We prove the existence and linear stability of quasi-periodic breather solutions in a 1d Hamiltonian lattice of identical, weakly-coupled, anharmonic oscillators with general on-site potentials and under the effect of long-ranged interaction, via de KAM technique. We prove the persistence of finite-dimensional tori which correspond in the uncoupled limit to N arbitrary lattice sites initially excited. The frequencies of the invariant tori of the perturbed system are only slightly deformed from the frequencies of the unperturbed tori.

Thesis (Ph. D.)--University of California, Riverside, 2009. / Includes abstract. Title from first page of PDF file (viewed March 11, 2010). Available via ProQuest Digital Dissertations. Includes bibliographical references (p. 170-172). Also issued in print.

Thesis (Ph. D.)--State University of New York at Binghamton, Thomas J. Watson School of Engineering and Applied Science, Department of Systems Science and Industrial Engineering, 2009. / Includes bibliographical references.

Thesis (Ph. D. in Mathematics)--Vanderbilt University, 2003. / Title from PDF title screen. Includes bibliographical references and index.

Runtime enforcement mechanisms are an important and well-employed method for ensuring an execution only exhibits acceptable behavior, as dictated by a security policy. Wherever interaction occurs between two or more parties that do not completely trust each other, it is most often the case that a runtime enforcement mechanism is between them in some form, monitoring the exchange. Considering the ubiquity of such scenarios in the computing world, there has been an increased effort to build formal models of runtime monitors that closely capture their capabilities so that their effectiveness can be analysed more precisely. While models have grown more faithful to their real-life counterparts, is- sues concerning complexity and manageability (a common concern for software engineers) of centralized policies remains to be fully addressed. The goal of this thesis is to provide a principled approach to policy construction that is modular, intuitive, and backed by formal methods. This thesis introduces a class of policy combinators adequate for use with runtime en- forcement policies and analyses a particular instance of them called Static Committee Com- binators (SCCs). SCCs present a model of policy composition where combinators act as committees that vote on events passing through the monitor. They were conceptualized in collaboration with Jay Ligatti and Daniel Lomsak. The general class of combinators are called Static Decision Combinators (SDCs), which share key features with SCCs such as allowing combinators to respond with alternative events when polled, in addition to re- sponding with grants or denials. SDCs treat the base-level policies they compose as black boxes, which helps decouple the system of combinators from the underlying policy model. The base policies could be modelled by automata but the combinators would not maintain their own state, being "static". This allows them to be easily defined and understood using truth tables, as well as analysed using logic tools. In addition to an analysis of SDCs and SCCs, we provide useful examples and a reusable combinator library.

Algorithms based on Markov chains are ubiquitous across scientific disciplines, as they provide a method for extracting statistical information about large, complicated systems. Although these algorithms may be applied to arbitrary graphs, many physical applications are more naturally studied under the restriction to regular lattices. We study several local Markov chains on lattices, exploring how small changes to some parameters can greatly influence efficiency of the algorithms. We begin by examining a natural Markov Chain that arises in the context of "monotonic surfaces", where some point on a surface is sightly raised or lowered each step, but with a greater rate of raising than lowering. We show that this chain is rapidly mixing (converges quickly to the equilibrium) using a coupling argument; the novelty of our proof is that it requires defining an exponentially increasing distance function on pairs of surfaces, allowing us to derive near optimal results in many settings. Next, we present new methods for lower bounding the time local chains may take to converge to equilibrium. For many models that we study, there seems to be a phase transition as a parameter is changed, so that the chain is rapidly mixing above a critical point and slow mixing below it. Unfortunately, it is not always possible to make this intuition rigorous. We present the first proofs of slow mixing for three sampling problems motivated by statistical physics and nanotechnology: independent sets on the triangular lattice (the hard-core lattice gas model), weighted even orientations of the two-dimensional Cartesian lattice (the 8-vertex model), and non-saturated Ising (tile-based self-assembly). Previous proofs of slow mixing for other models have been based on contour arguments that allow us prove that a bottleneck in the state space constricts the mixing. The standard contour arguments do not seem to apply to these problems, so we modify this approach by introducing the notion of "fat contours" that can have nontrivial area. We use these to prove that the local chains defined for these models are slow mixing. Finally, we study another important issue that arises in the context of phase transitions in physical systems, namely how the boundary of a lattice can affect the efficiency of the Markov chain. We examine a local chain on the perfect and near-perfect matchings of the square-octagon lattice, and show for one boundary condition the chain will mix in polynomial time, while for another it will mix exponentially slowly. Strikingly, the two boundary conditions only differ at four vertices. These are the first rigorous proofs of such a phenomenon on lattice graphs.

Advances in additive manufacturing technologies have brought a new paradigm shift to both design and manufacturing. There is a much bigger design space in which designers can achieve a level of complexity and customizability, which are infeasible using traditional manufacturing processes. One application of this technology is for fabrication of meso-scale lattice structures (MSLS). These types of structures are designed to have material where it is needed for specific applications. They are suitable for any weight-critical applications, particularly in industries where both low weight and high strength are desired. MSLS can easily have hundreds to thousands of individual strut, where the diameter of each strut can be treated as a design variable. As a result, the design process poses a computational challenge. Since the computational complexity of the design problem often scales exponentially with the number of design variables, topological optimization that requires multi-variable optimization algorithm is infeasible for large-scale problems. In previous research, a new method was presented for efficiently optimizing MSLS by utilizing a heuristic that reduces the multivariable optimization problem to a problem of only two variables. The method is called the Size Matching and Scaling (SMS) method, which combines solid-body analysis and predefined unit-cell library to generate the topology of the structure. However, the method lacks a systematic methodology to generate the initial ground geometry for the design process, which limits the previous implementations of the SMS method to only simple, axis-aligned structures. In this research, an augmented SMS method is presented. The augmented method includes the integration of free-mesh approach in generating the initial ground geometry. The software that embodies that ground geometry generation process is integrated to commercial CAD system that allows designer to set lattice size parameters through graphical user interface. In this thesis, the augmented method and the unit-cell library are applied to various design examples. The augmented SMS method can be applied effectively in the design of conformal lattice structure with highly optimized stiffness and volume for complex surface. Conformal lattice structures are those conformed to the shape of a part's surface and that can used to stiffen or strengthen a complex and curved surface. This design approach removes the need for a rigorous topology optimization, which is a main bottleneck in designing MSLS.

Thesis (Ph. D.)--University of Alabama at Birmingham, 2007. / Additional advisors: Richard Brown, Ioulia Karpechina, Ryoichi Kawai, Boris Kunin. Description based on contents viewed Feb. 5, 2008; title from title screen. Includes bibliographical references (p. 73-75).

Thesis (Ph.D.)--Mathematics, Georgia Institute of Technology, 2008. / Committee Chair: Trotter, William T.; Committee Member: Duke, Richard A.; Committee Member: Randall, Dana; Committee Member: Thomas, Robin; Committee Member: Yu, Xingxing

No description available.