Mathematical models of single neurons typically involve either differential equations, discrete maps, or some combination of both. A number of two-dimensional spiking neuron models that combine continuous dynamics with an instantaneous reset have been introduced in the literature. The models are capable of reproducing a variety of experimentally observed spiking patterns, and also have the advantage of being mathematically tractable. Here an analysis of the transverse stability of orbits in the phase plane leads to sufficient conditions on the model parameters for regular spiking to occur. The application of this method is illustrated by three examples, taken from existing models in the neuroscience literature. In the first two examples the model has no equilibrium states, and regular spiking follows directly. In the third example there are equilibrium points, and some additional quantitative arguments are given to prove that regular spiking occurs. / Graduate

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In this thesis we investigate cosmological models more general than the isotropic and homogeneous Friedmann-Lemaître models. We focus on cosmologies with one spatial degree of freedom, whose matter content consists of a perfect fluid and the cosmological constant. We formulate the Einstein field equations as a system of quasilinear first order partial differential equations, using scale-invariant variables. The primary goal is to study the dynamics in the two asymptotic regimes, i. e. near the initial singularity and at late times. We highlight the role of spatially homogeneous dynamics as the background dynamics, and analyze the inhomogeneous aspect of the dynamics. We perform a variety of numerical simulations to support our analysis and to explore new phenomena.