Let S ⊆ V be an arbitrary subset of vertices of a graph G = (V,E). The differential ∂(S) equals the difference between the cardinality of the set of vertices not in S but adjacent to vertices in S, and the cardinality of the set S. The differential of a graph G equals the maximum differential of any subset S of V . A set S is called a zero set if ∂(S) = 0. In this thesis we introduce the study of zero sets in graphs. We give proofs of the existence of zero sets in various kinds of graphs such as even order graphs, bipartite graphs, and graphs of maximum degree 3. We also give proofs regarding the existence of graphs which contain no zero sets and the construction of zero-free graphs from graphs which contain zero sets.

Discrete optical systems are a subgroup of periodic structures in which the evolution of a continuous electromagnetic field can be described by a discrete model. In this model, the total field is the sum of localized, discrete modes. Weakly coupled arrays of single mode channel waveguides have been known to fall into this class of systems since the late 1960's. Nonlinear discrete optics has received a considerable amount of interest in the last few years, triggered by the experimental realization of discrete solitons in a Kerr nonlinear AlGaAs waveguide array by H. Eisenberg and coworkers in 1998. In this work a detailed experimental investigation of discrete nonlinear wave propagation and the interactions between beams, including discrete solitons, in discrete systems is reported for the case of a strong Kerr nonlinearity. The possibility to completely overcome "discrete" diffraction and create highly localized solitons, in a scalar or vector geometry, as well as the limiting factors in the formation of such nonlinear waves is discussed. The reversal of the sign of diffraction over a range of propagation angles leads to the stability of plane waves in a material with positive nonlinearity. This behavior can not be found in continuous self-focusing materials where plane waves are unstable against perturbations. The stability of plane waves in the anomalous diffraction region, even at highest powers, has been experimentally verified. The interaction of high power beams and discrete solitons in arrays has been studied in detail. Of particular interest is the experimental verification of a theoretically predicted unique, all optical switching scheme, based on the interaction of a so called "blocker" soliton with a second beam. This switching method has been experimentally realized for both the coherent and incoherent case. Limitations of such schemes due to nonlinear losses at the required high powers are shown.

Using simulation techniques, performance of any proposed system can be tested for different scenarios with a generated model. However, it is difficult to rapidly create simulation models that will accurately represent the complexity of the system. In recent years, Object-Oriented Discrete-Event Simulation has emerged as the potential technology to implement rapid simulation schemes. A number of software based on programming languages like C++ and Java are available for carrying out Object Oriented Discrete-Event Simulation. These software packages establish a general framework for simulation in computer programs, but need to be further customized for desired end-use applications. In this thesis, a generic simulation library is created for the distributed Synchronous Parallel Environment for Emulation and Discrete-Event Simulation (SPEEDES). This library offers classes to model the functionality of servers, processes, resources, transporters, and decisions. The library is expected to produce efficient simulation models in less time and with a lesser amount of coding. The class hierarchy is modeled using the Unified Modeling Language (UML). To test the library, the existing SPEEDES Space Shuttle Model is enhanced and recreated. This enhanced model is successfully validated against the original Arena model.

Spatially discrete Nagumo equations have widespread physical applications, including modeling electrical impulses traveling through a demyelinated axon, an environment typical in multiple scle- rosis. We construct steady-state, single front solutions by employing a piecewise linear reaction term. Using a combination of Jacobi-Operator theory and the Sherman-Morrison formula we de- rive exact solutions in the cases of homogeneous and inhomogeneous diffusion. Solutions exist only under certain conditions outlined in their construction. The range of parameter values that satisfy these conditions constitutes the interval of propagation failure, determining under what circumstances a front becomes pinned in the media. Our exact solutions represent a very specific solution to the spatially discrete Nagumo equation. For example, we only consider inhomogeneous media with one defect present. We created an original script in MATLAB which algorithmically solves more general cases of the equation, including the case for multiple defects. The algorithmic solutions are then compared to known exact solutions to determine their validity.

In this paper we study the hierarchical structure of the 2-d polyominoes. We introduce a new infinite family of polyominoes which we prove tiles a strip. We discuss applications of algebra to tiling. We discuss the algorithmic decidability of tiling the infinite plane Z x Z given a finite set of polyominoes. We will then discuss tiling with rectangles. We will then get some new, and some analogous results concerning the possible hierarchical structure for the 3-d polycubes.

No / A discrete particle model is described which simulates bedload transport over a flat bed of a unimodal mixed-sized distribution of particles. Simple physical rules are applied to large numbers of discrete sediment grains moving within a unidirectional flow. The modelling assumptions and main algorithms of the bedload transport model are presented and discussed. Sediment particles are represented by smooth spheres, which move under the drag forces of a simulated fluid flow. Bedload mass-transport rates calculated by the model exhibit a low sensitivity to chosen model parameters. Comparisons of the calculated mass-transport rates with well-established empirical relationships are good, strongly suggesting that the discrete particle model has captured the essential elements of the system physics. This performance provides strong justification for future interrogation of the model to investigate details of the small-scale constituent processes which have hitherto been outside the reach of previous experimental and modelling investigations.

