Analysis of permutation distance-preserving mappings using graphs

- Swart, Theo G., Ferreira, Hendrik C.

**Authors:**Swart, Theo G. , Ferreira, Hendrik C.**Date:**2007**Subjects:**Mappings (Mathematics) , Permutations , Representations of graphs**Language:**English**Type:**Conference proceedings**Identifier:**http://hdl.handle.net/10210/15518 , uj:15670 , Citation: Swart, T.G. & Ferreira, H.C. 2007. Analysis of permutation distance-preserving mappings using graphs. In: Proceedings of the International Symposium on Communication Theory and its Applications, Ambleside, England, July 16-20, 2007**Description:**Abstract A new way of analyzing permutation distance preserving mappings is presented by making use of a graph representation. The properties necessary to make such graphs distance-preserving and how this relates to the total sum of distances that exist for such mappings, are investigated. This new knowledge is used to analyze previous constructions, as well as showing the existence or non-existence of simple algorithms for mappings attaining the upper bound on the sum of distances. Finally, two applications for such graphs are considered.**Full Text:**

**Authors:**Swart, Theo G. , Ferreira, Hendrik C.**Date:**2007**Subjects:**Mappings (Mathematics) , Permutations , Representations of graphs**Language:**English**Type:**Conference proceedings**Identifier:**http://hdl.handle.net/10210/15518 , uj:15670 , Citation: Swart, T.G. & Ferreira, H.C. 2007. Analysis of permutation distance-preserving mappings using graphs. In: Proceedings of the International Symposium on Communication Theory and its Applications, Ambleside, England, July 16-20, 2007**Description:**Abstract A new way of analyzing permutation distance preserving mappings is presented by making use of a graph representation. The properties necessary to make such graphs distance-preserving and how this relates to the total sum of distances that exist for such mappings, are investigated. This new knowledge is used to analyze previous constructions, as well as showing the existence or non-existence of simple algorithms for mappings attaining the upper bound on the sum of distances. Finally, two applications for such graphs are considered.**Full Text:**

Spectral shaping and distance mapping with permutation sequences

**Authors:**Ouahada, Khmaies Taher**Date:**2012-06-04**Subjects:**Mappings (Mathematics) , Error-correcting codes (Information theory) , Permutations**Type:**Thesis**Identifier:**uj:2344 , http://hdl.handle.net/10210/4800**Description:**D.Ing. , In this thesis we combined two techniques, namely a spectral shaping technique and a distance-preserving mapping technique to design new codes with both special spectrum shaping and error correction capabilities, in order to overcome certain communication problems like those that occur in a power-line communication channel. A new distance-preserving mapping construction based on graph theory is firstly presented. The k-cube graph construction from binary sequences to permutation sequences reached the upper bound on the sum of the Hamming distances for certain lengths of the permutation sequences and achieves the same sum of the Hamming distances as the best previously published constructions for most of the rest of the lengths. The k-cube graph construction is considered to be a simple and easy construction to understand the concept of mappings and especially the concept of a distance-reducing mapping.**Full Text:**

**Authors:**Ouahada, Khmaies Taher**Date:**2012-06-04**Subjects:**Mappings (Mathematics) , Error-correcting codes (Information theory) , Permutations**Type:**Thesis**Identifier:**uj:2344 , http://hdl.handle.net/10210/4800**Description:**D.Ing. , In this thesis we combined two techniques, namely a spectral shaping technique and a distance-preserving mapping technique to design new codes with both special spectrum shaping and error correction capabilities, in order to overcome certain communication problems like those that occur in a power-line communication channel. A new distance-preserving mapping construction based on graph theory is firstly presented. The k-cube graph construction from binary sequences to permutation sequences reached the upper bound on the sum of the Hamming distances for certain lengths of the permutation sequences and achieves the same sum of the Hamming distances as the best previously published constructions for most of the rest of the lengths. The k-cube graph construction is considered to be a simple and easy construction to understand the concept of mappings and especially the concept of a distance-reducing mapping.**Full Text:**

Kognitiewe kartering as strategie van wiskunde-onderrig aan leerders met 'n gesiggestremdheid

