This thesis demonstrates an alternative, concise but informative, method for representing group elements, which will prove particularly useful for the sporadic groups. It explains the theory behind symmetric presentations, and describes the algorithm for working with elements represented in this manner.

This thesis consists of two parts. In the first chapter, we give some results on modulation spaces. First the relationship between the classical spaces and the modulation spaces is established. It is proved that certain modulation spaces defined on R² lie in the BMO space. Another result is that the Zak transform, a discrete time-frequency transform, maps a modulation space into a higher dimensional modulation space. And by using these results, an uncertainty principle for Gabor frames via modulation spaces is obtained. In the second part, we deal with optimization of an increasing positively homogeneous functions on the unit simplex. The class of increasing positively homogeneous functions is one of the function classes obtained via min-type functions in the context of abstract convexity. The cutting angle method is used for the minimization of this type functions. The most important step of this method is the minimization of a function which is the maximum of a number of min-type functions on the unit simplex. We propose a numerical algorithm for the minimization of such functions on the unit simplex and we mathematically prove that this algorithm finds the exact solution of the minimization problem. Some experiments have been carried out and the results of the experiments have been presented.

In simulation-based design, a key challenge is to formulate and solve analysis problems efficiently to evaluate a large variety of design alternatives. The solution of analysis problems has benefited from advancements in commercial off-the-shelf math solvers and computational capabilities. However, the formulation of analysis problems is often a costly and laborious process. Traditional simulation templates used for representing analysis problems are typically brittle with respect to variations in artifact topology and the idealization decisions taken by analysts. These templates often require manual updates and "re-wiring" of the analysis knowledge embodied in them. This makes the use of traditional simulation templates ineffective for multi-disciplinary design and optimization problems. Based on these issues, this dissertation defines a special class of problems known as variable topology multi-body (VTMB) problems that characterizes the types of variations seen in design-analysis interoperability. This research thus primarily answers the following question: How can we improve the effectiveness of the analysis problem formulation process for VTMB problems? The knowledge composition methodology (KCM) presented in this dissertation answers this question by addressing the following research gaps: (1) the lack of formalization of the knowledge used by analysts in formulating simulation templates, and (2) the inability to leverage this knowledge to define model composition methods for formulating simulation templates. KCM overcomes these gaps by providing: (1) formal representation of analysis knowledge as modular, reusable, analyst-intelligible building blocks, (2) graph transformation-based methods to automatically compose simulation templates from these building blocks based on analyst idealization decisions, and (3) meta-models for representing advanced simulation templates VTMB design models, analysis models, and the idealization relationships between them. Applications of the KCM to thermo-mechanical analysis of multi-stratum printed wiring boards and multi-component chip packages demonstrate its effectiveness handling VTMB and idealization variations with significantly enhanced formulation efficiency (from several hours in existing methods to few minutes). In addition to enhancing the effectiveness of analysis problem formulation, KCM is envisioned to provide a foundational approach to model formulation for generalized variable topology problems.

The study explores Mathematics educators’ use of students’ out-of-school experiences in the teaching of Transformation Geometry. This thesis focuses on an analysis of the extent to which students’ out-of-school experiences are reflected in the actual teaching, textbook tasks and national examination items set and other resources used. Teachers’ teaching practices are expected to support students’ learning of concepts in mathematics. Freudenthal (1991) argues that students develop their mathematical understanding by working from contexts that make sense to them, contexts that are grounded in realistic settings. ZIMSEC Examiners Reports (2010; 2011) reveal a low student performance in the topic of Transformation Geometry in Zimbabwe, yet, the topic has a close relationship with the environment in which students live (Purpura, Baroody & Lonigan, 2013). Thus, the main purpose of the study is to explore Mathematics teachers’ use of students’ out-of-school experiences in the teaching of Transformation Geometry at secondary school level. The investigation encompassed; (a) teacher perceptions about transformation geometry concepts that have a close link with students’ out-of-school experiences, (b) how teachers are teaching transformation geometry in Zimbabwe’s rural secondary schools, (c) the extent to which students’ out-of-school experiences are incorporated in Transformation Geometry tasks, and (d) the extent to which transformation geometry, as reflected in the official textbooks and suggested teaching models, is linked to students’ out-of-school experiences. Consistent with the interpretive qualitative research paradigm the transcendental phenomenology was used as the research design. Semi-structured interviews, Lesson observations, document analysis and a test were used as data gathering instruments. Data analysis, mainly for qualitative data, involved coding and categorising emerging themes from the different data sources. The key epistemological assumption was derived from the notion that knowing reality is through understanding the experiences of others found in a phenomenon of interest (Yuksel & Yildirim, 2015). In this study, the phenomenon of interest was the teaching of Transformation Geometry in rural secondary schools. In the same light, it meant observing teachers teaching the topic of Transformation Geometry, listening to their perceptions about the topic during interviews, and considering how they plan for their teaching as well as how students are assessed in transformation geometry. The research site included 3 selected rural secondary schools; one Mission boarding high school, a Council run secondary school and a Government rural day secondary school. Purposive sampling technique was used carefully to come up with 3 different types of schools in a typical rural Zimbabwe. Purposive sampling technique was also used to choose the teacher participants, whereas learners who sat for the test were randomly selected from the ordinary level classes. The main criterion for including teacher participants was if they were currently teaching an Ordinary Level Mathematics class and had gained more experience in teaching Transformation Geometry. In total, six teachers and forty-five students were selected to participate in the study. Results from the study reveal that some teachers have limited knowledge on transformation geometry concepts embedded in students’ out-of-school experience. Using Freudenthal’s (1968) RME Model to judge their effectiveness in teaching, the implication is teaching and learning would fail to utilise contexts familiar with the students and hence can hardly promote mastery of transformation geometry concepts. Data results also reveal some disconnect between teaching practices as espoused in curriculum documents and actual teaching practice. Although policy stipulates that concepts must be developed starting from concrete situations and moving to the abstract concepts, teachers seem to prefer starting with the formal Mathematics, giving students definitions and procedures for carrying out the different geometric transformations. On the other hand, tasks in Transformation Geometry both at school level and the national examinations focus on testing learner’s ability to define and use procedures for performing specific transformations at the expense of testing for real understanding of concepts. In view of these findings the study recommends the revision of the school Mathematics curriculum emphasising pre-service programmes for teacher professional knowledge to be built on features of contemporary learning theory, such as RME theory. Such as a revision can include the need to plan instruction so that students build models and representations rather than apply already developed ones. / Curriculum and Instructional Studies / D. Ed. (Curriculum Studies)

Indiana University-Purdue University Indianapolis (IUPUI) / HEVC has achieved significant coding efficiency improvement beyond existing video coding standard by employing many new coding tools. Deblocking Filter, Sample Adaptive Offset and Adaptive Loop Filter for in-loop filtering are currently introduced for the HEVC standardization. However these filters are implemented in spatial domain despite the fact of temporal correlation within video sequences. To reduce the artifacts and better align object boundaries in video , a new algorithm in in-loop filtering is proposed. The proposed algorithm is implemented in HM-11.0 software. This proposed algorithm allows an average bitrate reduction of about 0.7% and improves the PSNR of the decoded frame by 0.05%, 0.30% and 0.35% in luminance and chroma.