The classification of some fuzzy subgroups of finite groups under a natural equivalence and its extension, with particular emphasis on the number of equivalence classes
- Authors: Ndiweni, Odilo
- Date: 2007
- Subjects: Fuzzy sets , Maximal functions , Finite groups , Equivalence classes (Set theory)
- Language: English
- Type: Thesis , Masters , M Sc (Mathematics)
- Identifier: vital:11587 , http://hdl.handle.net/10353/88 , Fuzzy sets , Maximal functions , Finite groups , Equivalence classes (Set theory)
- Description: In this thesis we use the natural equivalence of fuzzy subgroups studied by Murali and Makamba [25] to characterize fuzzy subgroups of some finite groups. We focus on the determination of the number of equivalence classes of fuzzy subgroups of some selected finite groups using this equivalence relation and its extension. Firstly we give a brief discussion on the theory of fuzzy sets and fuzzy subgroups. We prove a few properties of fuzzy sets and fuzzy subgroups. We then introduce the selected groups namely the symmetric group 3 S , dihedral group 4 D , the quaternion group Q8 , cyclic p-group pn G = Z/ , pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . We also present their subgroups structures and construct lattice diagrams of subgroups in order to study their maximal chains. We compute the number of maximal chains and give a brief explanation on how the maximal chains are used in the determination of the number of equivalence classes of fuzzy subgroups. In determining the number of equivalence classes of fuzzy subgroups of a group, we first list down all the maximal chains of the group. Secondly we pick any maximal chain and compute the number of distinct fuzzy subgroups represented by that maximal chain, expressing each fuzzy subgroup in the form of a keychain. Thereafter we pick the next maximal chain and count the number of equivalence classes of fuzzy subgroups not counted in the first chain. We proceed inductively until all the maximal chains have been exhausted. The total number of fuzzy subgroups obtained in all the maximal chains represents the number of equivalence classes of fuzzy subgroups for the entire group, (see sections 3.2.1, 3.2.2, 3.2.6, 3.2.8, 3.2.9, 3.2.15, 3.16 and 3.17 for the case of selected finite groups). We study, establish and prove the formulae for the number of maximal chains for the groups pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . To accomplish this, we use lattice diagrams of subgroups of these groups to identify the maximal chains. For instance, the group pn qm G = Z/ + Z/ would require the use of a 2- dimensional rectangular diagram (see section 3.2.18 and 5.3.5), while for the group pn qm r s G = Z/ + Z/ + Z/ we execute 3- dimensional lattice diagrams of subgroups (see section 5.4.2, 5.4.3, 5.4.4, 5.4.5 and 5.4.6). It is through these lattice diagrams that we identify routes through which to carry out the extensions. Since fuzzy subgroups represented by maximal chains are viewed as keychains, we give a brief discussion on the notion of keychains, pins and their extensions. We present propositions and proofs on why this counting technique is justifiable. We derive and prove formulae for the number of equivalence classes of the groups pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . We give a detailed explanation and illustrations on how this keychain extension principle works in Chapter Five. We conclude by giving specific illustrations on how we compute the number of equivalence classes of a fuzzy subgroup for the group p2 q2 r 2 G = Z/ + Z/ + Z/ from the number of fuzzy subgroups of the group p q r G = Z/ + Z/ + Z/ 1 2 2 . This illustrates a general technique of computing the number of fuzzy subgroups of G = Z/ + Z/ + Z/ from the number of fuzzy subgroups of 1 -1 = / + / + / pn qm r s G Z Z Z . Our illustration also shows two ways of extending from a lattice diagram of 1 G to that of G .