**Authors:**Van der Spuy, Janette**Date:**2012-09-05**Subjects:**Mappings (Mathematics) , Mathematics - Study and teaching (Secondary) - Audio-visual aids , Children with visual disabilities - Education - South Africa , Cognitive learning**Type:**Thesis**Identifier:**uj:3553 , http://hdl.handle.net/10210/6938**Description:**M.Ed. , This study is an investigation into cognitive mapping as strategy in the constructivistic approach to mathematics education to learners with a visual disability with the view to describe the change in pupils' thoughts on mathematical concepts, as well as their experiences during the process of cognitive mapping. The rationale for the investigation was derived from the shift in South African Mathematics teaching from traditional teaching to constructivistic (or problem-centered) teaching. As this implies a shift in paradigm, teachers will be in need of relevant constructivistic training to equip them with effective teaching strategies. The objective of this study is to examine cognitive mapping as a possible supportive strategy to constructivism . The study commences with a theoretical framework in which constructivism is clearly explicated. The principles of radical and social constructivism, the roots of which can be traced back to the epistemological theories of Piaget and Vygotsky, are explored. The constructivistic view of knowledge, with the relationship between public knowledge and the forming of personal knowledge, is discussed and extended to include the formation of mathematical knowledge. The focus then shifts to the concepts of instruction and learning and the role they play in the constructivistic paradigm. In the constructivistic view, learning implies cognitive restructuring, which is facilitated by assimilation and accommodation. The implications of this view of learning for instruction, and in particular mathematics instruction, is then discussed. This chapter concludes with the working definition the researcher has used to conduct the remainder of the study. The theoretical framework is structured furthermore to give background regarding cognitive mapping. According to the constructivistic approach, learning implies conceptual change. Cognitive maps externalise conceptual change by means of visual representations, and therefore it was decided to investigate them as a teaching strategy. Some definitions, as found in the literature, are given, and three types of maps are illustrated as examples. The different uses of cognitive maps, among which study strategy, lesson planning and means of evaluation, are discussed. A discussion on the different methods of constructing a map follows, with specific focus on how to include the whole class in the activity. The advantage of social interaction while constructing knowledge, is highlighted. Lastly, the advantages and disadvantages connected to cognitive mapping as teaching strategy, are discussed. The theoretical framework is complemented by a chapter on the design of the research, substantiating the choice of format and methods of data collection and analysis. The data is reported in the succeeding chapter, and examples of raw data from transcriptions, journals of the pupils and cognitive maps are presented. Finally, the consolidated data is interpreted. In the concluding chapter the findings of the study are discussed. The most significant findings of this study are: cognitive mapping, as mathematical teaching strategy, improved the understanding of grade nine learners, with a visual disability, of real numbers; the learners experienced the teaching strategy of cognitive mapping positively; the number of group members involved in the construction of a cognitive map, influenced.**Full Text:**

