The notions of partial Lagrangians, partial Noether operators and partial Euler-Lagrange equations are used in the construction of first integrals for ordinary differential equations (ODEs) that need not be derivable from variational principles. We obtain a Noetherlike theorem that provides the first integral by means of a formula which has the same structure as the Noether integral. However, the invariance condition for the determination of the partial Noether operators is different as we have a partial Lagrangian and as a result partial Euler-Lagrange equations. In order to investigate the effectiveness of the partial Lagrangian approach, some models such as the oscillator systems both linear and nonlinear, Emden and Ermakov-pinnery equations and the Hamiltonian system with two degrees of freedom are considered in this work. We study a general linear system of two second-order ODEs with variable coefficients. Note that, a Lagrangian exists for the special case only but, in general, the system under consideration does not have a standard Lagrangian. However, partial Lagrangians do exist for all such equations in the absence of Lagrangians. Firstly, we classify all the Noether and partial Noether operators for the case when the system admits a standard Lagrangian. We show that the first integrals that result due to the partial Noether approach is the same as for the Noether approach. First integrals are then constructed by the partial Noether approach for the general case when there is in general no Lagrangian for the system of two second-order ODEs with variable coefficients. We give an easy way of constructing first integrals for such systems by utilization of a partial Noether’s theorem with the help of partial Noether operators associated with a partial Lagrangian. Furthermore, we classify all the potential functions for which we construct first integrals for a system with two degrees of freedom. Moreover, the comparison of Lagrangian and partial Lagrangian approaches for the two degrees of freedom Lagrangian system is also given. In addition, we extend the idea of a partial Lagrangian for the perturbed ordinary differential equations. Several examples are constructed to illustrate the definition of a partial Lagrangian in the approximate situation. An approximate Noether-like theorem which gives the approximate first integrals for the perturbed ordinary differential equations without regard to a Lagrangian is deduced. We study the approximate partial Noether operators for a system of two coupled nonlinear oscillators and the approximate first integrals are obtained for both resonant and non-resonant cases. Finally, we construct the approximate first integrals for a system of two coupled van der Pol oscillators with linear diffusive coupling. Since the system mentioned above does not satisfy a standard Lagrangian, the approximate first integrals are still constructed by invoking an approximate Noether-like theorem with the help of approximate partial Noether operators. This approach can give rise to further studies in the construction of approximate first integrals for perturbed equations without a variational principle.

Master of Science / Department of Mathematics / Ivan Blank / Charles N. Moore / This paper will examine some mathematical properties and models of inhomogeneous materials. By deriving models for elastic energy and heat flow we are able to establish equations that arise in the study of divergence form uniformly elliptic partial differential equations. In the late 1950's DeGiorgi and Nash showed that weak solutions to our partial differential equation lie in the Holder class. After fixing the dimension of the space, the Holder exponent guaranteed by this work depends only on the ratio of the eigenvalues. In this paper we will look at a specific geometry and show that the Holder exponent of the actual solutions is bounded away from zero independent of the eigenvalues.

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The problem of this study is that of developing a testing procedure for multiple-choice tests which would increase the relationship between test scores and a criterion. The procedure investigated in this research was one in which subjects took a multiple-choice test but were required to continue responding on each item until the correct answer was obtained. The total number of responses was used as the score on the test. The purpose of this research was to investigate the possibility of increasing predictability by changing the procedure of administering the test, rather than changing the test itself.

