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.

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This thesis work is about commutativity which is a very important topic in mathematics, physics, engineering and many other fields. Two processes are said to be commutative if the order of "operation" of these processes does not matter. A typical example of two processes in real life that are not commutative is the process of opening the door and the process of going through the door. In mathematics, it is well known that matrix multiplication is not always commutative. Commutating operators play an essential role in mathematics, physics engineering and many other fields. A typical example of the importance of commutativity comes from signal processing. Signals pass through filters (often called operators on a Hilbert space by mathematicians) and commutativity of two operators corresponds to having the same result even when filters are interchanged. Many important relations in mathematics, physics and engineering are represented by operators satisfying a number of commutation relations. In chapter two of this thesis we treat commutativity of monomials of operatos satisfying certain commutation relations in relation to one-dimensional dynamical systems. We derive explicit conditions for commutativity of the said monomials in relation to the existence of periodic points of certain one-dimensional dynamical systems. In chapter three, we treat the crossed product algebra for the algebra of piecewise constant functions on given set, describe the commutant of this algebra of functions which happens to be the maximal commutative subalgebra of the crossed product containing this algebra. In chapter four, we give a characterization of the commutant for the algebra of piecewise constant functions on the real line, by comparing commutants for a non decreasing sequence of algebras.

by Shu-yu Hsu. / "July 1999." / Thesis (Ph.D.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (p. 40-42). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

Mean curvature flow evolves isometrically immersed base Riemannian manifolds M in the direction of their mean curvature in an ambient manifold M. We consider the classical solutions to the mean curvature flow. If the base manifold M is compact, the short time existence and uniqueness of the mean curvature flow are well-known. For complete noncompact isometrically immersed hypersurfaces M (uniformly local lipschitz) in Euclidean space, the short time existence was established by Ecker and Huisken in [10]. The short time existence and the uniqueness of the solutions to the mean curvature flow of complete isometrically immersed manifolds of arbitrary codimensions in the Euclidean space are still open questions. In this thesis, we solve the uniqueness problem affirmatively for the mean curvature flow of general codimensions and general ambient manifolds. More precisely, let (M, g) be a complete Riemannian manifold of dimension n such that the curvature and its covariant derivatives up to order 2 are bounded and the injectivity radius is bounded from below by a positive constant, we prove that the solution of the mean curvature flow with bounded second fundamental form on an isometrically immersed manifold M (may be of high codimension) is unique. In the second part of the thesis, inspired by the Ricci flow, we prove the pseudolocality theorem of mean curvature flow. As a consequence, we obtain the strong uniqueness theorem, which removes the boundedness assumption of the second fundamental form of the solution in the uniqueness theorem (only assume the second fundamental form of the initial submanifold is bounded). / Yin, Le. / "July 2007." / Adviser: Leung Nai-Chung. / Source: Dissertation Abstracts International, Volume: 69-01, Section: B, page: 0357. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (p. 65-68). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

Many biological problems can be reduced to the description of a food chain model or a food web. In these systems, the biodiversity and coexistence of all species are vital issues to discuss. Three ecological models have been proposed in case of the existence of a reserved area, in order to understand multi-species interactions so as to prevent the slow extinction of some endangered species and to test the stability when the length of the food chain and size of the web models are increased. It is taken that the environment has been divided into two disjoint regions, namely, unreserved and reserved zones, where a predator is not allowed to enter the latter. The first model describes a four species food chain predator-prey model with prey refuge (prey in the reserved zone, prey in the unreserved zone, predator and top predator), with the predator being entirely dependent on the prey in the unprotected area. The second model addresses the same problem, but in addition, a third component in the chain partially depends on the prey in the unreserved zone. Finally, the last model investigates a four species food web system with a prey refuge and in this case, the fourth component can also feed directly on the prey in the unreserved zone. The boundedness, existence and uniqueness of the solutions of the proposed models are established. The local and global dynamical behaviours are investigated, with the persistence conditions of the models being elicited. The local bifurcation near each of the equilibrium points is obtained. The numerical simulations in MATLABR are used to study the influence of the existence of the reserved zone on the dynamical behaviour of the proposed models. It has been concluded that the role of the reserved area could be beneficial for the survival and stabilising of multi-species interactions.

We present a novel approach to parameter estimation of systems with complicated dynamics, as well as evidence for the existence of a universal power law that enables us to quantify the dependence of global geometry on small changes in the parameters of the system. This power law gives rise to what seems to be a new dynamical system invariant.