**Authors:**Van der Spuy, Janette**Date:**2012-09-05**Subjects:**Mappings (Mathematics) , Mathematics - Study and teaching (Secondary) - Audio-visual aids , Children with visual disabilities - Education - South Africa , Cognitive learning**Type:**Thesis**Identifier:**uj:3553 , http://hdl.handle.net/10210/6938**Description:**M.Ed. , This study is an investigation into cognitive mapping as strategy in the constructivistic approach to mathematics education to learners with a visual disability with the view to describe the change in pupils' thoughts on mathematical concepts, as well as their experiences during the process of cognitive mapping. The rationale for the investigation was derived from the shift in South African Mathematics teaching from traditional teaching to constructivistic (or problem-centered) teaching. As this implies a shift in paradigm, teachers will be in need of relevant constructivistic training to equip them with effective teaching strategies. The objective of this study is to examine cognitive mapping as a possible supportive strategy to constructivism . The study commences with a theoretical framework in which constructivism is clearly explicated. The principles of radical and social constructivism, the roots of which can be traced back to the epistemological theories of Piaget and Vygotsky, are explored. The constructivistic view of knowledge, with the relationship between public knowledge and the forming of personal knowledge, is discussed and extended to include the formation of mathematical knowledge. The focus then shifts to the concepts of instruction and learning and the role they play in the constructivistic paradigm. In the constructivistic view, learning implies cognitive restructuring, which is facilitated by assimilation and accommodation. The implications of this view of learning for instruction, and in particular mathematics instruction, is then discussed. This chapter concludes with the working definition the researcher has used to conduct the remainder of the study. The theoretical framework is structured furthermore to give background regarding cognitive mapping. According to the constructivistic approach, learning implies conceptual change. Cognitive maps externalise conceptual change by means of visual representations, and therefore it was decided to investigate them as a teaching strategy. Some definitions, as found in the literature, are given, and three types of maps are illustrated as examples. The different uses of cognitive maps, among which study strategy, lesson planning and means of evaluation, are discussed. A discussion on the different methods of constructing a map follows, with specific focus on how to include the whole class in the activity. The advantage of social interaction while constructing knowledge, is highlighted. Lastly, the advantages and disadvantages connected to cognitive mapping as teaching strategy, are discussed. The theoretical framework is complemented by a chapter on the design of the research, substantiating the choice of format and methods of data collection and analysis. The data is reported in the succeeding chapter, and examples of raw data from transcriptions, journals of the pupils and cognitive maps are presented. Finally, the consolidated data is interpreted. In the concluding chapter the findings of the study are discussed. The most significant findings of this study are: cognitive mapping, as mathematical teaching strategy, improved the understanding of grade nine learners, with a visual disability, of real numbers; the learners experienced the teaching strategy of cognitive mapping positively; the number of group members involved in the construction of a cognitive map, influenced.**Full Text:**

The control of chaotic maps

**Authors:**Hoffman, Lance Douglas**Date:**2012-09-04**Subjects:**Control theory , Chaotic behavior in systems , Mappings (Mathematics)**Type:**Thesis**Identifier:**uj:3451 , http://hdl.handle.net/10210/6845**Description:**2003 , Some important ideas froni classical control theory are introduced with the intention of applying them to chaotic dynamical systems, in particular the coupled logistic equations. The structure of this dissertation is such that a strong foundation in control theory is first established before introducing the coupled logistic map or the methods of control and targetting in chaotic systems. In chapter 1 some aspects of classical control theory are reviewed. Continuous- and discrete-time dynamical systems are introduced and the existence and uniquendss criteria for the continuous case are explored via Lipschitz continuity. The matrix form of an inhomogeneous linear differential equation is presented and several properties of the associated transition matrix are discussed. Several linear algebraic ideas, most notably the Cayley-Hamilton theorem, are employed to explore the important concepts of controllability and observability in linear systems. The stabilisability problem is thoroughly investigated. Finally, the neighbourhood properties of continuous nonlinear dynamical systems with reference to controllability, stability and noise are established. Chapter 2 places emphasis on canonical forms, pole assignments and state observers. The decomposition of a general system into distinct components is facilitated by the general structure theorem, which is proved. The pole placement problem is described and the correspondence between the stabilisability of a system and the placement of poles is noted by the use'of a socalled feedback matrix. Lastly, the notion of a state observer, with reference to some dynamic feedback law, is introduced. The dynamics of the coupled logistic equations are studied in chapter 3. The fixed points of the map are calculated and the subsequent dynamical consequences explored. Using methods introduced in earlier chapters, the stability of the map is investigated. Using the so-called variational equations, the Lyapunov exponents are computed and used to classify, the motion of the system for the parameter values r and a. This chapter concludes with a discussion of the basins of attraction and critical curves associated with the coupled logistic equations. It is in chapter 4 that the models for controlling chaos are instantiated. The famous Ott-Grebogi- Yorke (OGY) method for controlling chaos is explained and related to the pole placement problem, discussed previously. The theory is extended to study the control of periodic orbits with periods greater than one.**Full Text:**