- Full Text:
- Date Issued: 2007
- Authors: Ndiweni, Odilo
- Date: 2007
- Subjects: Fuzzy sets , Maximal functions , Finite groups , Equivalence classes (Set theory)
- Language: English
- Type: Thesis , Masters , M Sc (Mathematics)
- Identifier: vital:11587 , http://hdl.handle.net/10353/88 , Fuzzy sets , Maximal functions , Finite groups , Equivalence classes (Set theory)
- Description: In this thesis we use the natural equivalence of fuzzy subgroups studied by Murali and Makamba [25] to characterize fuzzy subgroups of some finite groups. We focus on the determination of the number of equivalence classes of fuzzy subgroups of some selected finite groups using this equivalence relation and its extension. Firstly we give a brief discussion on the theory of fuzzy sets and fuzzy subgroups. We prove a few properties of fuzzy sets and fuzzy subgroups. We then introduce the selected groups namely the symmetric group 3 S , dihedral group 4 D , the quaternion group Q8 , cyclic p-group pn G = Z/ , pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . We also present their subgroups structures and construct lattice diagrams of subgroups in order to study their maximal chains. We compute the number of maximal chains and give a brief explanation on how the maximal chains are used in the determination of the number of equivalence classes of fuzzy subgroups. In determining the number of equivalence classes of fuzzy subgroups of a group, we first list down all the maximal chains of the group. Secondly we pick any maximal chain and compute the number of distinct fuzzy subgroups represented by that maximal chain, expressing each fuzzy subgroup in the form of a keychain. Thereafter we pick the next maximal chain and count the number of equivalence classes of fuzzy subgroups not counted in the first chain. We proceed inductively until all the maximal chains have been exhausted. The total number of fuzzy subgroups obtained in all the maximal chains represents the number of equivalence classes of fuzzy subgroups for the entire group, (see sections 3.2.1, 3.2.2, 3.2.6, 3.2.8, 3.2.9, 3.2.15, 3.16 and 3.17 for the case of selected finite groups). We study, establish and prove the formulae for the number of maximal chains for the groups pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . To accomplish this, we use lattice diagrams of subgroups of these groups to identify the maximal chains. For instance, the group pn qm G = Z/ + Z/ would require the use of a 2- dimensional rectangular diagram (see section 3.2.18 and 5.3.5), while for the group pn qm r s G = Z/ + Z/ + Z/ we execute 3- dimensional lattice diagrams of subgroups (see section 5.4.2, 5.4.3, 5.4.4, 5.4.5 and 5.4.6). It is through these lattice diagrams that we identify routes through which to carry out the extensions. Since fuzzy subgroups represented by maximal chains are viewed as keychains, we give a brief discussion on the notion of keychains, pins and their extensions. We present propositions and proofs on why this counting technique is justifiable. We derive and prove formulae for the number of equivalence classes of the groups pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . We give a detailed explanation and illustrations on how this keychain extension principle works in Chapter Five. We conclude by giving specific illustrations on how we compute the number of equivalence classes of a fuzzy subgroup for the group p2 q2 r 2 G = Z/ + Z/ + Z/ from the number of fuzzy subgroups of the group p q r G = Z/ + Z/ + Z/ 1 2 2 . This illustrates a general technique of computing the number of fuzzy subgroups of G = Z/ + Z/ + Z/ from the number of fuzzy subgroups of 1 -1 = / + / + / pn qm r s G Z Z Z . Our illustration also shows two ways of extending from a lattice diagram of 1 G to that of G .
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- Date Issued: 2007
Post-harvest Physiology and Technology: AGH 313
- Authors: Maphaha, M F , Eiasu, B
- Date: 2001-06
- Language: English
- Type: Examination paper
- Identifier: vital:17579 , http://hdl.handle.net/10353/d1009925
- Description: Post-harvest Physiology and Technology: AGH 313, degree examination June 2011.
- Full Text: false
- Date Issued: 2001-06
- Authors: Maphaha, M F , Eiasu, B
- Date: 2001-06
- Language: English
- Type: Examination paper
- Identifier: vital:17579 , http://hdl.handle.net/10353/d1009925
- Description: Post-harvest Physiology and Technology: AGH 313, degree examination June 2011.
- Full Text: false
- Date Issued: 2001-06
Research Techniques in Pasture Managament: AGP 602
- Authors: Sikhalazo, Dube , Dziba, L
- Date: 2000-11
- Language: English
- Type: Examination paper
- Identifier: vital:17709 , http://hdl.handle.net/10353/d1010114
- Description: Research Techniques in Pasture Managament: AGP 602, degree examination November 2009.
