Theses and Dissertations (Mathematical Sciences)http://hdl.handle.net/10500/30172018-04-20T19:42:03Z2018-04-20T19:42:03ZTopics on z-ideals of commutative ringsTlharesakgosi, Batsilehttp://hdl.handle.net/10500/236192018-03-07T10:02:13Z2017-08-01T00:00:00ZTopics on z-ideals of commutative rings
Tlharesakgosi, Batsile
The first few chapters of the dissertation will catalogue what is known regarding z-ideals in
commutative rings with identity. Some special attention will be paid to z-ideals in function
rings to show how the presence of the topological description simplifies z-covers of arbitrary
ideals. Conditions in an f-ring that ensure that the sum of z-ideals is a z-ideal will be given.
In the latter part of the dissertation I will generalise a result in higher order z-ideals and
introduce a notion of higher order d-ideals
2017-08-01T00:00:00ZGene expression programming for logic circuit designMasimula, Steven Mandlahttp://hdl.handle.net/10500/236172018-03-07T09:56:10Z2017-10-01T00:00:00ZGene expression programming for logic circuit design
Masimula, Steven Mandla
Finding an optimal solution for the logic circuit design problem is challenging and time-consuming especially
for complex logic circuits. As the number of logic gates increases the task of designing optimal logic circuits
extends beyond human capability. A number of evolutionary algorithms have been invented to tackle a range
of optimisation problems, including logic circuit design. This dissertation explores two of these evolutionary
algorithms i.e. Gene Expression Programming (GEP) and Multi Expression Programming (MEP) with the
aim of integrating their strengths into a new Genetic Programming (GP) algorithm. GEP was invented by
Candida Ferreira in 1999 and published in 2001 [8]. The GEP algorithm inherits the advantages of the Genetic
Algorithm (GA) and GP, and it uses a simple encoding method to solve complex problems [6, 32]. While
GEP emerged as powerful due to its simplicity in implementation and
exibility in genetic operations, it is
not without weaknesses. Some of these inherent weaknesses are discussed in [1, 6, 21]. Like GEP, MEP is a
GP-variant that uses linear chromosomes of xed length [23]. A unique feature of MEP is its ability to store
multiple solutions of a problem in a single chromosome. MEP also has an ability to implement code-reuse which
is achieved through its representation which allow multiple references to a single sub-structure.
This dissertation proposes a new GP algorithm, Improved Gene Expression Programming (IGEP) which im-
proves the performance of the traditional GEP by combining the code-reuse capability and simplicity of gene encoding method from MEP and GEP, respectively. The results obtained using the IGEP and the traditional
GEP show that the two algorithms are comparable in terms of the success rate when applied on simple problems
such as basic logic functions. However, for complex problems such as one-bit Full Adder (FA) and AND-OR
Arithmetic Logic Unit (ALU) the IGEP performs better than the traditional GEP due to the code-reuse in IGEP
2017-10-01T00:00:00ZMathematical symbolisation: challenges and instructional strategies for Limpopo Province secondary school learnersMutodi, Paulhttp://hdl.handle.net/10500/231542017-09-13T05:47:52Z2016-09-01T00:00:00ZMathematical symbolisation: challenges and instructional strategies for Limpopo Province secondary school learners
Mutodi, Paul
This study reports on an investigation into the manner in which mathematical symbols influence learners’ understanding of mathematical concepts. The study was conducted in Greater Sekhukhune and Capricorn districts of Limpopo Province, South Africa. Multistage sampling (for the district), simple random sampling (for the schools), purposive sampling (for the teachers) and stratified random sampling with proportional allocation (for the learners) were used. The study was conducted in six schools randomly selected from rural, semi-urban and urban settings. A sample of 565 FET learners and 15 FET band mathematics teachers participated in the study. This study is guided by four interrelated constructivist theories: symbol sense, algebraic insight, APOS and procept theories. The research instruments for the study consist of questionnaires and interviews. A mixed method approach that was predominantly qualitative was employed. An