**Authors:**Hoffman, Lance Douglas**Date:**2012-09-04**Subjects:**Control theory , Chaotic behavior in systems , Mappings (Mathematics)**Type:**Thesis**Identifier:**uj:3451 , http://hdl.handle.net/10210/6845**Description:**2003 , Some important ideas froni classical control theory are introduced with the intention of applying them to chaotic dynamical systems, in particular the coupled logistic equations. The structure of this dissertation is such that a strong foundation in control theory is first established before introducing the coupled logistic map or the methods of control and targetting in chaotic systems. In chapter 1 some aspects of classical control theory are reviewed. Continuous- and discrete-time dynamical systems are introduced and the existence and uniquendss criteria for the continuous case are explored via Lipschitz continuity. The matrix form of an inhomogeneous linear differential equation is presented and several properties of the associated transition matrix are discussed. Several linear algebraic ideas, most notably the Cayley-Hamilton theorem, are employed to explore the important concepts of controllability and observability in linear systems. The stabilisability problem is thoroughly investigated. Finally, the neighbourhood properties of continuous nonlinear dynamical systems with reference to controllability, stability and noise are established. Chapter 2 places emphasis on canonical forms, pole assignments and state observers. The decomposition of a general system into distinct components is facilitated by the general structure theorem, which is proved. The pole placement problem is described and the correspondence between the stabilisability of a system and the placement of poles is noted by the use'of a socalled feedback matrix. Lastly, the notion of a state observer, with reference to some dynamic feedback law, is introduced. The dynamics of the coupled logistic equations are studied in chapter 3. The fixed points of the map are calculated and the subsequent dynamical consequences explored. Using methods introduced in earlier chapters, the stability of the map is investigated. Using the so-called variational equations, the Lyapunov exponents are computed and used to classify, the motion of the system for the parameter values r and a. This chapter concludes with a discussion of the basins of attraction and critical curves associated with the coupled logistic equations. It is in chapter 4 that the models for controlling chaos are instantiated. The famous Ott-Grebogi- Yorke (OGY) method for controlling chaos is explained and related to the pole placement problem, discussed previously. The theory is extended to study the control of periodic orbits with periods greater than one.**Full Text:**

Using graphs for the analysis and construction of permutation distance-preserving mappings

- Swart, Theo G., Ferreira, Hendrik C., Ouahada, Khmaies

**Authors:**Swart, Theo G. , Ferreira, Hendrik C. , Ouahada, Khmaies**Date:**2008**Subjects:**Code construction , Distance-preserving mappings , Representation of graphs , Permutations , Mappings (Mathematics)**Language:**English**Type:**Article**Identifier:**http://hdl.handle.net/10210/20030 , uj:16060 , ISSN: 0018-9448 , Citation: Swart, T.G., Ferreira H.C. & Ouahada, K. 2008. Using graphs for the analysis and construction of permutation distance-preserving mappings. IEEE Transactions on Information Theory, 54(2):910-916.**Description:**Abstract: A new way of looking at permutation distance-preserving mappings (DPMs) is presented by making use of a graph representation. The properties necessary to make such a graph distance-preserving, are also investigated. Further, this new knowledge is used to analyze previous constructions, as well as to construct a new general mapping algorithm for a previous multilevel construction.**Full Text:**

**Authors:**Swart, Theo G. , Ferreira, Hendrik C. , Ouahada, Khmaies**Date:**2008**Subjects:**Code construction , Distance-preserving mappings , Representation of graphs , Permutations , Mappings (Mathematics)**Language:**English**Type:**Article**Identifier:**http://hdl.handle.net/10210/20030 , uj:16060 , ISSN: 0018-9448 , Citation: Swart, T.G., Ferreira H.C. & Ouahada, K. 2008. Using graphs for the analysis and construction of permutation distance-preserving mappings. IEEE Transactions on Information Theory, 54(2):910-916.**Description:**Abstract: A new way of looking at permutation distance-preserving mappings (DPMs) is presented by making use of a graph representation. The properties necessary to make such a graph distance-preserving, are also investigated. Further, this new knowledge is used to analyze previous constructions, as well as to construct a new general mapping algorithm for a previous multilevel construction.**Full Text:**