- Full Text: false
- Date Issued: 2000-11
- Authors: Sikhalazo, Dube , Dziba, L
- Date: 2000-11
- Language: English
- Type: Examination paper
- Identifier: vital:17709 , http://hdl.handle.net/10353/d1010114
- Description: Research Techniques in Pasture Managament: AGP 602, degree examination November 2009.
- Full Text: false
- Date Issued: 2000-11
Analysis of decision making in smallholder irrigation practice: a case study of Shiloh and Zanyokwe irrigation schemes in Central Eastern Cape, South Africa
- Authors: Isaac, Agholor Azikiwe
- Subjects: Irrigation farming -- South Africa -- Eastern Cape , Farms, Small -- South Africa -- Eastern Cape , Agricultural extension work -- South Africa -- Eastern Cape , Crops -- Irrigation -- South Africa -- Eastern Cape
- Language: English
- Type: Thesis , Doctoral , PhD (Agricultural Economics)
- Identifier: vital:11215 , http://hdl.handle.net/10353/d1019766 , Irrigation farming -- South Africa -- Eastern Cape , Farms, Small -- South Africa -- Eastern Cape , Agricultural extension work -- South Africa -- Eastern Cape , Crops -- Irrigation -- South Africa -- Eastern Cape
- Description: The study was conducted in Zanyokwe and Shiloh smallholder irrigation schemes located in Eastern Cape Province at Amathole and Chris Hani districts respectively. The choice of Zanyokwe and Shiloh smallholder irrigation scheme for this study is mainly supported by the fact that it had a substantial level of crop farming activity taking place especially at Zanyokwe while the Shiloh smallholder irrigation specialises dairy farming. The study examined decision making in smallholder irrigation practice with particular reference to Shiloh and Zanyokwe irrigation schemes. The general objective of the thesis was to analyse and model the determinants of SIS farmer‟s decision making. The specific objectives of the study are as follows: to investigate the determinants of decision making among smallholder irrigation farmers; to examine the relationship between household and farm characteristics and institutional factors that explain decision making in smallholder irrigation scheme; assess the contribution of smallholder irrigation farming to household food security; and determine the production and marketing constraints of smallholder farmers‟ in both schemes. The theoretical and conceptual framework of the study gave a detailed discussion on the determinants of decision making of households. The theories used to understand household behaviour under different assumptions were variously discussed. Comprehensive illustrations of analytical framework of the study were also conceptualised. This study used a survey design, quantitative and qualitative research methodologies involving the use of questionnaires and focus group discussions. The data was coded and analysed using the Statistical Package for Social Science (SPSS). However, frequencies, percentages, bar and pie chart was also computed to describe the data. In consideration of the conceptual framework of the study, the agricultural household model was adopted to analyse smallholder farmer‟s household decision making. Twenty one explanatory variables identified in the conceptual framework of the study were discussed and some of these identified variables were incorporated into the model. The logistic regression model was used as a method of analysis because it can estimate the probability of a certain event occurring and it accommodates a lot of variables which can be ranked in order to illustrate which variables are significant. In the binary logistic model used, seven variables (farm experience, size of farmland, land rights/PTO, water sufficiency, farm asset, market information and production variation) out of the twelve predictor variables were found to have significant effect on influencing household decision making in Shiloh smallholder irrigation scheme, while five variables (gender, age, education, road distance and extension access) were not significant. Of the seven significant variables, four had positive signs (land rights/PTO, water sufficiency and market information); which means that an increase in either of these variables may be associated with an increase in household decision making in Shiloh. The other three predictor variables (farm experience, farm asset and product variation) had negative signs; this means an increase in either of these variables may be associated with a decrease in decision making. In Zanyokwe, six variables (farm experience, land rights/PTO, water sufficiency, farm asset, market information and production variation) out of the twelve predictor variables were found to have significant impact on influencing household decision making, while six variables (gender, age, education, size of farm land, road distance and extension access) were not significant. Of the six significant variables, two had positive signs (water sufficiency and farm asset); which means that an increase in either of these variables may be associated with an increase in household decision making in Zanyokwe. The other four predictor variables (farm experience, land rights/PTO, market information and product variation) had negative signs; this means an increase in either of these variables may be associated with a decrease in decision making. The study concludes that smallholder agriculture is essential for employment generation and food security of households. It is apparent that household food security will not be achieved without giving attention to the role played by smallholders‟ farmers in South Africa. It is pertinent to promulgate an efficient policy programme to address the diversity of smallholders‟ situations and identify the main constraints on investment. Therefore, all spheres of government, the private sector and NGOs should consider investment in smallholder agriculture through coordinated strategies and political support. This study also recommends that government should develop a strategic Smallholder Investment Plan which would improve investments in smallholder agriculture.