The Grassmannian space GC(2, 4) embedded in CP5 as the Klein quadric of twistor theory has a natural interpretation in terms of the geometry of “round” 2-spheres in S4. The incidence of two lines in CP3 corresponds to the contact properties of two 2- spheres, where contact is generalized from tangency to include “half-tangency:” 2-spheres may be in contact at two isolated points. There is a connection between the contact properties of 2-spheres and soliton geometry through the classical Ribaucour and Darboux transformations. The transformation theory of surfaces in S4 is investigated using the recently developed theory of “Discrete Differential Geometry” with results leading to the conclusion that the discrete conformal maps into C of Hertrich-Jeromin, McIntosh, Norman and Pedit may be defined in terms a discrete integrable system employing halftangency in S4.

<p> The topic of study within includes the development and application of nonlinear control technologies on sampled systems. Discrete control structures are introduced that expand on existing differential geometric and predictive control methods. The differential geometric techniques are described from the error trajectory context, which are typically only derived for continuous application. The discrete error trajectory controllers introduced have one of two configurations. The first configuration requires satisfaction of the error trajectory objective at the next sampling interval through prediction of system behaviour over the controller sampling interval. This objective found limited success and it is observed that satisfaction of the error trajectory objective at discrete intervals does not generally result in the intended response. The second configuration minimizes the integrated distance from the error manifold defined by the error trajectory objective over the entire controller sampling interval. It is observed that this integrated error trajectory controller best emulates the intent of the continuous controller in the discrete domain. Techniques borrowed from predictive control are incorporated into the integrated error trajectory controller such as input move suppression and constraints to produce an optimal error trajectory controller, further improving performance.</p> <p> The predictive control method introduced utilizes a transformation of the input space. The differentiating property of input transformation predictive control (ITPC) from other methods is the prediction technique that is capable of estimating the future behaviour of nonlinear systems through elementary matrix operations similar to the dynamic matrix control (DMC) prediction technique. This is achieved by separation of the steady state and dynamic system properties and the introduction of an intermediate state prediction layer. This allows for the nonlinear prediction of system behaviour without the need to numerically integrate the system model.</p> <p> Two example systems are used to demonstrate application of the discrete error trajectory and ITPC on nonlinear controllers. Performance for these control structures is compared to technologies accepted within the control community for a broad range for characteristics including, computation efficiency, design effort and other nonlinear performance criteria, with favourable results.</p> / Thesis / Master of Engineering (MEngr)

The Internet of Things (IoT) is an important and expanding technology used for a large variety of applications to monitor and automate processes. The aim of this thesis is to present a way to find and assign locally unique IDs to access points (APs) in enormous wireless IoT systems where mobile tags are traversing the network and communicating with multiple APs simultaneously. This is done in order to improve the robustness of the system and increase the battery time of the tags. The resulting algorithm is based on transforming the problem into a graph coloring problem and solving it using approximate methods. Two metaheuristics: Simulated annealing and tabu search were implemented and compared for this purpose. Both of these showed similar results and neither was clearly superior to the other. Furthermore, the presented algorithm can also exclude nodes from the coloring based on the results in order to ensure a proper solution that also satisfies a robustness criterion. A metric was also created in order for a user to intuitively evaluate the quality of a given solution. The algorithm was tested and evaluated on a system of 222 APs for which it produced good results.

Yes / Market competition is fierce and has been intensified due to globalization, therefore companies have experienced increase pressure to improve cycle and delivery times and achieve a high level of customization. Lean principles have been designed and implemented to respond to market changes. However, these principles are commonly aimed to aid low-mix high-volume (LMHV) manufacturers. This paper aims to develop a systematic approach to implement a lean framework in a high-mix low-volume (HMLV) manufacturing environment. An HMLV manufacturing environment, currently produces a variety of products which differs in terms of shape and size and different sequence of operation and as a much smaller batch size. Additionally, analysis of the manufacturing assembly shows there is a substantial variation in cycle and changeover time from product to product. In this approach, an action based study has been completed. This study uses data collection methods to be applied in order to calculate timings to be used in a Discrete Event Simulation (DES). The simulation uses lean tools to study material movement and facility layout to minimize bottlenecks and eliminate waste from the process. An introduction of lean supermarket is considered within the simulation whereby an operator withdraws products in an specific amount needed by a downstream process before introducing it across the assembly floor and an action plan is created. As a result, a lean manufacturing simulation study has been implemented to evaluate the effects of a supermarket concept within the current manufacturing environment. Initial results show a variation within the cycle and setup times for each of the different products due to the nature of the process. The paper, is limited to applying the approach to a singular production line and a singular manufacturing plant. The practical implications to implement this approach into a manufacturing environment is that there should be a need from the company. This approach should be driven by senior members within the organization as the resistance to change would increase the risk of failure. Another implication of the proposed approach is to ensure the accuracy of the data collected and to introduce a series of briefs within each stage of the approach. This is important as all stakeholders would need to be kept up-to-date with the project. However, the approach is applicable to any organization and can be applied to any sector. This paper, develops a systematic approach to implement a lean framework in a high-mix low-volume (HMLV) manufacturing environment. The approach is validated in an automotive manufacturing organization competing in global markets.