Laser based mapping of an unknown environment

- Corregedor, Antonio Rodrigues

**Authors:**Corregedor, Antonio Rodrigues**Date:**2014-03-17**Subjects:**Laser based mapping , Algorithms , Simulation methods & models , Mappings (Mathematics) , Environmental mapping , Machine learning**Type:**Thesis**Identifier:**uj:4341 , http://hdl.handle.net/10210/9690**Description:**M.Ing. (Electrical and Electronic Engineering) , This dissertation deals with the mapping of an unknown environment. Mapping of an environment can be accomplished by asking the question “What is in my world?” whilst moving through the environment. Once the objects occupying the ‘world’ have been discovered, the locations of these objects are stored somewhere (for example on paper), so that the environment can be navigated at a later stage. In the context of robots, a map provides the robot with a certain degree of “intelligence”. Several different types of applications are available for robots with “intelligence”; ranging from mining applications, to search and rescue situations, to surveillance applications and recognisance applications. The research hypothesis posed by this dissertation is as follows: Produce a human readable map for an unknown defined structured environment using a single laser range finder (LRF). The focus was on mapping environments resembling mine tunnels. In mine tunnel environments sensors, such as wheel odometers, can fail. This failure makes it advantageous to be able to create a map of the environment with the data obtained solely from the LRF. For this dissertation, the following restrictions were placed on the environment being mapped. It had to be structured (i.e. the environment could be described by simple geometric primitives such as lines); it had to be static (the only entity allowed to move in the environment was the LRF to obtain data); and the environment had to be defined (i.e. have a starting and ending point). During the course of this Masters research, it was discovered that in order to create a human readable map, one has to determine the accurate localisation of the sensor in the environment whilst mapping. The described scenario is a typical problem in mapping and is referred to as the ‘simultaneous localisation and mapping (SLAM) problem’. This dissertation shows results when mapping was done with – and without – accurate localisation. The final approach used to create the human readable map consisted of determining scan matched odometry (based on a feature matching and ICP algorithm). The scan matched odometry is incorporated into a grid-based SLAM technique that utilises a particle filter to accurately determine the position of the sensor in the environment, in order to create a human readable map of the environment. The algorithm used (as described) was able to close loops (i.e. the mapping algorithm was able to handle the sensor returning to its starting point) and it produced satisfactory results for the types of environments as required by the scope of this dissertation.**Full Text:**

**Authors:**Corregedor, Antonio Rodrigues**Date:**2014-03-17**Subjects:**Laser based mapping , Algorithms , Simulation methods & models , Mappings (Mathematics) , Environmental mapping , Machine learning**Type:**Thesis**Identifier:**uj:4341 , http://hdl.handle.net/10210/9690**Description:**M.Ing. (Electrical and Electronic Engineering) , This dissertation deals with the mapping of an unknown environment. Mapping of an environment can be accomplished by asking the question “What is in my world?” whilst moving through the environment. Once the objects occupying the ‘world’ have been discovered, the locations of these objects are stored somewhere (for example on paper), so that the environment can be navigated at a later stage. In the context of robots, a map provides the robot with a certain degree of “intelligence”. Several different types of applications are available for robots with “intelligence”; ranging from mining applications, to search and rescue situations, to surveillance applications and recognisance applications. The research hypothesis posed by this dissertation is as follows: Produce a human readable map for an unknown defined structured environment using a single laser range finder (LRF). The focus was on mapping environments resembling mine tunnels. In mine tunnel environments sensors, such as wheel odometers, can fail. This failure makes it advantageous to be able to create a map of the environment with the data obtained solely from the LRF. For this dissertation, the following restrictions were placed on the environment being mapped. It had to be structured (i.e. the environment could be described by simple geometric primitives such as lines); it had to be static (the only entity allowed to move in the environment was the LRF to obtain data); and the environment had to be defined (i.e. have a starting and ending point). During the course of this Masters research, it was discovered that in order to create a human readable map, one has to determine the accurate localisation of the sensor in the environment whilst mapping. The described scenario is a typical problem in mapping and is referred to as the ‘simultaneous localisation and mapping (SLAM) problem’. This dissertation shows results when mapping was done with – and without – accurate localisation. The final approach used to create the human readable map consisted of determining scan matched odometry (based on a feature matching and ICP algorithm). The scan matched odometry is incorporated into a grid-based SLAM technique that utilises a particle filter to accurately determine the position of the sensor in the environment, in order to create a human readable map of the environment. The algorithm used (as described) was able to close loops (i.e. the mapping algorithm was able to handle the sensor returning to its starting point) and it produced satisfactory results for the types of environments as required by the scope of this dissertation.**Full Text:**

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