- Full Text:
- Authors: Isaac, Agholor Azikiwe
- Subjects: Irrigation farming -- South Africa -- Eastern Cape , Farms, Small -- South Africa -- Eastern Cape , Agricultural extension work -- South Africa -- Eastern Cape , Crops -- Irrigation -- South Africa -- Eastern Cape
- Language: English
- Type: Thesis , Doctoral , PhD (Agricultural Economics)
- Identifier: vital:11215 , http://hdl.handle.net/10353/d1019766 , Irrigation farming -- South Africa -- Eastern Cape , Farms, Small -- South Africa -- Eastern Cape , Agricultural extension work -- South Africa -- Eastern Cape , Crops -- Irrigation -- South Africa -- Eastern Cape
- Description: The study was conducted in Zanyokwe and Shiloh smallholder irrigation schemes located in Eastern Cape Province at Amathole and Chris Hani districts respectively. The choice of Zanyokwe and Shiloh smallholder irrigation scheme for this study is mainly supported by the fact that it had a substantial level of crop farming activity taking place especially at Zanyokwe while the Shiloh smallholder irrigation specialises dairy farming. The study examined decision making in smallholder irrigation practice with particular reference to Shiloh and Zanyokwe irrigation schemes. The general objective of the thesis was to analyse and model the determinants of SIS farmer‟s decision making. The specific objectives of the study are as follows: to investigate the determinants of decision making among smallholder irrigation farmers; to examine the relationship between household and farm characteristics and institutional factors that explain decision making in smallholder irrigation scheme; assess the contribution of smallholder irrigation farming to household food security; and determine the production and marketing constraints of smallholder farmers‟ in both schemes. The theoretical and conceptual framework of the study gave a detailed discussion on the determinants of decision making of households. The theories used to understand household behaviour under different assumptions were variously discussed. Comprehensive illustrations of analytical framework of the study were also conceptualised. This study used a survey design, quantitative and qualitative research methodologies involving the use of questionnaires and focus group discussions. The data was coded and analysed using the Statistical Package for Social Science (SPSS). However, frequencies, percentages, bar and pie chart was also computed to describe the data. In consideration of the conceptual framework of the study, the agricultural household model was adopted to analyse smallholder farmer‟s household decision making. Twenty one explanatory variables identified in the conceptual framework of the study were discussed and some of these identified variables were incorporated into the model. The logistic regression model was used as a method of analysis because it can estimate the probability of a certain event occurring and it accommodates a lot of variables which can be ranked in order to illustrate which variables are significant. In the binary logistic model used, seven variables (farm experience, size of farmland, land rights/PTO, water sufficiency, farm asset, market information and production variation) out of the twelve predictor variables were found to have significant effect on influencing household decision making in Shiloh smallholder irrigation scheme, while five variables (gender, age, education, road distance and extension access) were not significant. Of the seven significant variables, four had positive signs (land rights/PTO, water sufficiency and market information); which means that an increase in either of these variables may be associated with an increase in household decision making in Shiloh. The other three predictor variables (farm experience, farm asset and product variation) had negative signs; this means an increase in either of these variables may be associated with a decrease in decision making. In Zanyokwe, six variables (farm experience, land rights/PTO, water sufficiency, farm asset, market information and production variation) out of the twelve predictor variables were found to have significant impact on influencing household decision making, while six variables (gender, age, education, size of farm land, road distance and extension access) were not significant. Of the six significant variables, two had positive signs (water sufficiency and farm asset); which means that an increase in either of these variables may be associated with an increase in household decision making in Zanyokwe. The other four predictor variables (farm experience, land rights/PTO, market information and product variation) had negative signs; this means an increase in either of these variables may be associated with a decrease in decision making. The study concludes that smallholder agriculture is essential for employment generation and food security of households. It is apparent that household food security will not be achieved without giving attention to the role played by smallholders‟ farmers in South Africa. It is pertinent to promulgate an efficient policy programme to address the diversity of smallholders‟ situations and identify the main constraints on investment. Therefore, all spheres of government, the private sector and NGOs should consider investment in smallholder agriculture through coordinated strategies and political support. This study also recommends that government should develop a strategic Smallholder Investment Plan which would improve investments in smallholder agriculture.