analysis of learners’ difficulties with mathematical symbols produced three (3) clusters. The main cluster consists of 236 (41.6%) learners who indicate that they experience severe challenges with mathematical symbols compared to 108 (19.1%) learners who indicated that they could confidently handle and manipulate mathematical symbols with understanding. Six (6) categories of challenges with mathematical symbols emerged from learners’ encounters with mathematical symbols: reading mathematical text and symbols, prior knowledge, time allocated for mathematical classes and activities, lack of symbol sense and problem contexts and pedagogical approaches to mathematical symbolisation. Two sets of theme classes related to learners’ difficulties with mathematical symbols and instructional strategies emerged. Learners lack symbol sense for mathematical concepts and algebraic insight for problem solving. Learners stick to procedurally driven symbols at the expense of conceptual and contextual understanding. From a pedagogical perspective teachers indicated that they face the following difficulties when teaching: the challenge of introducing unfamiliar notation in a new topic; reading, writing and verbalising symbols; signifier and signified connections; and teaching both symbolisation and conceptual understanding simultaneously. The study recommends teachers to use strategies such as informed choice of subject matter and a pedagogical approach in which concepts are understood before they are symbolised.
2016-09-01T00:00:00ZVariants of P-frames and associated ringsNsayi, Jissy Nsondehttp://hdl.handle.net/10500/217952017-12-07T10:44:26Z2015-12-01T00:00:00ZVariants of P-frames and associated rings
Nsayi, Jissy Nsonde
We study variants of P-frames and associated rings, which can be viewed as natural
generalizations of the classical variants of P-spaces and associated rings. To be more
precise, we de ne quasi m-rings to be those rings in which every prime d-ideal is either
maximal or minimal. For a completely regular frame L, if the ring RL of real-valued
continuous functions of L is a quasi m-ring, we say L is a quasi cozero complemented
frame. These frames are less restricted than the cozero complemented frames. Using
these frames we study some properties of what are called quasi m-spaces, and observe
that the property of being a quasi m-space is inherited by cozero subspaces, dense z-
embedded subspaces, and regular-closed subspaces among normal quasi m-space.
M. Henriksen, J. Mart nez and R. G. Woods have de ned a Tychono space X to be a
quasi P-space in case every prime z-ideal of C(X) is either minimal or maximal. We call a
point I of L a quasi P-point if every prime z-ideal of RL contained in the maximal ideal
associated with I is either maximal or minimal. If all points of L are quasi P-points, we
say L is a quasi P-frame. This is a conservative de nition in the sense that X is a quasi
P-space if and only if the frame OX is a quasi P-frame. We characterize these frames
in terms of cozero elements, and, among cozero complemented frames, give a su cient
condition for a frame to be a quasi P-frame.
A Tychono space X is called a weak almost P-space if for every two zero-sets E and
F of X with IntE IntF, there is a nowhere dense zero-set H of X such that E F [H.
We present the pointfree version of weakly almost P-spaces. We de ne weakly regular
rings by a condition characterizing the rings C(X) for weak almost P-spaces X. We
show that a reduced f-ring is weakly regular if and only if every prime z-ideal in it which contains only zero-divisors is a d-ideal. We characterize the frames L for which the ring
RL of real-valued continuous functions on L is weakly regular.
We introduce the notions of boundary frames and boundary rings, and use them to
give another ring-theoretic characterization of boundary spaces. We show that X is a
boundary space if and only if C(X) is a boundary ring.
A Tychono space whose Stone- Cech compacti cation is a nite union of closed subspaces
each of which is an F-space is said to be nitely an F-space. Among normal spaces,
S. Larson gave a characterization of these spaces in terms of properties of function rings
C(X). By extending this notion to frames, we show that the normality restriction can
actually be dropped, even in spaces, and thus we sharpen Larson's result.
2015-12-01T00:00:00Z