Modern Field-Programmable Gate Arrays (FPGAs) are no longer used to implement small “glue logic” circuitries. The high-density of reconfigurable logic resources in today’s FPGAs enable the implementation of large systems in a single chip. FPGAs are highly flexible devices; their functionality can be altered by simply loading a new binary file in their configuration memory. While the flexibility of FPGAs is comparable to General-Purpose Processors (GPPs), in the sense that different functions can be performed using the same hardware, the performance gain that can be achieved using FPGAs can be orders of magnitudes higher as FPGAs offer the ability for customisation of parallel computational architectures. Dynamic Partial Reconfiguration (DPR) allows for changing the functionality of certain blocks on the chip while the rest of the FPGA is operational. DPR has sparked the interest of researchers to explore new computational platforms where computational tasks are off-loaded from a main CPU to be executed using dedicated reconfigurable hardware accelerators configured on demand at run-time. By having a battery of custom accelerators which can be swapped in and out of the FPGA at runtime, a higher computational density can be achieved compared to static systems where the accelerators are bound to fixed locations within the chip. Furthermore, the ability of relocating these accelerators across several locations on the chip allows for the implementation of adaptive systems which can mitigate emerging faults in the FPGA chip when operating in harsh environments. By porting the appropriate fault mitigation techniques in such computational platforms, the advantages of FPGAs can be harnessed in different applications in space and military electronics where FPGAs are usually seen as unreliable devices due to their sensitivity to radiation and extreme environmental conditions. In light of the above, this thesis investigates the deployment of DPR as: 1) a method for enhancing performance by efficient exploitation of the FPGA resources, and 2) a method for enhancing the reliability of systems intended to operate in harsh environments. Achieving optimal performance in such systems requires an efficient internal configuration management system to manage the reconfiguration and execution of the reconfigurable modules in the FPGA. In addition, the system needs to support “fault-resilience” features by integrating parameterisable fault detection and recovery capabilities to meet the reliability standard of fault-tolerant applications. This thesis addresses all the design and implementation aspects of an Internal Configuration Manger (ICM) which supports a novel bitstream relocation model to enable the placement of relocatable accelerators across several locations on the FPGA chip. In addition to supporting all the configuration capabilities required to implement a Reconfigurable Operating System (ROS), the proposed ICM also supports the novel multiple-clone configuration technique which allows for cloning several instances of the same hardware accelerator at the same time resulting in much shorter configuration time compared to traditional configuration techniques. A faulttolerant (FT) version of the proposed ICM which supports a comprehensive faultrecovery scheme is also introduced in this thesis. The proposed FT-ICM is designed with a much smaller area footprint compared to Triple Modular Redundancy (TMR) hardening techniques while keeping a comparable level of fault-resilience. The capabilities of the proposed ICM system are demonstrated with two novel applications. The first application demonstrates a proof-of-concept reliable FPGA server solution used for executing encryption/decryption queries. The proposed server deploys bitstream relocation and modular redundancy to mitigate both permanent and transient faults in the device. It also deploys a novel Built-In Self- Test (BIST) diagnosis scheme, specifically designed to detect emerging permanent faults in the system at run-time. The second application is a data mining application where DPR is used to increase the computational density of a system used to implement the Frequent Itemset Mining (FIM) problem.

This body of work examines the plausibility of applying partial di erential equations and time-fractional partial di erential equations to images. The standard di usion equation is coupled with a nonlinear cubic source term of the Fitzhugh-Nagumo type to obtain a model with di usive properties and a binarizing e ect due to the source term. We examine the e ects of applying this model to a class of images known as document images; images that largely comprise text. The e ects of this model result in a binarization process that is competitive with the state-of-the-art techniques. Further to this application, we provide a stability analysis of the method as well as high-performance implementation on general purpose graphical processing units. The model is extended to include time derivatives to a fractional order which a ords us another degree of control over this process and the nature of the fractionality is discussed indicating the change in dynamics brought about by this generalization. We apply a semi-discrete method derived by hybridizing the Laplace transform and two discretization methods: nite-di erences and Chebyshev collocation. These hybrid techniques are coupled with a quasi-linearization process to allow for the application of the Laplace transform, a linear operator, to a nonlinear equation of fractional order in the temporal domain. A thorough analysis of these methods is provided giving rise to conditions for solvability. The merits and demerits of the methods are discussed indicating the appropriateness of each method.

A research report submitted to the Faculty of Science, University of the Witwatersrand, in ful lment of the requirements for the degree of Master of Science. School of Mathematics Johannesburg February 15, 2016 / We obtain a solution of the Black-Scholes equation with a non-smooth bound- ary condition using symmetry methods. The Black-Scholes equation along with its boundary condition are rst transformed into the one dimensional heat equation and an initial condition respectively. We then nd an appro- priate general symmetry generator of the heat equation using symmetries of the heat equation and the fundamental solution of the heat equation. The method we use to nd the symmetry generator is such that the boundary condition is left invariant and yet the symmetry can still be used to solve the heat equation. We then use the help of Mathematica to nd the solution to the heat equation. Then the solution is then transformed backwards to a solution of the Black-Scholes equation using the same change of variables that were used for the forward transformations. The solution is then nally checked if it satis es the boundary condition of the Black-Scholes equation.