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Climate smart soil management: a win-win response to climate change and food security challenges
- Authors: Mnkeni, Pearson
- Subjects: Food security , Population growth , Soil degration , Climate change , Global warming , Conservation agriculture , Organic materials
- Language: English
- Type: Inaugural lecture
- Identifier: vital:11980 , http://hdl.handle.net/10353/d1011255 , Food security , Population growth , Soil degration , Climate change , Global warming , Conservation agriculture , Organic materials
- Description: Sub-Saharan Africa faces a major food security challenge as a result of projected fast increases in population growth and continuing declining per capita food availability. This calls for accelerated increases in productivity to meet expected increases in food demand. However, the soils from which the extra production is to come from are highly degraded, especially in South Africa where a large proportion of the land is ranked as having high degradation potential. This is compounded by the increasing climate change challenge which will render more land unfavourable for production. The climate change is mainly caused by global warming believed to be a result of increasing greenhouse gas emissions. The link between soil carbon, food security, and climate change will be explained in this paper. It will be shown that the high degradation status of South African soils is related to their low organic carbon contents. Efforts to restore their productivity must include strategies to minimize further loss of organic matter and encouraging carbon sequestration. Some interventions investigated with the help of my students and collaborators are presented. They include use of farmer available organic materials that can be applied to soils to improve soil carbon sequestration and fertility status; use of cyanobacteria to improve soil carbon sequestration and soil biogeochemical performance; and the adoption of conservation agriculture.
- Full Text:
- Authors: Mnkeni, Pearson
- Subjects: Food security , Population growth , Soil degration , Climate change , Global warming , Conservation agriculture , Organic materials
- Language: English
- Type: Inaugural lecture
- Identifier: vital:11980 , http://hdl.handle.net/10353/d1011255 , Food security , Population growth , Soil degration , Climate change , Global warming , Conservation agriculture , Organic materials
- Description: Sub-Saharan Africa faces a major food security challenge as a result of projected fast increases in population growth and continuing declining per capita food availability. This calls for accelerated increases in productivity to meet expected increases in food demand. However, the soils from which the extra production is to come from are highly degraded, especially in South Africa where a large proportion of the land is ranked as having high degradation potential. This is compounded by the increasing climate change challenge which will render more land unfavourable for production. The climate change is mainly caused by global warming believed to be a result of increasing greenhouse gas emissions. The link between soil carbon, food security, and climate change will be explained in this paper. It will be shown that the high degradation status of South African soils is related to their low organic carbon contents. Efforts to restore their productivity must include strategies to minimize further loss of organic matter and encouraging carbon sequestration. Some interventions investigated with the help of my students and collaborators are presented. They include use of farmer available organic materials that can be applied to soils to improve soil carbon sequestration and fertility status; use of cyanobacteria to improve soil carbon sequestration and soil biogeochemical performance; and the adoption of conservation agriculture.
- Full Text: